Daily Papers

Solving mechanics problems using numerical methods requires comprehensive intelligent capability of retrieving relevant knowledge and theory, constructing and executing codes, analyzing the results, a task that has thus far mainly been reserved for humans. While emerging AI methods can provide effective approaches to solve end-to-end problems, for instance via the use of deep surrogate models or various data analytics strategies, they often lack physical intuition since knowledge is baked into the parametric complement through training, offering less flexibility when it comes to incorporating mathematical or physical insights. By leveraging diverse capabilities of multiple dynamically interacting large language models (LLMs), we can overcome the limitations of conventional approaches and develop a new class of physics-inspired generative machine learning platform, here referred to as MechAgents. A set of AI agents can solve mechanics tasks, here demonstrated for elasticity problems, via autonomous collaborations. A two-agent team can effectively write, execute and self-correct code, in order to apply finite element methods to solve classical elasticity problems in various flavors (different boundary conditions, domain geometries, meshes, small/finite deformation and linear/hyper-elastic constitutive laws, and others). For more complex tasks, we construct a larger group of agents with enhanced division of labor among planning, formulating, coding, executing and criticizing the process and results. The agents mutually correct each other to improve the overall team-work performance in understanding, formulating and validating the solution. Our framework shows the potential of synergizing the intelligence of language models, the reliability of physics-based modeling, and the dynamic collaborations among diverse agents, opening novel avenues for automation of solving engineering problems.

Visual planning represents a crucial facet of human intelligence, especially in tasks that require complex spatial reasoning and navigation. Yet, in machine learning, this inherently visual problem is often tackled through a verbal-centric lens. While recent research demonstrates the promise of fully visual approaches, they suffer from significant computational inefficiency due to the step-by-step planning-by-generation paradigm. In this work, we present EAR, an editing-as-reasoning paradigm that reformulates visual planning as a single-step image transformation. To isolate intrinsic reasoning from visual recognition, we employ abstract puzzles as probing tasks and introduce AMAZE, a procedurally generated dataset that features the classical Maze and Queen problems, covering distinct, complementary forms of visual planning. The abstract nature of AMAZE also facilitates automatic evaluation of autoregressive and diffusion-based models in terms of both pixel-wise fidelity and logical validity. We assess leading proprietary and open-source editing models. The results show that they all struggle in the zero-shot setting, finetuning on basic scales enables remarkable generalization to larger in-domain scales and out-of-domain scales and geometries. However, our best model that runs on high-end hardware fails to match the zero-shot efficiency of human solvers, highlighting a persistent gap in neural visual reasoning.

Nonrigid registration is conventionally divided into point set registration, which aligns sparse geometries, and image registration, which aligns continuous intensity fields on regular grids. However, this dichotomy creates a critical bottleneck for emerging scientific data, such as spatial transcriptomics, where high-dimensional vector-valued functions, e.g., gene expression, are defined on irregular, sparse manifolds. Consequently, researchers currently face a forced choice: either sacrifice single-cell resolution via voxelization to utilize image-based tools, or ignore the critical functional signal to utilize geometric tools. To resolve this dilemma, we propose Domain Elastic Transform (DET), a grid-free probabilistic framework that unifies geometric and functional alignment. By treating data as functions on irregular domains, DET registers high-dimensional signals directly without binning. We formulate the problem within a rigorous Bayesian framework, modeling domain deformation as an elastic motion guided by a joint spatial-functional likelihood. The method is fully unsupervised and scalable, utilizing feature-sensitive downsampling to handle massive atlases. We demonstrate that DET achieves 92\% topological preservation on MERFISH data where state-of-the-art optimal transport methods struggle (<5\%), and successfully registers whole-embryo Stereo-seq atlases across developmental stages -- a task involving massive scale and complex nonrigid growth. The implementation of DET is available on {https://github.com/ohirose/bcpd} (since Mar, 2025).

In aerodynamics, accurately modeling subsonic compressible flow over airfoils is critical for aircraft design. However, solving the governing nonlinear perturbation velocity potential equation presents computational challenges. Traditional approaches often rely on linearized equations or finite, truncated domains, which introduce non-negligible errors and limit applicability in real-world scenarios. In this study, we propose a novel framework utilizing Physics-Informed Neural Networks (PINNs) to solve the full nonlinear compressible potential equation in an unbounded (infinite) domain. We address the unbounded-domain and convergence challenges inherent in standard PINNs by incorporating a coordinate transformation and embedding physical asymptotic constraints directly into the network architecture. Furthermore, we employ a Multi-Stage PINN (MS-PINN) approach to iteratively minimize residuals, achieving solution accuracy approaching machine precision. We validate this framework by simulating flow over circular and elliptical geometries, comparing our results against traditional finite-domain and linearized solutions. Our findings quantify the noticeable discrepancies introduced by domain truncation and linearization, particularly at higher Mach numbers, and demonstrate that this new framework is a robust, high-fidelity tool for computational fluid dynamics.

Machine learning approaches for solving partial differential equations require learning mappings between function spaces. While convolutional or graph neural networks are constrained to discretized functions, neural operators present a promising milestone toward mapping functions directly. Despite impressive results they still face challenges with respect to the domain geometry and typically rely on some form of discretization. In order to alleviate such limitations, we present CORAL, a new method that leverages coordinate-based networks for solving PDEs on general geometries. CORAL is designed to remove constraints on the input mesh, making it applicable to any spatial sampling and geometry. Its ability extends to diverse problem domains, including PDE solving, spatio-temporal forecasting, and inverse problems like geometric design. CORAL demonstrates robust performance across multiple resolutions and performs well in both convex and non-convex domains, surpassing or performing on par with state-of-the-art models.

Deep learning surrogate models have shown promise in solving partial differential equations (PDEs). Among them, the Fourier neural operator (FNO) achieves good accuracy, and is significantly faster compared to numerical solvers, on a variety of PDEs, such as fluid flows. However, the FNO uses the Fast Fourier transform (FFT), which is limited to rectangular domains with uniform grids. In this work, we propose a new framework, viz., geo-FNO, to solve PDEs on arbitrary geometries. Geo-FNO learns to deform the input (physical) domain, which may be irregular, into a latent space with a uniform grid. The FNO model with the FFT is applied in the latent space. The resulting geo-FNO model has both the computation efficiency of FFT and the flexibility of handling arbitrary geometries. Our geo-FNO is also flexible in terms of its input formats, viz., point clouds, meshes, and design parameters are all valid inputs. We consider a variety of PDEs such as the Elasticity, Plasticity, Euler's, and Navier-Stokes equations, and both forward modeling and inverse design problems. Geo-FNO is 10^5 times faster than the standard numerical solvers and twice more accurate compared to direct interpolation on existing ML-based PDE solvers such as the standard FNO.

Neural operators have become increasingly popular in solving partial differential equations (PDEs) due to their superior capability to capture intricate mappings between function spaces over complex domains. However, the data-hungry nature of operator learning inevitably poses a bottleneck for their widespread applications. At the core of the challenge lies the absence of transferability of neural operators to new geometries. To tackle this issue, we propose operator learning with domain decomposition, a local-to-global framework to solve PDEs on arbitrary geometries. Under this framework, we devise an iterative scheme Schwarz Neural Inference (SNI). This scheme allows for partitioning of the problem domain into smaller subdomains, on which local problems can be solved with neural operators, and stitching local solutions to construct a global solution. Additionally, we provide a theoretical analysis of the convergence rate and error bound. We conduct extensive experiments on several representative PDEs with diverse boundary conditions and achieve remarkable geometry generalization compared to alternative methods. These analysis and experiments demonstrate the proposed framework's potential in addressing challenges related to geometry generalization and data efficiency.

Transformers have empowered many milestones across various fields and have recently been applied to solve partial differential equations (PDEs). However, since PDEs are typically discretized into large-scale meshes with complex geometries, it is challenging for Transformers to capture intricate physical correlations directly from massive individual points. Going beyond superficial and unwieldy meshes, we present Transolver based on a more foundational idea, which is learning intrinsic physical states hidden behind discretized geometries. Specifically, we propose a new Physics-Attention to adaptively split the discretized domain into a series of learnable slices of flexible shapes, where mesh points under similar physical states will be ascribed to the same slice. By calculating attention to physics-aware tokens encoded from slices, Transovler can effectively capture intricate physical correlations under complex geometrics, which also empowers the solver with endogenetic geometry-general modeling capacity and can be efficiently computed in linear complexity. Transolver achieves consistent state-of-the-art with 22% relative gain across six standard benchmarks and also excels in large-scale industrial simulations, including car and airfoil designs. Code is available at https://github.com/thuml/Transolver.

A key scalability challenge in neural solvers for industrial-scale physics simulations is efficiently capturing both fine-grained local interactions and long-range global dependencies across millions of spatial elements. We introduce the Multi-Scale Patch Transformer (MSPT), an architecture that combines local point attention within patches with global attention to coarse patch-level representations. To partition the input domain into spatially-coherent patches, we employ ball trees, which handle irregular geometries efficiently. This dual-scale design enables MSPT to scale to millions of points on a single GPU. We validate our method on standard PDE benchmarks (elasticity, plasticity, fluid dynamics, porous flow) and large-scale aerodynamic datasets (ShapeNet-Car, Ahmed-ML), achieving state-of-the-art accuracy with substantially lower memory footprint and computational cost.

Reconstructing 3D objects into editable programs is pivotal for applications like reverse engineering and shape editing. However, existing methods often rely on limited domain-specific languages (DSLs) and small-scale datasets, restricting their ability to model complex geometries and structures. To address these challenges, we introduce MeshCoder, a novel framework that reconstructs complex 3D objects from point clouds into editable Blender Python scripts. We develop a comprehensive set of expressive Blender Python APIs capable of synthesizing intricate geometries. Leveraging these APIs, we construct a large-scale paired object-code dataset, where the code for each object is decomposed into distinct semantic parts. Subsequently, we train a multimodal large language model (LLM) that translates 3D point cloud into executable Blender Python scripts. Our approach not only achieves superior performance in shape-to-code reconstruction tasks but also facilitates intuitive geometric and topological editing through convenient code modifications. Furthermore, our code-based representation enhances the reasoning capabilities of LLMs in 3D shape understanding tasks. Together, these contributions establish MeshCoder as a powerful and flexible solution for programmatic 3D shape reconstruction and understanding.

Large-scale text-guided image diffusion models have shown astonishing results in text-to-image (T2I) generation. However, applying these models to synthesize textures for 3D geometries remains challenging due to the domain gap between 2D images and textures on a 3D surface. Early works that used a projecting-and-inpainting approach managed to preserve generation diversity but often resulted in noticeable artifacts and style inconsistencies. While recent methods have attempted to address these inconsistencies, they often introduce other issues, such as blurring, over-saturation, or over-smoothing. To overcome these challenges, we propose a novel text-to-texture synthesis framework that leverages pretrained diffusion models. We first introduce a local attention reweighing mechanism in the self-attention layers to guide the model in concentrating on spatial-correlated patches across different views, thereby enhancing local details while preserving cross-view consistency. Additionally, we propose a novel latent space merge pipeline, which further ensures consistency across different viewpoints without sacrificing too much diversity. Our method significantly outperforms existing state-of-the-art techniques regarding texture consistency and visual quality, while delivering results much faster than distillation-based methods. Importantly, our framework does not require additional training or fine-tuning, making it highly adaptable to a wide range of models available on public platforms.

We introduce a diffusion-based framework that performs aligned novel view image and geometry generation via a warping-and-inpainting methodology. Unlike prior methods that require dense posed images or pose-embedded generative models limited to in-domain views, our method leverages off-the-shelf geometry predictors to predict partial geometries viewed from reference images, and formulates novel-view synthesis as an inpainting task for both image and geometry. To ensure accurate alignment between generated images and geometry, we propose cross-modal attention distillation, where attention maps from the image diffusion branch are injected into a parallel geometry diffusion branch during both training and inference. This multi-task approach achieves synergistic effects, facilitating geometrically robust image synthesis as well as well-defined geometry prediction. We further introduce proximity-based mesh conditioning to integrate depth and normal cues, interpolating between point cloud and filtering erroneously predicted geometry from influencing the generation process. Empirically, our method achieves high-fidelity extrapolative view synthesis on both image and geometry across a range of unseen scenes, delivers competitive reconstruction quality under interpolation settings, and produces geometrically aligned colored point clouds for comprehensive 3D completion. Project page is available at https://cvlab-kaist.github.io/MoAI.

We propose PAniC-3D, a system to reconstruct stylized 3D character heads directly from illustrated (p)ortraits of (ani)me (c)haracters. Our anime-style domain poses unique challenges to single-view reconstruction; compared to natural images of human heads, character portrait illustrations have hair and accessories with more complex and diverse geometry, and are shaded with non-photorealistic contour lines. In addition, there is a lack of both 3D model and portrait illustration data suitable to train and evaluate this ambiguous stylized reconstruction task. Facing these challenges, our proposed PAniC-3D architecture crosses the illustration-to-3D domain gap with a line-filling model, and represents sophisticated geometries with a volumetric radiance field. We train our system with two large new datasets (11.2k Vroid 3D models, 1k Vtuber portrait illustrations), and evaluate on a novel AnimeRecon benchmark of illustration-to-3D pairs. PAniC-3D significantly outperforms baseline methods, and provides data to establish the task of stylized reconstruction from portrait illustrations.

The adaptation of large-scale Vision-Language Models (VLMs) like CLIP to downstream tasks with extremely limited data -- specifically in the one-shot regime -- is often hindered by a significant "Stability-Plasticity" dilemma. While efficient caching mechanisms have been introduced by training-free methods such as Tip-Adapter, these approaches often function as local Nadaraya-Watson estimators. Such estimators are characterized by inherent boundary bias and a lack of global structural regularization. In this paper, ReHARK (Refined Hybrid Adaptive RBF Kernels) is proposed as a synergistic training-free framework that reinterprets few-shot adaptation through global proximal regularization in a Reproducing Kernel Hilbert Space (RKHS). A multistage refinement pipeline is introduced, consisting of: (1) Hybrid Prior Construction, where zero-shot textual knowledge from CLIP and GPT-3 is fused with visual class prototypes to form a robust semantic-visual anchor; (2) Support Set Augmentation (Bridging), where intermediate samples are generated to smooth the transition between visual and textual modalities; (3) Adaptive Distribution Rectification, where test feature statistics are aligned with the augmented support set to mitigate domain shifts; and (4) Multi-Scale RBF Kernels, where an ensemble of kernels is employed to capture complex feature geometries across diverse scales. Superior stability and accuracy are demonstrated through extensive experiments on 11 diverse benchmarks. A new state-of-the-art for one-shot adaptation is established by ReHARK, which achieves an average accuracy of 65.83%, significantly outperforming existing baselines. Code is available at https://github.com/Jahid12012021/ReHARK.

Data heterogeneity in federated learning, characterized by a significant misalignment between local and global distributions, leads to divergent local optimization directions and hinders global model training. Existing studies mainly focus on optimizing local updates or global aggregation, but these indirect approaches demonstrate instability when handling highly heterogeneous data distributions, especially in scenarios where label skew and domain skew coexist. To address this, we propose a geometry-guided data generation method that centers on simulating the global embedding distribution locally. We first introduce the concept of the geometric shape of an embedding distribution and then address the challenge of obtaining global geometric shapes under privacy constraints. Subsequently, we propose GGEUR, which leverages global geometric shapes to guide the generation of new samples, enabling a closer approximation to the ideal global distribution. In single-domain scenarios, we augment samples based on global geometric shapes to enhance model generalization; in multi-domain scenarios, we further employ class prototypes to simulate the global distribution across domains. Extensive experimental results demonstrate that our method significantly enhances the performance of existing approaches in handling highly heterogeneous data, including scenarios with label skew, domain skew, and their coexistence. Code published at: https://github.com/WeiDai-David/2025CVPR_GGEUR

LiDAR-based 3D detectors need large datasets for training, yet they struggle to generalize to novel domains. Domain Generalization (DG) aims to mitigate this by training detectors that are invariant to such domain shifts. Current DG approaches exclusively rely on global geometric features (point cloud Cartesian coordinates) as input features. Over-reliance on these global geometric features can, however, cause 3D detectors to prioritize object location and absolute position, resulting in poor cross-domain performance. To mitigate this, we propose to exploit explicit local point cloud structure for DG, in particular by encoding point cloud neighborhoods with Gaussian blobs, GBlobs. Our proposed formulation is highly efficient and requires no additional parameters. Without any bells and whistles, simply by integrating GBlobs in existing detectors, we beat the current state-of-the-art in challenging single-source DG benchmarks by over 21 mAP (Waymo->KITTI), 13 mAP (KITTI->Waymo), and 12 mAP (nuScenes->KITTI), without sacrificing in-domain performance. Additionally, GBlobs demonstrate exceptional performance in multi-source DG, surpassing the current state-of-the-art by 17, 12, and 5 mAP on Waymo, KITTI, and ONCE, respectively.

Large Language Models (LLMs) have demonstrated impressive capabilities in a wide range of code generation tasks. However, generating code for certain domains remains challenging. One such domain is Computer-Aided Design (CAD) program, where the goal is to produce scripted parametric models that define object geometry for precise design and manufacturing applications. A key challenge in LLM-based CAD program generation is the limited geometric complexity of generated shapes compared to those found in real-world industrial designs. This shortfall is in part due to the lack of diversity in the available CAD program training data. To address this, we propose a novel data augmentation paradigm that prompts an LLM to generate CAD programs conditioned on a reference surface program and a modeling procedure - an idea inspired by practices in industrial design. By varying the reference surface using a collection of organic shapes, our method enriches the geometric distribution of generated CAD models. In particular, it introduces edges and faces defined by spline-based curvature, which are typically missing or underrepresented in existing open-source CAD program datasets. Experiments show that our method produces CAD samples with significantly greater geometric diversity and a higher resemblance to industry-grade CAD designs in terms of the proportion of organic shape primitives. This enhancement makes our CAD data augmentation approach a useful tool for training LLMs and other deep learning models in CAD generation.

We introduce Geometric Neural Operators (GNPs) for accounting for geometric contributions in data-driven deep learning of operators. We show how GNPs can be used (i) to estimate geometric properties, such as the metric and curvatures, (ii) to approximate Partial Differential Equations (PDEs) on manifolds, (iii) learn solution maps for Laplace-Beltrami (LB) operators, and (iv) to solve Bayesian inverse problems for identifying manifold shapes. The methods allow for handling geometries of general shape including point-cloud representations. The developed GNPs provide approaches for incorporating the roles of geometry in data-driven learning of operators.

The Distance Geometry Problem (DGP) seeks to find positions for a set of points in geometric space when some distances between pairs of these points are known. The so-called discretization assumptions allow to discretize the search space of DGP instances. In this paper, we study the Discretizable Molecular Distance Geometry Problem whose feasible solutions provide a discretization scheme for the DGP. We propose the first constraint programming formulations as well as a set of checks for proving infeasibility, domain reduction techniques, symmetry breaking constraints and valid inequalities. Our computational results indicate that our formulations outperform the state-of-the-art integer programming formulations, both for feasible and infeasible instances.

Collections of probability distributions arise in a variety of applications ranging from user activity pattern analysis to brain connectomics. In practice these distributions can be defined over diverse domain types including finite intervals, circles, cylinders, spheres, other manifolds, and graphs. This paper introduces an approach for detecting differences between two collections of distributions over such general domains. To this end, we propose the intrinsic slicing construction that yields a novel class of Wasserstein distances on manifolds and graphs. These distances are Hilbert embeddable, allowing us to reduce the distribution collection comparison problem to a more familiar mean testing problem in a Hilbert space. We provide two testing procedures one based on resampling and another on combining p-values from coordinate-wise tests. Our experiments in various synthetic and real data settings show that the resulting tests are powerful and the p-values are well-calibrated.

We introduce GeoDANO, a geometric vision-language model (VLM) with a domain-agnostic vision encoder, for solving plane geometry problems. Although VLMs have been employed for solving geometry problems, their ability to recognize geometric features remains insufficiently analyzed. To address this gap, we propose a benchmark that evaluates the recognition of visual geometric features, including primitives such as dots and lines, and relations such as orthogonality. Our preliminary study shows that vision encoders often used in general-purpose VLMs, e.g., OpenCLIP, fail to detect these features and struggle to generalize across domains. We develop GeoCLIP, a CLIP based model trained on synthetic geometric diagram-caption pairs to overcome the limitation. Benchmark results show that GeoCLIP outperforms existing vision encoders in recognizing geometric features. We then propose our VLM, GeoDANO, which augments GeoCLIP with a domain adaptation strategy for unseen diagram styles. GeoDANO outperforms specialized methods for plane geometry problems and GPT-4o on MathVerse.

Domain generalization (DG) aims to tackle the distribution shift between training domains and unknown target domains. Generating new domains is one of the most effective approaches, yet its performance gain depends on the distribution discrepancy between the generated and target domains. Distributionally robust optimization is promising to tackle distribution discrepancy by exploring domains in an uncertainty set. However, the uncertainty set may be overwhelmingly large, leading to low-confidence prediction in DG. It is because a large uncertainty set could introduce domains containing semantically different factors from training domains. To address this issue, we propose to perform a moderately distributional exploration (MODE) for domain generalization. Specifically, MODE performs distribution exploration in an uncertainty subset that shares the same semantic factors with the training domains. We show that MODE can endow models with provable generalization performance on unknown target domains. The experimental results show that MODE achieves competitive performance compared to state-of-the-art baselines.

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently nonEuclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field.

While scaling laws have transformed natural language processing and computer vision, 3D point cloud understanding has yet to reach that stage. This can be attributed to both the comparatively smaller scale of 3D datasets, as well as the disparate sources of the data itself. Point clouds are captured by diverse sensors (e.g., depth cameras, LiDAR) across varied domains (e.g., indoor, outdoor), each introducing unique scanning patterns, sampling densities, and semantic biases. Such domain heterogeneity poses a major barrier towards training unified models at scale, especially under the realistic constraint that domain labels are typically inaccessible at inference time. In this work, we propose Point-MoE, a Mixture-of-Experts architecture designed to enable large-scale, cross-domain generalization in 3D perception. We show that standard point cloud backbones degrade significantly in performance when trained on mixed-domain data, whereas Point-MoE with a simple top-k routing strategy can automatically specialize experts, even without access to domain labels. Our experiments demonstrate that Point-MoE not only outperforms strong multi-domain baselines but also generalizes better to unseen domains. This work highlights a scalable path forward for 3D understanding: letting the model discover structure in diverse 3D data, rather than imposing it via manual curation or domain supervision.

Domain Generalization (DG) is a critical area that focuses on developing models capable of performing well on data from unseen distributions, which is essential for real-world applications. Existing approaches primarily concentrate on learning domain-invariant features, which assume that a model robust to variations in the source domains will generalize well to unseen target domains. However, these approaches neglect a deeper analysis at the parameter level, which makes the model hard to explicitly differentiate between parameters sensitive to domain shifts and those robust, potentially hindering its overall ability to generalize. In order to address these limitations, we first build a covariance-based parameter sensitivity analysis framework to quantify the sensitivity of each parameter in a model to domain shifts. By computing the covariance of parameter gradients across multiple source domains, we can identify parameters that are more susceptible to domain variations, which serves as our theoretical foundation. Based on this, we propose Domain-Sensitive Parameter Regularization (DSP-Reg), a principled framework that guides model optimization by a soft regularization technique that encourages the model to rely more on domain-invariant parameters while suppressing those that are domain-specific. This approach provides a more granular control over the model's learning process, leading to improved robustness and generalization to unseen domains. Extensive experiments on benchmarks, such as PACS, VLCS, OfficeHome, and DomainNet, demonstrate that DSP-Reg outperforms state-of-the-art approaches, achieving an average accuracy of 66.7\% and surpassing all baselines.

We introduce Domain-specific Masks for Generalization, a model for improving both in-domain and out-of-domain generalization performance. For domain generalization, the goal is to learn from a set of source domains to produce a single model that will best generalize to an unseen target domain. As such, many prior approaches focus on learning representations which persist across all source domains with the assumption that these domain agnostic representations will generalize well. However, often individual domains contain characteristics which are unique and when leveraged can significantly aid in-domain recognition performance. To produce a model which best generalizes to both seen and unseen domains, we propose learning domain specific masks. The masks are encouraged to learn a balance of domain-invariant and domain-specific features, thus enabling a model which can benefit from the predictive power of specialized features while retaining the universal applicability of domain-invariant features. We demonstrate competitive performance compared to naive baselines and state-of-the-art methods on both PACS and DomainNet.

Domain Generalization (DG) is a challenging task in machine learning that requires a coherent ability to comprehend shifts across various domains through extraction of domain-invariant features. DG performance is typically evaluated by performing image classification in domains of various image styles. However, current methodology lacks quantitative understanding about shifts in stylistic domain, and relies on a vast amount of pre-training data, such as ImageNet1K, which are predominantly in photo-realistic style with weakly supervised class labels. Such a data-driven practice could potentially result in spurious correlation and inflated performance on DG benchmarks. In this paper, we introduce a new DG paradigm to address these risks. We first introduce two new quantitative measures ICV and IDD to describe domain shifts in terms of consistency of classes within one domain and similarity between two stylistic domains. We then present SuperMarioDomains (SMD), a novel synthetic multi-domain dataset sampled from video game scenes with more consistent classes and sufficient dissimilarity compared to ImageNet1K. We demonstrate our DG method SMOS. SMOS first uses SMD to train a precursor model, which is then used to ground the training on a DG benchmark. We observe that SMOS contributes to state-of-the-art performance across five DG benchmarks, gaining large improvements to performances on abstract domains along with on-par or slight improvements to those on photo-realistic domains. Our qualitative analysis suggests that these improvements can be attributed to reduced distributional divergence between originally distant domains. Our data are available at https://github.com/fpsluozi/SMD-SMOS .

The numerical solution of partial differential equations (PDEs) is difficult, having led to a century of research so far. Recently, there have been pushes to build neural--numerical hybrid solvers, which piggy-backs the modern trend towards fully end-to-end learned systems. Most works so far can only generalize over a subset of properties to which a generic solver would be faced, including: resolution, topology, geometry, boundary conditions, domain discretization regularity, dimensionality, etc. In this work, we build a solver, satisfying these properties, where all the components are based on neural message passing, replacing all heuristically designed components in the computation graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. In order to encourage stability in training autoregressive models, we put forward a method that is based on the principle of zero-stability, posing stability as a domain adaptation problem. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.

Large Language Models frequently hallucinate in precision-critical domains such as technical diagramming and mechanical design, where outputs must satisfy strict geometric constraints. We study open-ended geometric synthesis from natural language: translating free-form descriptions into precise constructions whose entities must simultaneously satisfy dozens of interacting constraints. To make this tractable, we release PyGeoX, a programmable geometric DSL that compiles declarative constraints into a differentiable loss, and PyGeoX-Bench, a stratified suite of 300 problems with per-constraint verifiable rewards. Using PyGeoX as a verifier, we identify a failure mode we call Outlier Gradient Masking: under global-norm rewards (any scheme that aggregates residuals through a single norm, for example, exp(-MSE)), a single outlier constraint can nullify the learning signal across all others. To address this, we propose Saturating Additive Rewards (SAR), which decompose the reward into bounded per-constraint terms, preserving partial progress and ensuring consistent gradients even under severe violations. Against MSE-based rewards, the natural baseline for geometry solvers, SAR improves the hard-tier solving rate by 2.3times, and the resulting 8B model is competitive with much larger frontier systems on this benchmark. We release the engine, benchmark, and data at https://github.com/Huawei-AI4Math/PyGeoX.

Distribution shift presents a significant challenge in machine learning, where models often underperform during the test stage when faced with a different distribution than the one they were trained on. This paper focuses on domain shifts, which occur when the model is applied to new domains that are different from the ones it was trained on, and propose a new approach called D^3G. Unlike previous methods that aim to learn a single model that is domain invariant, D^3G leverages domain similarities based on domain metadata to learn domain-specific models. Concretely, D^3G learns a set of training-domain-specific functions during the training stage and reweights them based on domain relations during the test stage. These domain relations can be directly obtained and learned from domain metadata. Under mild assumptions, we theoretically prove that using domain relations to reweight training-domain-specific functions achieves stronger out-of-domain generalization compared to the conventional averaging approach. Empirically, we evaluate the effectiveness of D^3G using real-world datasets for tasks such as temperature regression, land use classification, and molecule-protein binding affinity prediction. Our results show that D^3G consistently outperforms state-of-the-art methods.

Handling out-of-distribution samples is a long-lasting challenge for deep visual models. In particular, domain generalization (DG) is one of the most relevant tasks that aims to train a model with a generalization capability on novel domains. Most existing DG approaches share the same philosophy to minimize the discrepancy between domains by finding the domain-invariant representations. On the contrary, our proposed method called POEM acquires a strong DG capability by learning domain-invariant and domain-specific representations and polarizing them. Specifically, POEM cotrains category-classifying and domain-classifying embeddings while regularizing them to be orthogonal via minimizing the cosine-similarity between their features, i.e., the polarization of embeddings. The clear separation of embeddings suppresses domain-specific features in the domain-invariant embeddings. The concept of POEM shows a unique direction to enhance the domain robustness of representations that brings considerable and consistent performance gains when combined with existing DG methods. Extensive simulation results in popular DG benchmarks with the PACS, VLCS, OfficeHome, TerraIncognita, and DomainNet datasets show that POEM indeed facilitates the category-classifying embedding to be more domain-invariant.

Multi-domain graph pre-training integrates knowledge from diverse domains to enhance performance in the target domains, which is crucial for building graph foundation models. Despite initial success, existing solutions often fall short of answering a fundamental question: how is knowledge integrated or transferred across domains? This theoretical limitation motivates us to rethink the consistency and transferability between model pre-training and domain adaptation. In this paper, we propose a fresh Riemannian geometry perspective, whose core idea is to merge any graph dataset into a unified, smooth Riemannian manifold, enabling a systematic understanding of knowledge integration and transfer. To achieve this, our key contribution is the theoretical establishment of neural manifold gluing, which first characterizes local geometry using an adaptive orthogonal frame and then "glues" the local pieces together into a coherent whole. Building on this theory, we present the GraphGlue framework, which supports batched pre-training with EMA prototyping and provides a transferability measure based on geometric consistence. Extensive experiments demonstrate its superior performance across diverse graph domains. Moreover, we empirically validated GraphGlue's geometric scaling law, showing that larger quantities of datasets improve model transferability by producing a smoother manifold. Codes are available at https://github.com/RiemannGraph/GraphGlue.

Domain adaptation aims to mitigate distribution shifts among different domains. However, traditional formulations are mostly limited to categorical domains, greatly simplifying nuanced domain relationships in the real world. In this work, we tackle a generalization with taxonomy-structured domains, which formalizes domains with nested, hierarchical similarity structures such as animal species and product catalogs. We build on the classic adversarial framework and introduce a novel taxonomist, which competes with the adversarial discriminator to preserve the taxonomy information. The equilibrium recovers the classic adversarial domain adaptation's solution if given a non-informative domain taxonomy (e.g., a flat taxonomy where all leaf nodes connect to the root node) while yielding non-trivial results with other taxonomies. Empirically, our method achieves state-of-the-art performance on both synthetic and real-world datasets with successful adaptation. Code is available at https://github.com/Wang-ML-Lab/TSDA.

Do contemporary diffusion models preserve the class geometry of hyperspherical data? Standard diffusion models rely on isotropic Gaussian noise in the forward process, inherently favoring Euclidean spaces. However, many real-world problems involve non-Euclidean distributions, such as hyperspherical manifolds, where class-specific patterns are governed by angular geometry within hypercones. When modeled in Euclidean space, these angular subtleties are lost, leading to suboptimal generative performance. To address this limitation, we introduce HyperSphereDiff to align hyperspherical structures with directional noise, preserving class geometry and effectively capturing angular uncertainty. We demonstrate both theoretically and empirically that this approach aligns the generative process with the intrinsic geometry of hyperspherical data, resulting in more accurate and geometry-aware generative models. We evaluate our framework on four object datasets and two face datasets, showing that incorporating angular uncertainty better preserves the underlying hyperspherical manifold. Resources are available at: {https://github.com/IAB-IITJ/Harmonizing-Geometry-and-Uncertainty-Diffusion-with-Hyperspheres/}

When learning simulations for modeling physical phenomena in industrial designs, geometrical variabilities are of prime interest. While classical regression techniques prove effective for parameterized geometries, practical scenarios often involve the absence of shape parametrization during the inference stage, leaving us with only mesh discretizations as available data. Learning simulations from such mesh-based representations poses significant challenges, with recent advances relying heavily on deep graph neural networks to overcome the limitations of conventional machine learning approaches. Despite their promising results, graph neural networks exhibit certain drawbacks, including their dependency on extensive datasets and limitations in providing built-in predictive uncertainties or handling large meshes. In this work, we propose a machine learning method that do not rely on graph neural networks. Complex geometrical shapes and variations with fixed topology are dealt with using well-known mesh morphing onto a common support, combined with classical dimensionality reduction techniques and Gaussian processes. The proposed methodology can easily deal with large meshes without the need for explicit shape parameterization and provides crucial predictive uncertainties, which are essential for informed decision-making. In the considered numerical experiments, the proposed method is competitive with respect to existing graph neural networks, regarding training efficiency and accuracy of the predictions.

Computer-aided design (CAD) is the digital construction of 2D and 3D objects, and is central to a wide range of engineering and manufacturing applications like automobile and aviation. Despite its importance, CAD modeling remains largely a time-intensive, manual task. Recent works have attempted to automate this process with small transformer-based models and handcrafted CAD sequence representations. However, there has been little effort to leverage the potential of large language models (LLMs) for sequential CAD design. In this work, we introduce a new large-scale dataset of more than 170k CAD models annotated with high-quality, human-like descriptions generated with our pipeline based on GPT-4.1. Using this dataset, we fine-tune powerful code-LLMs to generate CAD sequences represented in a JSON-based format from natural language descriptions, demonstrating the viability and effectiveness of this approach for text-conditioned CAD generation. Because simple metrics often fail to reflect the quality of generated objects, we introduce geometric and topological metrics based on sphericity, mean curvature, and Euler characteristic to provide richer structural insights. Our experiments and ablation studies on both synthetic and human-annotated data demonstrate that CADmium is able to automate CAD design, drastically speeding up the design of new objects. The dataset, code, and fine-tuned models are available online.

Domain Generalization (DG) is a fundamental challenge for machine learning models, which aims to improve model generalization on various domains. Previous methods focus on generating domain invariant features from various source domains. However, we argue that the domain variantions also contain useful information, ie, classification-aware information, for downstream tasks, which has been largely ignored. Different from learning domain invariant features from source domains, we decouple the input images into Domain Expert Features and noise. The proposed domain expert features lie in a learned latent space where the images in each domain can be classified independently, enabling the implicit use of classification-aware domain variations. Based on the analysis, we proposed a novel paradigm called Domain Disentanglement Network (DDN) to disentangle the domain expert features from the source domain images and aggregate the source domain expert features for representing the target test domain. We also propound a new contrastive learning method to guide the domain expert features to form a more balanced and separable feature space. Experiments on the widely-used benchmarks of PACS, VLCS, OfficeHome, DomainNet, and TerraIncognita demonstrate the competitive performance of our method compared to the recently proposed alternatives.

Graph Domain Adaptation (GDA) typically uses adversarial learning to align graph embeddings in Euclidean space. However, this paradigm suffers from two critical challenges: Structural Degeneration, where hierarchical and semantic representations are entangled, and Optimization Instability, which arises from oscillatory dynamics of minimax adversarial training. To tackle these issues, we propose DisRFM, a geometry-aware GDA framework that unifies Riemannian embedding and flow-based transport. First, to overcome structural degeneration, we embed graphs into a Riemannian manifold. By adopting polar coordinates, we explicitly disentangle structure (radius) from semantics (angle). Then, we enforce topology preservation through radial Wasserstein alignment and semantic discrimination via angular clustering, thereby preventing feature entanglement and collapse. Second, we address the instability of adversarial alignment by using Riemannian flow matching. This method learns a smooth vector field to guide source features toward the target along geodesic paths, guaranteeing stable convergence. The geometric constraints further guide the flow to maintain the disentangled structure during transport. Theoretically, we prove the asymptotic stability of the flow matching and derive a tighter bound for the target risk. Extensive experiments demonstrate that DisRFM consistently outperforms state-of-the-art methods.

Neural representations of 3D data have been widely adopted across various applications, particularly in recent work leveraging coordinate-based networks to model scalar or vector fields. However, these approaches face inherent challenges, such as handling thin structures and non-watertight geometries, which limit their flexibility and accuracy. In contrast, we propose a novel geometric data representation that models geometry as distributions-a powerful representation that makes no assumptions about surface genus, connectivity, or boundary conditions. Our approach uses diffusion models with a novel network architecture to learn surface point distributions, capturing fine-grained geometric details. We evaluate our representation qualitatively and quantitatively across various object types, demonstrating its effectiveness in achieving high geometric fidelity. Additionally, we explore applications using our representation, such as textured mesh representation, neural surface compression, dynamic object modeling, and rendering, highlighting its potential to advance 3D geometric learning.

Dimensionality reduction is a fundamental task that aims to simplify complex data by reducing its feature dimensionality while preserving essential patterns, with core applications in data analysis and visualisation. To preserve the underlying data structure, multi-dimensional scaling (MDS) methods focus on preserving pairwise dissimilarities, such as distances. They optimise the embedding to have pairwise distances as close as possible to the data dissimilarities. However, the current standard is limited to embedding data in Riemannian manifolds. Motivated by the lack of asymmetry in the Riemannian metric of the embedding space, this paper extends the MDS problem to a natural asymmetric generalisation of Riemannian manifolds called Finsler manifolds. Inspired by Euclidean space, we define a canonical Finsler space for embedding asymmetric data. Due to its simplicity with respect to geodesics, data representation in this space is both intuitive and simple to analyse. We demonstrate that our generalisation benefits from the same theoretical convergence guarantees. We reveal the effectiveness of our Finsler embedding across various types of non-symmetric data, highlighting its value in applications such as data visualisation, dimensionality reduction, directed graph embedding, and link prediction.

Domain adaptation framework of GANs has achieved great progress in recent years as a main successful approach of training contemporary GANs in the case of very limited training data. In this work, we significantly improve this framework by proposing an extremely compact parameter space for fine-tuning the generator. We introduce a novel domain-modulation technique that allows to optimize only 6 thousand-dimensional vector instead of 30 million weights of StyleGAN2 to adapt to a target domain. We apply this parameterization to the state-of-art domain adaptation methods and show that it has almost the same expressiveness as the full parameter space. Additionally, we propose a new regularization loss that considerably enhances the diversity of the fine-tuned generator. Inspired by the reduction in the size of the optimizing parameter space we consider the problem of multi-domain adaptation of GANs, i.e. setting when the same model can adapt to several domains depending on the input query. We propose the HyperDomainNet that is a hypernetwork that predicts our parameterization given the target domain. We empirically confirm that it can successfully learn a number of domains at once and may even generalize to unseen domains. Source code can be found at https://github.com/MACderRu/HyperDomainNet

The very challenging task of learning solution operators of PDEs on arbitrary domains accurately and efficiently is of vital importance to engineering and industrial simulations. Despite the existence of many operator learning algorithms to approximate such PDEs, we find that accurate models are not necessarily computationally efficient and vice versa. We address this issue by proposing a geometry aware operator transformer (GAOT) for learning PDEs on arbitrary domains. GAOT combines novel multiscale attentional graph neural operator encoders and decoders, together with geometry embeddings and (vision) transformer processors to accurately map information about the domain and the inputs into a robust approximation of the PDE solution. Multiple innovations in the implementation of GAOT also ensure computational efficiency and scalability. We demonstrate this significant gain in both accuracy and efficiency of GAOT over several baselines on a large number of learning tasks from a diverse set of PDEs, including achieving state of the art performance on a large scale three-dimensional industrial CFD dataset.

Recent advances in 3D X-ray Computed Tomographic (CT) sensors have stimulated research efforts to unveil the extremely complex micro-scale processes that control the activity of soil microorganisms. Voxel-based description (up to hundreds millions voxels) of the pore space can be extracted, from grey level 3D CT scanner images, by means of simple image processing tools. Classical methods for numerical simulation of biological dynamics using mesh of voxels, such as Lattice Boltzmann Model (LBM), are too much time consuming. Thus, the use of more compact and reliable geometrical representations of pore space can drastically decrease the computational cost of the simulations. Several recent works propose basic analytic volume primitives (e.g. spheres, generalized cylinders, ellipsoids) to define a piece-wise approximation of pore space for numerical simulation of draining, diffusion and microbial decomposition. Such approaches work well but the drawback is that it generates approximation errors. In the present work, we study another alternative where pore space is described by means of geometrically relevant connected subsets of voxels (regions) computed from the curvilinear skeleton. Indeed, many works use the curvilinear skeleton (3D medial axis) for analyzing and partitioning 3D shapes within various domains (medicine, material sciences, petroleum engineering, etc.) but only a few ones in soil sciences. Within the context of soil sciences, most studies dealing with 3D medial axis focus on the determination of pore throats. Here, we segment pore space using curvilinear skeleton in order to achieve numerical simulation of microbial decomposition (including diffusion processes). We validate simulation outputs by comparison with other methods using different pore space geometrical representations (balls, voxels).

Large language models have seen widespread adoption in math problem-solving. However, in geometry problems that usually require visual aids for better understanding, even the most advanced multi-modal models currently still face challenges in effectively using image information. High-quality data is crucial for enhancing the geometric capabilities of multi-modal models, yet existing open-source datasets and related efforts are either too challenging for direct model learning or suffer from misalignment between text and images. To overcome this issue, we introduce a novel pipeline that leverages GPT-4 and GPT-4V to generate relatively basic geometry problems with aligned text and images, facilitating model learning. We have produced a dataset of 4.9K geometry problems and combined it with 19K open-source data to form our GeoGPT4V dataset. Experimental results demonstrate that the GeoGPT4V dataset significantly improves the geometry performance of various models on the MathVista and MathVision benchmarks. The code is available at https://github.com/Lanyu0303/GeoGPT4V_Project

Can one inject new concepts into an already trained generative model, while respecting its existing structure and knowledge? We propose a new task - domain expansion - to address this. Given a pretrained generator and novel (but related) domains, we expand the generator to jointly model all domains, old and new, harmoniously. First, we note the generator contains a meaningful, pretrained latent space. Is it possible to minimally perturb this hard-earned representation, while maximally representing the new domains? Interestingly, we find that the latent space offers unused, "dormant" directions, which do not affect the output. This provides an opportunity: By "repurposing" these directions, we can represent new domains without perturbing the original representation. In fact, we find that pretrained generators have the capacity to add several - even hundreds - of new domains! Using our expansion method, one "expanded" model can supersede numerous domain-specific models, without expanding the model size. Additionally, a single expanded generator natively supports smooth transitions between domains, as well as composition of domains. Code and project page available at https://yotamnitzan.github.io/domain-expansion/.

Large language models (LLMs) have achieved remarkable success and demonstrated superior performance across various tasks, including natural language processing (NLP), weather forecasting, biological protein folding, text generation, and solving mathematical problems. However, many real-world data exhibit highly non-Euclidean latent hierarchical anatomy, such as protein networks, transportation networks, financial networks, brain networks, and linguistic structures or syntactic trees in natural languages. Effectively learning intrinsic semantic entailment and hierarchical relationships from these raw, unstructured input data using LLMs remains an underexplored area. Due to its effectiveness in modeling tree-like hierarchical structures, hyperbolic geometry -- a non-Euclidean space -- has rapidly gained popularity as an expressive latent representation space for complex data modeling across domains such as graphs, images, languages, and multi-modal data. Here, we provide a comprehensive and contextual exposition of recent advancements in LLMs that leverage hyperbolic geometry as a representation space to enhance semantic representation learning and multi-scale reasoning. Specifically, the paper presents a taxonomy of the principal techniques of Hyperbolic LLMs (HypLLMs) in terms of four main categories: (1) hyperbolic LLMs through exp/log maps; (2) hyperbolic fine-tuned models; (3) fully hyperbolic LLMs, and (4) hyperbolic state-space models. We also explore crucial potential applications and outline future research directions. A repository of key papers, models, datasets, and code implementations is available at https://github.com/sarangp2402/Hyperbolic-LLM-Models/tree/main.

Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects--the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra--which have been loosely referred to as "positive geometries". The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic singularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. In this paper we initiate an exploration of "positive geometries" and "canonical forms" as objects of study in their own right in a more general mathematical setting. We give a precise definition of positive geometries and canonical forms, introduce general methods for finding forms for more complicated positive geometries from simpler ones, and present numerous examples of positive geometries in projective spaces, Grassmannians, and toric, cluster and flag varieties. We also illustrate a number of strategies for computing canonical forms which yield interesting representations for the forms associated with wide classes of positive geometries, ranging from the simplest Amplituhedra to new expressions for the volume of arbitrary convex polytopes.

We address the computational challenges of solving parametric PDEs with non parametrized geometric variations and non-reducible problems, such as those involving shocks and discontinuities of variable positions. Traditional dimensionality reduction methods like POD struggle with these scenarios due to slowly decaying Kolmogorov widths. To overcome this, we propose a novel non-linear dimensionality reduction technique to reduce the required modes for representation. The non-linear reduction is obtained through a POD after applying a transformation on the fields, which we call optimal mappings, and is a solution to an optimization problem in infinite dimension. The proposed learning framework combines morphing techniques, non-linear dimensionality reduction, and Gaussian Process Regression (GPR). The problem is reformulated on a reference geometry before applying the dimensionality reduction. Our method learns both the optimal mapping, and the solution fields, using a series of GPR models, enabling efficient and accurate modeling of complex parametric PDEs with geometrical variability. The results obtained concur with current state-of-the-art models. We mainly compare our method with the winning solution of the ML4CFD NeurIPS 2024 competition.

Domain shift refers to the well known problem that a model trained in one source domain performs poorly when applied to a target domain with different statistics. {Domain Generalization} (DG) techniques attempt to alleviate this issue by producing models which by design generalize well to novel testing domains. We propose a novel {meta-learning} method for domain generalization. Rather than designing a specific model that is robust to domain shift as in most previous DG work, we propose a model agnostic training procedure for DG. Our algorithm simulates train/test domain shift during training by synthesizing virtual testing domains within each mini-batch. The meta-optimization objective requires that steps to improve training domain performance should also improve testing domain performance. This meta-learning procedure trains models with good generalization ability to novel domains. We evaluate our method and achieve state of the art results on a recent cross-domain image classification benchmark, as well demonstrating its potential on two classic reinforcement learning tasks.

Machine learning models fail to perform when facing out-of-distribution (OOD) domains, a challenging task known as domain generalization (DG). In this work, we develop a novel DG training strategy, we call PGrad, to learn a robust gradient direction, improving models' generalization ability on unseen domains. The proposed gradient aggregates the principal directions of a sampled roll-out optimization trajectory that measures the training dynamics across all training domains. PGrad's gradient design forces the DG training to ignore domain-dependent noise signals and updates all training domains with a robust direction covering main components of parameter dynamics. We further improve PGrad via bijection-based computational refinement and directional plus length-based calibrations. Our theoretical proof connects PGrad to the spectral analysis of Hessian in training neural networks. Experiments on DomainBed and WILDS benchmarks demonstrate that our approach effectively enables robust DG optimization and leads to smoothly decreased loss curves. Empirically, PGrad achieves competitive results across seven datasets, demonstrating its efficacy across both synthetic and real-world distributional shifts. Code is available at https://github.com/QData/PGrad.

Domain generalization (DG) aims to learn domain-generalizable models from one or multiple source domains that can perform well in unseen target domains. Despite its recent progress, most existing work suffers from the misalignment between the difficulty level of training samples and the capability of contemporarily trained models, leading to over-fitting or under-fitting in the trained generalization model. We design MoDify, a Momentum Difficulty framework that tackles the misalignment by balancing the seesaw between the model's capability and the samples' difficulties along the training process. MoDify consists of two novel designs that collaborate to fight against the misalignment while learning domain-generalizable models. The first is MoDify-based Data Augmentation which exploits an RGB Shuffle technique to generate difficulty-aware training samples on the fly. The second is MoDify-based Network Optimization which dynamically schedules the training samples for balanced and smooth learning with appropriate difficulty. Without bells and whistles, a simple implementation of MoDify achieves superior performance across multiple benchmarks. In addition, MoDify can complement existing methods as a plug-in, and it is generic and can work for different visual recognition tasks.

Given appropriate representations of the semantic relations between carpenter and wood and between mason and stone (for example, vectors in a vector space model), a suitable algorithm should be able to recognize that these relations are highly similar (carpenter is to wood as mason is to stone; the relations are analogous). Likewise, with representations of dog, house, and kennel, an algorithm should be able to recognize that the semantic composition of dog and house, dog house, is highly similar to kennel (dog house and kennel are synonymous). It seems that these two tasks, recognizing relations and compositions, are closely connected. However, up to now, the best models for relations are significantly different from the best models for compositions. In this paper, we introduce a dual-space model that unifies these two tasks. This model matches the performance of the best previous models for relations and compositions. The dual-space model consists of a space for measuring domain similarity and a space for measuring function similarity. Carpenter and wood share the same domain, the domain of carpentry. Mason and stone share the same domain, the domain of masonry. Carpenter and mason share the same function, the function of artisans. Wood and stone share the same function, the function of materials. In the composition dog house, kennel has some domain overlap with both dog and house (the domains of pets and buildings). The function of kennel is similar to the function of house (the function of shelters). By combining domain and function similarities in various ways, we can model relations, compositions, and other aspects of semantics.

We present GenesisGeo, an automated theorem prover in Euclidean geometry. We have open-sourced a large-scale geometry dataset of 21.8 million geometric problems, over 3 million of which contain auxiliary constructions. Specially, we significantly accelerate the symbolic deduction engine DDARN by 120x through theorem matching, combined with a C++ implementation of its core components. Furthermore, we build our neuro-symbolic prover, GenesisGeo, upon Qwen3-0.6B-Base, which solves 24 of 30 problems (IMO silver medal level) in the IMO-AG-30 benchmark using a single model, and achieves 26 problems (IMO gold medal level) with a dual-model ensemble.

This is the first paper in a series of work we have accomplished over the past three years. In this paper, we have constructed a consistent formal plane geometry system. This will serve as a crucial bridge between IMO-level plane geometry challenges and readable AI automated reasoning. Within this formal framework, we have been able to seamlessly integrate modern AI models with our formal system. AI is now capable of providing deductive reasoning solutions to IMO-level plane geometry problems, just like handling other natural languages, and these proofs are readable, traceable, and verifiable. We propose the geometry formalization theory (GFT) to guide the development of the geometry formal system. Based on the GFT, we have established the FormalGeo, which consists of 88 geometric predicates and 196 theorems. It can represent, validate, and solve IMO-level geometry problems. we also have crafted the FGPS (formal geometry problem solver) in Python. It serves as both an interactive assistant for verifying problem-solving processes and an automated problem solver. We've annotated the formalgeo7k and formalgeo-imo datasets. The former contains 6,981 (expand to 133,818 through data augmentation) geometry problems, while the latter includes 18 (expand to 2,627 and continuously increasing) IMO-level challenging geometry problems. All annotated problems include detailed formal language descriptions and solutions. Implementation of the formal system and experiments validate the correctness and utility of the GFT. The backward depth-first search method only yields a 2.42% problem-solving failure rate, and we can incorporate deep learning techniques to achieve lower one. The source code of FGPS and datasets are available at https://github.com/BitSecret/FGPS.

Human visual perception can easily generalize to out-of-distributed visual data, which is far beyond the capability of modern machine learning models. Domain generalization (DG) aims to close this gap, with existing DG methods mainly focusing on the loss function design. In this paper, we propose to explore an orthogonal direction, i.e., the design of the backbone architecture. It is motivated by an empirical finding that transformer-based models trained with empirical risk minimization (ERM) outperform CNN-based models employing state-of-the-art (SOTA) DG algorithms on multiple DG datasets. We develop a formal framework to characterize a network's robustness to distribution shifts by studying its architecture's alignment with the correlations in the dataset. This analysis guides us to propose a novel DG model built upon vision transformers, namely Generalizable Mixture-of-Experts (GMoE). Extensive experiments on DomainBed demonstrate that GMoE trained with ERM outperforms SOTA DG baselines by a large margin. Moreover, GMoE is complementary to existing DG methods and its performance is substantially improved when trained with DG algorithms.

We characterise sets of points of exceptional Lie incidence geometries, that is, the natural geometries arising from spherical buildings of exceptional types F_4, E_6, E_7, E_8 and G_2, that form a line using the opposition relation. With that, we obtain a classification of so-called ``geometric lines'' in many of these geometries. Furthermore, our results lead to a characterisation of geometric lines in finite exceptional Lie incidence geometries as minimal blocking sets, that is, point sets of the size of a line admitting no object opposite to all of their members, in most cases, and we classify all exceptions. As a further consequence, we obtain a characterisation of automorphisms of exceptional spherical buildings as certain opposition preserving maps.

Empirically validating new 3D-printing related algorithms and implementations requires testing data representative of inputs encountered in the wild. An ideal benchmarking dataset should not only draw from the same distribution of shapes people print in terms of class (e.g., toys, mechanisms, jewelry), representation type (e.g., triangle soup meshes) and complexity (e.g., number of facets), but should also capture problems and artifacts endemic to 3D printing models (e.g., self-intersections, non-manifoldness). We observe that the contextual and geometric characteristics of 3D printing models differ significantly from those used for computer graphics applications, not to mention standard models (e.g., Stanford bunny, Armadillo, Fertility). We present a new dataset of 10,000 models collected from an online 3D printing model-sharing database. Via analysis of both geometric (e.g., triangle aspect ratios, manifoldness) and contextual (e.g., licenses, tags, classes) characteristics, we demonstrate that this dataset represents a more concise summary of real-world models used for 3D printing compared to existing datasets. To facilitate future research endeavors, we also present an online query interface to select subsets of the dataset according to project-specific characteristics. The complete dataset and per-model statistical data are freely available to the public.

Rapid growth of high-dimensional datasets in fields such as single-cell RNA sequencing and spatial genomics has led to unprecedented opportunities for scientific discovery, but it also presents unique computational and statistical challenges. Traditional methods struggle with geometry-aware data generation, interpolation along meaningful trajectories, and transporting populations via feasible paths. To address these issues, we introduce Geometry-Aware Generative Autoencoder (GAGA), a novel framework that combines extensible manifold learning with generative modeling. GAGA constructs a neural network embedding space that respects the intrinsic geometries discovered by manifold learning and learns a novel warped Riemannian metric on the data space. This warped metric is derived from both the points on the data manifold and negative samples off the manifold, allowing it to characterize a meaningful geometry across the entire latent space. Using this metric, GAGA can uniformly sample points on the manifold, generate points along geodesics, and interpolate between populations across the learned manifold using geodesic-guided flows. GAGA shows competitive performance in simulated and real-world datasets, including a 30% improvement over the state-of-the-art methods in single-cell population-level trajectory inference.

We introduce methods for obtaining pretrained Geometric Neural Operators (GNPs) that can serve as basal foundation models for use in obtaining geometric features. These can be used within data processing pipelines for machine learning tasks and numerical methods. We show how our GNPs can be trained to learn robust latent representations for the differential geometry of point-clouds to provide estimates of metric, curvature, and other shape-related features. We demonstrate how our pre-trained GNPs can be used (i) to estimate the geometric properties of surfaces of arbitrary shape and topologies with robustness in the presence of noise, (ii) to approximate solutions of geometric partial differential equations (PDEs) on manifolds, and (iii) to solve equations for shape deformations such as curvature driven flows. We release codes and weights for using GNPs in the package geo_neural_op. This allows for incorporating our pre-trained GNPs as components for reuse within existing and new data processing pipelines. The GNPs also can be used as part of numerical solvers involving geometry or as part of methods for performing inference and other geometric tasks.

Recent advances in vision-language models (VLMs) have demonstrated remarkable zero-shot capabilities, yet adapting these models to specialized domains remains a significant challenge. Building on recent theoretical insights suggesting that independently trained VLMs are related by a canonical transformation, we extend this understanding to the concept of domains. We hypothesize that image features across disparate domains are related by a canonicalized geometric transformation that can be recovered using a small set of anchors. Few-shot classification provides a natural setting for this alignment, as the limited labeled samples serve as the anchors required to estimate this transformation. Motivated by this hypothesis, we introduce BiCLIP, a framework that applies a targeted transformation to multimodal features to enhance cross-modal alignment. Our approach is characterized by its extreme simplicity and low parameter footprint. Extensive evaluations across 11 standard benchmarks, including EuroSAT, DTD, and FGVCAircraft, demonstrate that BiCLIP consistently achieves state-of-the-art results. Furthermore, we provide empirical verification of existing geometric findings by analyzing the orthogonality and angular distribution of the learned transformations, confirming that structured alignment is the key to robust domain adaptation. Code is available at https://github.com/QuantitativeImagingLaboratory/BilinearCLIP

Graph foundation models represent a transformative paradigm for learning transferable representations across diverse graph domains. Recent methods leverage large language models to unify graph and text modalities into a shared representation space using contrastive learning. However, systematic evaluations reveal significant performance degradation at structural boundaries where distinct topological patterns converge, with accuracy losses exceeding 20 percentage points. This issue arises from a key limitation: current methods assume all graph structures can be encoded within a single Euclidean space. In reality, tree structures require hyperbolic geometry to preserve hierarchical branching, while cyclic patterns depend on spherical geometry for closure properties. At structural boundaries, nodes experience conflicting geometric constraints that uniform encoding spaces cannot resolve. This raises a crucial challenge: Can alignment frameworks be designed to respect the intrinsic geometric diversity of graph structures? We introduce GraphShaper, a geometry-aware framework that enhances graph encoding through multi-geometric specialization. Our approach employs expert networks tailored to different geometric spaces, dynamically computing fusion weights to adaptively integrate geometric properties based on local structural characteristics. This adaptive fusion preserves structural integrity before alignment with text embeddings. Extensive experiments demonstrate that GraphShaper achieves 9.47\% accuracy improvements on citation networks and 7.63\% on social networks in zero-shot settings.

Domain Generalization (DG) aims to learn representations that remain robust under out-of-distribution (OOD) shifts and generalize effectively to unseen target domains. While recent invariant learning strategies and architectural advances have achieved strong performance, explicitly discovering a structured domain-invariant subspace through second-order statistics remains underexplored. In this work, we propose CPCANet, a novel framework grounded in Common Principal Component Analysis (CPCA), which unrolls the iterative Flury-Gautschi (FG) algorithm into fully differentiable neural layers. This approach integrates the statistical properties of CPCA into an end-to-end trainable framework, enforcing the discovery of a shared subspace across diverse domains while preserving interpretability. Experiments on four standard DG benchmarks demonstrate that CPCANet achieves state-of-the-art (SOTA) performance in zero-shot transfer. Moreover, CPCANet is architecture-agnostic and requires no dataset-specific tuning, providing a simple and efficient approach to learning robust representations under distribution shift. Code is available at https://github.com/wish44165/CPCANet.

Domain generalization (DG) methods aim to achieve generalizability to an unseen target domain by using only training data from the source domains. Although a variety of DG methods have been proposed, a recent study shows that under a fair evaluation protocol, called DomainBed, the simple empirical risk minimization (ERM) approach works comparable to or even outperforms previous methods. Unfortunately, simply solving ERM on a complex, non-convex loss function can easily lead to sub-optimal generalizability by seeking sharp minima. In this paper, we theoretically show that finding flat minima results in a smaller domain generalization gap. We also propose a simple yet effective method, named Stochastic Weight Averaging Densely (SWAD), to find flat minima. SWAD finds flatter minima and suffers less from overfitting than does the vanilla SWA by a dense and overfit-aware stochastic weight sampling strategy. SWAD shows state-of-the-art performances on five DG benchmarks, namely PACS, VLCS, OfficeHome, TerraIncognita, and DomainNet, with consistent and large margins of +1.6% averagely on out-of-domain accuracy. We also compare SWAD with conventional generalization methods, such as data augmentation and consistency regularization methods, to verify that the remarkable performance improvements are originated from by seeking flat minima, not from better in-domain generalizability. Last but not least, SWAD is readily adaptable to existing DG methods without modification; the combination of SWAD and an existing DG method further improves DG performances. Source code is available at https://github.com/khanrc/swad.

Plane Geometry Diagram Synthesis has been a crucial task in computer graphics, with applications ranging from educational tools to AI-driven mathematical reasoning. Traditionally, we rely on manual tools (e.g., Matplotlib and GeoGebra) to generate precise diagrams, but this usually requires huge, complicated calculations. Recently, researchers start to work on model-based methods (e.g., Stable Diffusion and GPT5) to automatically generate diagrams, saving operational cost but usually suffering from limited realism and insufficient accuracy. In this paper, we propose a novel framework GeoSDF, to automatically generate diagrams efficiently and accurately with Signed Distance Field (SDF). Specifically, we first represent geometric elements (e.g., points, segments, and circles) in the SDF, then construct a series of constraint functions to represent geometric relationships. Next, we optimize those constructed constraint functions to get an optimized field of both elements and constraints. Finally, by rendering the optimized field, we can obtain the synthesized diagram. In our GeoSDF, we define a symbolic language to represent geometric elements and constraints, and our synthesized geometry diagrams can be self-verified in the SDF, ensuring both mathematical accuracy and visual plausibility. In experiments, through both qualitative and quantitative analysis, GeoSDF synthesized both normal high-school level and IMO-level geometry diagrams. We achieve 88.67\% synthesis accuracy by human evaluation in the IMO problem set. Furthermore, we obtain a very high accuracy of solving geometry problems (over 95\% while the current SOTA accuracy is around 75%) by leveraging our self-verification property. All of these demonstrate the advantage of GeoSDF, paving the way for more sophisticated, accurate, and flexible generation of geometric diagrams for a wide array of applications.

We present two new classes of algorithms for efficient field integration on graphs encoding point clouds. The first class, SeparatorFactorization(SF), leverages the bounded genus of point cloud mesh graphs, while the second class, RFDiffusion(RFD), uses popular epsilon-nearest-neighbor graph representations for point clouds. Both can be viewed as providing the functionality of Fast Multipole Methods (FMMs), which have had a tremendous impact on efficient integration, but for non-Euclidean spaces. We focus on geometries induced by distributions of walk lengths between points (e.g., shortest-path distance). We provide an extensive theoretical analysis of our algorithms, obtaining new results in structural graph theory as a byproduct. We also perform exhaustive empirical evaluation, including on-surface interpolation for rigid and deformable objects (particularly for mesh-dynamics modeling), Wasserstein distance computations for point clouds, and the Gromov-Wasserstein variant.

Domain Generalization aims to develop models that can generalize to novel and unseen data distributions. In this work, we study how model architectures and pre-training objectives impact feature richness and propose a method to effectively leverage them for domain generalization. Specifically, given a pre-trained feature space, we first discover latent domain structures, referred to as pseudo-domains, that capture domain-specific variations in an unsupervised manner. Next, we augment existing classifiers with these complementary pseudo-domain representations making them more amenable to diverse unseen test domains. We analyze how different pre-training feature spaces differ in the domain-specific variances they capture. Our empirical studies reveal that features from diffusion models excel at separating domains in the absence of explicit domain labels and capture nuanced domain-specific information. On 5 datasets, we show that our very simple framework improves generalization to unseen domains by a maximum test accuracy improvement of over 4% compared to the standard baseline Empirical Risk Minimization (ERM). Crucially, our method outperforms most algorithms that access domain labels during training.

Geometric diagrams are critical in conveying mathematical and scientific concepts, yet traditional diagram generation methods are often manual and resource-intensive. While text-to-image generation has made strides in photorealistic imagery, creating accurate geometric diagrams remains a challenge due to the need for precise spatial relationships and the scarcity of geometry-specific datasets. This paper presents MagicGeo, a training-free framework for generating geometric diagrams from textual descriptions. MagicGeo formulates the diagram generation process as a coordinate optimization problem, ensuring geometric correctness through a formal language solver, and then employs coordinate-aware generation. The framework leverages the strong language translation capability of large language models, while formal mathematical solving ensures geometric correctness. We further introduce MagicGeoBench, a benchmark dataset of 220 geometric diagram descriptions, and demonstrate that MagicGeo outperforms current methods in both qualitative and quantitative evaluations. This work provides a scalable, accurate solution for automated diagram generation, with significant implications for educational and academic applications.

Recent advancements in implicit 3D representations and generative models have markedly propelled the field of 3D object generation forward. However, it remains a significant challenge to accurately model geometries with defined sharp features under parametric controls, which is crucial in fields like industrial design and manufacturing. To bridge this gap, we introduce a framework that employs Large Language Models (LLMs) to generate text-driven 3D shapes, manipulating 3D software via program synthesis. We present 3D-PreMise, a dataset specifically tailored for 3D parametric modeling of industrial shapes, designed to explore state-of-the-art LLMs within our proposed pipeline. Our work reveals effective generation strategies and delves into the self-correction capabilities of LLMs using a visual interface. Our work highlights both the potential and limitations of LLMs in 3D parametric modeling for industrial applications.

We present MoGe, a powerful model for recovering 3D geometry from monocular open-domain images. Given a single image, our model directly predicts a 3D point map of the captured scene with an affine-invariant representation, which is agnostic to true global scale and shift. This new representation precludes ambiguous supervision in training and facilitate effective geometry learning. Furthermore, we propose a set of novel global and local geometry supervisions that empower the model to learn high-quality geometry. These include a robust, optimal, and efficient point cloud alignment solver for accurate global shape learning, and a multi-scale local geometry loss promoting precise local geometry supervision. We train our model on a large, mixed dataset and demonstrate its strong generalizability and high accuracy. In our comprehensive evaluation on diverse unseen datasets, our model significantly outperforms state-of-the-art methods across all tasks, including monocular estimation of 3D point map, depth map, and camera field of view. Code and models can be found on our project page.

In this work we develop a novel domain splitting strategy for the solution of partial differential equations. Focusing on a uniform discretization of the d-dimensional advection-diffusion equation, our proposal is a two-level algorithm that merges the solutions obtained from the discretization of the equation over highly anisotropic submeshes to compute an initial approximation of the fine solution. The algorithm then iteratively refines the initial guess by leveraging the structure of the residual. Performing costly calculations on anisotropic submeshes enable us to reduce the dimensionality of the problem by one, and the merging process, which involves the computation of solutions over disjoint domains, allows for parallel implementation.

The ability to deploy robots that can operate safely in diverse environments is crucial for developing embodied intelligent agents. As a community, we have made tremendous progress in within-domain LiDAR semantic segmentation. However, do these methods generalize across domains? To answer this question, we design the first experimental setup for studying domain generalization (DG) for LiDAR semantic segmentation (DG-LSS). Our results confirm a significant gap between methods, evaluated in a cross-domain setting: for example, a model trained on the source dataset (SemanticKITTI) obtains 26.53 mIoU on the target data, compared to 48.49 mIoU obtained by the model trained on the target domain (nuScenes). To tackle this gap, we propose the first method specifically designed for DG-LSS, which obtains 34.88 mIoU on the target domain, outperforming all baselines. Our method augments a sparse-convolutional encoder-decoder 3D segmentation network with an additional, dense 2D convolutional decoder that learns to classify a birds-eye view of the point cloud. This simple auxiliary task encourages the 3D network to learn features that are robust to sensor placement shifts and resolution, and are transferable across domains. With this work, we aim to inspire the community to develop and evaluate future models in such cross-domain conditions.

The problem of domain generalization is to learn from multiple training domains, and extract a domain-agnostic model that can then be applied to an unseen domain. Domain generalization (DG) has a clear motivation in contexts where there are target domains with distinct characteristics, yet sparse data for training. For example recognition in sketch images, which are distinctly more abstract and rarer than photos. Nevertheless, DG methods have primarily been evaluated on photo-only benchmarks focusing on alleviating the dataset bias where both problems of domain distinctiveness and data sparsity can be minimal. We argue that these benchmarks are overly straightforward, and show that simple deep learning baselines perform surprisingly well on them. In this paper, we make two main contributions: Firstly, we build upon the favorable domain shift-robust properties of deep learning methods, and develop a low-rank parameterized CNN model for end-to-end DG learning. Secondly, we develop a DG benchmark dataset covering photo, sketch, cartoon and painting domains. This is both more practically relevant, and harder (bigger domain shift) than existing benchmarks. The results show that our method outperforms existing DG alternatives, and our dataset provides a more significant DG challenge to drive future research.

Generative models, especially diffusion models (DMs), have achieved promising results for generating feature-rich geometries and advancing foundational science problems such as molecule design. Inspired by the recent huge success of Stable (latent) Diffusion models, we propose a novel and principled method for 3D molecule generation named Geometric Latent Diffusion Models (GeoLDM). GeoLDM is the first latent DM model for the molecular geometry domain, composed of autoencoders encoding structures into continuous latent codes and DMs operating in the latent space. Our key innovation is that for modeling the 3D molecular geometries, we capture its critical roto-translational equivariance constraints by building a point-structured latent space with both invariant scalars and equivariant tensors. Extensive experiments demonstrate that GeoLDM can consistently achieve better performance on multiple molecule generation benchmarks, with up to 7\% improvement for the valid percentage of large biomolecules. Results also demonstrate GeoLDM's higher capacity for controllable generation thanks to the latent modeling. Code is provided at https://github.com/MinkaiXu/GeoLDM.

This paper investigates the 3D domain generalization (3DDG) ability of large 3D models based on prevalent prompt learning. Recent works demonstrate the performances of 3D point cloud recognition can be boosted remarkably by parameter-efficient prompt tuning. However, we observe that the improvement on downstream tasks comes at the expense of a severe drop in 3D domain generalization. To resolve this challenge, we present a comprehensive regulation framework that allows the learnable prompts to actively interact with the well-learned general knowledge in large 3D models to maintain good generalization. Specifically, the proposed framework imposes multiple explicit constraints on the prompt learning trajectory by maximizing the mutual agreement between task-specific predictions and task-agnostic knowledge. We design the regulation framework as a plug-and-play module to embed into existing representative large 3D models. Surprisingly, our method not only realizes consistently increasing generalization ability but also enhances task-specific 3D recognition performances across various 3DDG benchmarks by a clear margin. Considering the lack of study and evaluation on 3DDG, we also create three new benchmarks, namely base-to-new, cross-dataset and few-shot generalization benchmarks, to enrich the field and inspire future research. Code and benchmarks are available at https://github.com/auniquesun/Point-PRC.

Large language models (LLMs) have shown remarkable proficiency in human-level reasoning and generation capabilities, which encourages extensive research on their application in mathematical problem solving. However, current work has been largely focused on text-based mathematical problems, with limited investigation in problems involving geometric information. Addressing this gap, we aim to enable LLMs to solve geometric problems by understanding image input. We first analyze the limitations of current Multimodal Large Language Models (MLLMs) in this area: they struggle to accurately comprehending basic geometric elements and their relationships. To overcome these challenges, we take advantage of the unique characteristics of geometric problems (such as unique geometric logical form, and geometric scalability) and the capacity of the textual LLMs to build an enriched multimodal geometry dataset based on existing data. The augmented dataset, Geo170K, contains more than 170K geometric image-caption and question-answer pairs. Utilizing our constructed Geo170K dataset, we develop G-LLaVA, which demonstrates exceptional performance in solving geometric problems, significantly outperforming GPT-4-V on the MathVista benchmark with only 7B parameters.

Recent advances in Neural Fields have enabled powerful, discretization-invariant methods for learning neural operators that approximate solutions of Partial Differential Equations (PDEs) on general geometries. Building on these developments, we introduce enf2enf, an encoder--decoder methodology for predicting steady-state Partial Differential Equations with non-parameterized geometric variability, based on recently proposed Equivariant Neural Field architectures. In enf2enf, input geometries are encoded into latent point cloud embeddings that inherently preserve geometric grounding and capture local phenomena. The resulting representations are then combined with global parameters and directly decoded into continuous output fields, thus efficiently modeling the coupling between geometry and physics. By leveraging the inductive biases of locality and translation invariance, our approach is able to capture fine-scale physical features as well as complex shape variations, thereby enhancing generalization and physical compliance. Extensive experiments on a high-fidelity aerodynamic dataset, a hyper-elastic material benchmark, and multi-element airfoil geometries, demonstrate that the proposed model achieves superior or competitive performance compared to state-of-the-art graph based, operator learning, and neural field methods. Notably, our method supports real time inference and zero-shot super-resolution, enabling efficient training on low-resolution meshes while maintaining high accuracy on full-scale discretizations.

The aim of this article is to introduce a new methodology for constructing morphings between shapes that have identical topology. This morphing is obtained by deforming a reference shape, through the resolution of a sequence of linear elasticity equations, onto the target shape. In particular, our approach does not assume any knowledge of a boundary parametrization. Furthermore, we demonstrate how constraints can be imposed on specific points, lines and surfaces in the reference domain to ensure alignment with their counterparts in the target domain after morphing. Additionally, we show how the proposed methodology can be integrated in an offline and online paradigm, which is useful in reduced-order modeling scenarii involving variable shapes. This framework facilitates the efficient computation of the morphings in various geometric configurations, thus improving the versatility and applicability of the approach. The methodology is illustrated on the regression problem of the drag and lift coefficients of airfoils of non-parameterized variable shapes.

The objective of domain generalization (DG) is to enhance the transferability of the model learned from a source domain to unobserved domains. To prevent overfitting to a specific domain, Sharpness-Aware Minimization (SAM) reduces source domain's loss sharpness. Although SAM variants have delivered significant improvements in DG, we highlight that there's still potential for improvement in generalizing to unknown domains through the exploration on data space. This paper introduces an objective rooted in both parameter and data perturbed regions for domain generalization, coined Unknown Domain Inconsistency Minimization (UDIM). UDIM reduces the loss landscape inconsistency between source domain and unknown domains. As unknown domains are inaccessible, these domains are empirically crafted by perturbing instances from the source domain dataset. In particular, by aligning the loss landscape acquired in the source domain to the loss landscape of perturbed domains, we expect to achieve generalization grounded on these flat minima for the unknown domains. Theoretically, we validate that merging SAM optimization with the UDIM objective establishes an upper bound for the true objective of the DG task. In an empirical aspect, UDIM consistently outperforms SAM variants across multiple DG benchmark datasets. Notably, UDIM shows statistically significant improvements in scenarios with more restrictive domain information, underscoring UDIM's generalization capability in unseen domains. Our code is available at https://github.com/SJShin-AI/UDIM.

The increasing availability of large-scale global datasets has generated a demand for scalable spatial prediction methods defined on spherical domains. Classical spatial models that rely on Euclidean distance representations are inappropriate for spherical data because planar projections distort geodesic distances and spatial neighborhood structures, while traditional kriging-based prediction methods are often computationally prohibitive for massive datasets. To address these challenges, we propose a Spherical DeepKriging framework for spatial prediction on S^2. The proposed approach constructs a flexible prediction model by integrating thin-plate spline (TPS) basis functions defined intrinsically on the sphere. Simulation studies and real data analyses are presented to demonstrate the superior predictive performance of the proposed method.

In this paper we show that visual diffusion models can serve as effective geometric solvers: they can directly reason about geometric problems by working in pixel space. We first demonstrate this on the Inscribed Square Problem, a long-standing problem in geometry that asks whether every Jordan curve contains four points forming a square. We then extend the approach to two other well-known hard geometric problems: the Steiner Tree Problem and the Simple Polygon Problem. Our method treats each problem instance as an image and trains a standard visual diffusion model that transforms Gaussian noise into an image representing a valid approximate solution that closely matches the exact one. The model learns to transform noisy geometric structures into correct configurations, effectively recasting geometric reasoning as image generation. Unlike prior work that necessitates specialized architectures and domain-specific adaptations when applying diffusion to parametric geometric representations, we employ a standard visual diffusion model that operates on the visual representation of the problem. This simplicity highlights a surprising bridge between generative modeling and geometric problem solving. Beyond the specific problems studied here, our results point toward a broader paradigm: operating in image space provides a general and practical framework for approximating notoriously hard problems, and opens the door to tackling a far wider class of challenging geometric tasks.

The aerodynamic coefficients of aircrafts are significantly impacted by its geometry, especially when the angle of attack (AoA) is large. In the field of aerodynamics, traditional polynomial-based parameterization uses as few parameters as possible to describe the geometry of an airfoil. However, because the 3D geometry of a wing is more complicated than the 2D airfoil, polynomial-based parameterizations have difficulty in accurately representing the entire shape of a wing in 3D space. Existing deep learning-based methods can extract massive latent neural representations for the shape of 2D airfoils or 2D slices of wings. Recent studies highlight that directly taking geometric features as inputs to the neural networks can improve the accuracy of predicted aerodynamic coefficients. Motivated by geometry theory, we propose to incorporate Riemannian geometric features for learning Coefficient of Pressure (CP) distributions on wing surfaces. Our method calculates geometric features (Riemannian metric, connection, and curvature) and further inputs the geometric features, coordinates and flight conditions into a deep learning model to predict the CP distribution. Experimental results show that our method, compared to state-of-the-art Deep Attention Network (DAN), reduces the predicted mean square error (MSE) of CP by an average of 8.41% for the DLR-F11 aircraft test set.

Geometric spatial reasoning forms the foundation of many applications in artificial intelligence, yet the ability of large language models (LLMs) to operate over geometric spatial information expressed in procedural code remains underexplored. In this paper, we address this gap by formalizing the Program-to-Geometry task, which challenges models to translate programmatic drawing code into accurate and abstract geometric reasoning. To evaluate this capability, we present GeoGramBench, a benchmark of 500 carefully refined problems organized by a tailored three-level taxonomy that considers geometric complexity rather than traditional mathematical reasoning complexity. Our comprehensive evaluation of 17 frontier LLMs reveals consistent and pronounced deficiencies: even the most advanced models achieve less than 50% accuracy at the highest abstraction level. These results highlight the unique challenges posed by program-driven spatial reasoning and establish GeoGramBench as a valuable resource for advancing research in symbolic-to-spatial geometric reasoning. Project page: https://github.com/LiAuto-DSR/GeoGramBench.

Generative models have shown great promise in generating 3D geometric systems, which is a fundamental problem in many natural science domains such as molecule and protein design. However, existing approaches only operate on static structures, neglecting the fact that physical systems are always dynamic in nature. In this work, we propose geometric trajectory diffusion models (GeoTDM), the first diffusion model for modeling the temporal distribution of 3D geometric trajectories. Modeling such distribution is challenging as it requires capturing both the complex spatial interactions with physical symmetries and temporal correspondence encapsulated in the dynamics. We theoretically justify that diffusion models with equivariant temporal kernels can lead to density with desired symmetry, and develop a novel transition kernel leveraging SE(3)-equivariant spatial convolution and temporal attention. Furthermore, to induce an expressive trajectory distribution for conditional generation, we introduce a generalized learnable geometric prior into the forward diffusion process to enhance temporal conditioning. We conduct extensive experiments on both unconditional and conditional generation in various scenarios, including physical simulation, molecular dynamics, and pedestrian motion. Empirical results on a wide suite of metrics demonstrate that GeoTDM can generate realistic geometric trajectories with significantly higher quality.

Vision-language models (VLMs) often struggle with geometric reasoning due to their limited perception of fundamental diagram elements. To tackle this challenge, we introduce GeoPerceive, a benchmark comprising diagram instances paired with domain-specific language (DSL) representations, along with an efficient automatic data generation pipeline. This design enables the isolated evaluation of geometric perception independently from reasoning. To exploit the data provided by GeoPerceive for enhancing the geometric perception capabilities of VLMs, we propose GeoDPO, a translator-guided reinforcement learning (RL) framework. GeoDPO employs an NL-to-DSL translator, which is trained on synthetic pairs generated by the data engine of GeoPerceive, to bridge natural language and DSL. This translator facilitates the computation of fine-grained, DSL-level scores, which serve as reward signals in reinforcement learning. We assess GeoDPO on both in-domain and out-of-domain datasets, spanning tasks in geometric perception as well as downstream reasoning. Experimental results demonstrate that, while supervised fine-tuning (SFT) offers only marginal improvements and may even impair performance in out-of-domain scenarios, GeoDPO achieves substantial gains: +26.5% on in-domain data, +8.0% on out-of-domain data, and +39.0% on downstream reasoning tasks. These findings underscore the superior performance and generalization ability of GeoDPO over SFT. All codes are released at https://github.com/Longin-Yu/GeoPerceive to ensure reproducibility.

Domain generalization (DG) is proposed to deal with the issue of domain shift, which occurs when statistical differences exist between source and target domains. However, most current methods do not account for a common realistic scenario where the source and target domains have different classes. To overcome this deficiency, open set domain generalization (OSDG) then emerges as a more practical setting to recognize unseen classes in unseen domains. An intuitive approach is to use multiple one-vs-all classifiers to define decision boundaries for each class and reject the outliers as unknown. However, the significant class imbalance between positive and negative samples often causes the boundaries biased towards positive ones, resulting in misclassification for known samples in the unseen target domain. In this paper, we propose a novel meta-learning-based framework called dualistic MEta-learning with joint DomaIn-Class matching (MEDIC), which considers gradient matching towards inter-domain and inter-class splits simultaneously to find a generalizable boundary balanced for all tasks. Experimental results demonstrate that MEDIC not only outperforms previous methods in open set scenarios, but also maintains competitive close set generalization ability at the same time. Our code is available at https://github.com/zzwdx/MEDIC.

Domain generalization (DG) aims to learn a robust model from source domains that generalize well on unseen target domains. Recent studies focus on generating novel domain samples or features to diversify distributions complementary to source domains. Yet, these approaches can hardly deal with the restriction that the samples synthesized from various domains can cause semantic distortion. In this paper, we propose an online one-stage Cross Contrasting Feature Perturbation (CCFP) framework to simulate domain shift by generating perturbed features in the latent space while regularizing the model prediction against domain shift. Different from the previous fixed synthesizing strategy, we design modules with learnable feature perturbations and semantic consistency constraints. In contrast to prior work, our method does not use any generative-based models or domain labels. We conduct extensive experiments on a standard DomainBed benchmark with a strict evaluation protocol for a fair comparison. Comprehensive experiments show that our method outperforms the previous state-of-the-art, and quantitative analyses illustrate that our approach can alleviate the domain shift problem in out-of-distribution (OOD) scenarios.

We propose MoGe-2, an advanced open-domain geometry estimation model that recovers a metric scale 3D point map of a scene from a single image. Our method builds upon the recent monocular geometry estimation approach, MoGe, which predicts affine-invariant point maps with unknown scales. We explore effective strategies to extend MoGe for metric geometry prediction without compromising the relative geometry accuracy provided by the affine-invariant point representation. Additionally, we discover that noise and errors in real data diminish fine-grained detail in the predicted geometry. We address this by developing a unified data refinement approach that filters and completes real data from different sources using sharp synthetic labels, significantly enhancing the granularity of the reconstructed geometry while maintaining the overall accuracy. We train our model on a large corpus of mixed datasets and conducted comprehensive evaluations, demonstrating its superior performance in achieving accurate relative geometry, precise metric scale, and fine-grained detail recovery -- capabilities that no previous methods have simultaneously achieved.

Medical images often exhibit distribution shifts due to variations in imaging protocols and scanners across different medical centers. Domain Generalization (DG) methods aim to train models on source domains that can generalize to unseen target domains. Recently, the segment anything model (SAM) has demonstrated strong generalization capabilities due to its prompt-based design, and has gained significant attention in image segmentation tasks. Existing SAM-based approaches attempt to address the need for manual prompts by introducing prompt generators that automatically generate these prompts. However, we argue that auto-generated prompts may not be sufficiently accurate under distribution shifts, potentially leading to incorrect predictions that still require manual verification and correction by clinicians. To address this challenge, we propose a method for 2D medical image segmentation called Self-Correcting SAM (CoSAM). Our approach begins by generating coarse masks using SAM in a prompt-free manner, providing prior prompts for the subsequent stages, and eliminating the need for prompt generators. To automatically refine these coarse masks, we introduce a generalized error decoder that simulates the correction process typically performed by clinicians. Furthermore, we generate diverse prompts as feedback based on the corrected masks, which are used to iteratively refine the predictions within a self-correcting loop, enhancing the generalization performance of our model. Extensive experiments on two medical image segmentation benchmarks across multiple scenarios demonstrate the superiority of CoSAM over state-of-the-art SAM-based methods.

There has been an influx of biomedical domain-specific language models, showing language models pre-trained on biomedical text perform better on biomedical domain benchmarks than those trained on general domain text corpora such as Wikipedia and Books. Yet, most works do not study the factors affecting each domain language application deeply. Additionally, the study of model size on domain-specific models has been mostly missing. We empirically study and evaluate several factors that can affect performance on domain language applications, such as the sub-word vocabulary set, model size, pre-training corpus, and domain transfer. We show consistent improvements on benchmarks with our larger BioMegatron model trained on a larger domain corpus, contributing to our understanding of domain language model applications. We demonstrate noticeable improvements over the previous state-of-the-art (SOTA) on standard biomedical NLP benchmarks of named entity recognition, relation extraction, and question answering. Model checkpoints and code are available at [https://ngc.nvidia.com] and [https://github.com/NVIDIA/NeMo].

Recent unsupervised domain adaptation methods have utilized vicinal space between the source and target domains. However, the equilibrium collapse of labels, a problem where the source labels are dominant over the target labels in the predictions of vicinal instances, has never been addressed. In this paper, we propose an instance-wise minimax strategy that minimizes the entropy of high uncertainty instances in the vicinal space to tackle the stated problem. We divide the vicinal space into two subspaces through the solution of the minimax problem: contrastive space and consensus space. In the contrastive space, inter-domain discrepancy is mitigated by constraining instances to have contrastive views and labels, and the consensus space reduces the confusion between intra-domain categories. The effectiveness of our method is demonstrated on public benchmarks, including Office-31, Office-Home, and VisDA-C, achieving state-of-the-art performances. We further show that our method outperforms the current state-of-the-art methods on PACS, which indicates that our instance-wise approach works well for multi-source domain adaptation as well. Code is available at https://github.com/NaJaeMin92/CoVi.

Equation discovery from data is a core challenge in machine learning for science, requiring the recovery of concise symbolic expressions that govern complex physical and geometric phenomena. Recent approaches with large language models (LLMs) show promise in symbolic regression, but their success often hinges on memorized formulas or overly simplified functional forms. Existing benchmarks exacerbate this limitation: they focus on scalar functions, ignore domain grounding, and rely on brittle string-matching based metrics that fail to capture scientific equivalence. We introduce SurfaceBench, first comprehensive benchmark for symbolic surface discovery. SurfaceBench comprises 183 tasks across 15 categories of symbolic complexity, spanning explicit, implicit, and parametric equation representation forms. Each task includes ground-truth equations, variable semantics, and synthetically sampled three dimensional data. Unlike prior SR datasets, our tasks reflect surface-level structure, resist LLM memorization through novel symbolic compositions, and are grounded in scientific domains such as fluid dynamics, robotics, electromagnetics, and geometry. To evaluate equation discovery quality, we pair symbolic checks with geometry-aware metrics such as Chamfer and Hausdorff distances, capturing both algebraic fidelity and spatial reconstruction accuracy. Our experiments reveal that state-of-the-art frameworks, while occasionally successful on specific families, struggle to generalize across representation types and surface complexities. SurfaceBench thus establishes a challenging and diagnostic testbed that bridges symbolic reasoning with geometric reconstruction, enabling principled benchmarking of progress in compositional generalization, data-driven scientific induction, and geometry-aware reasoning with LLMs. We release the code here: https://github.com/Sanchit-404/surfacebench

Graph neural networks are emerging as promising methods for modeling molecular graphs, in which nodes and edges correspond to atoms and chemical bonds, respectively. Recent studies show that when 3D molecular geometries, such as bond lengths and angles, are available, molecular property prediction tasks can be made more accurate. However, computing of 3D molecular geometries requires quantum calculations that are computationally prohibitive. For example, accurate calculation of 3D geometries of a small molecule requires hours of computing time using density functional theory (DFT). Here, we propose to predict the ground-state 3D geometries from molecular graphs using machine learning methods. To make this feasible, we develop a benchmark, known as Molecule3D, that includes a dataset with precise ground-state geometries of approximately 4 million molecules derived from DFT. We also provide a set of software tools for data processing, splitting, training, and evaluation, etc. Specifically, we propose to assess the error and validity of predicted geometries using four metrics. We implement two baseline methods that either predict the pairwise distance between atoms or atom coordinates in 3D space. Experimental results show that, compared with generating 3D geometries with RDKit, our method can achieve comparable prediction accuracy but with much smaller computational costs. Our Molecule3D is available as a module of the MoleculeX software library (https://github.com/divelab/MoleculeX).

In this paper, we present CAD2Program, a new method for reconstructing 3D parametric models from 2D CAD drawings. Our proposed method is inspired by recent successes in vision-language models (VLMs), and departs from traditional methods which rely on task-specific data representations and/or algorithms. Specifically, on the input side, we simply treat the 2D CAD drawing as a raster image, regardless of its original format, and encode the image with a standard ViT model. We show that such an encoding scheme achieves competitive performance against existing methods that operate on vector-graphics inputs, while imposing substantially fewer restrictions on the 2D drawings. On the output side, our method auto-regressively predicts a general-purpose language describing 3D parametric models in text form. Compared to other sequence modeling methods for CAD which use domain-specific sequence representations with fixed-size slots, our text-based representation is more flexible, and can be easily extended to arbitrary geometric entities and semantic or functional properties. Experimental results on a large-scale dataset of cabinet models demonstrate the effectiveness of our method.

Normalization layers are crucial for deep learning, but their Euclidean formulations are inadequate for data on manifolds. On the other hand, many Riemannian manifolds in machine learning admit gyro-structures, enabling principled extensions of Euclidean neural networks to non-Euclidean domains. Inspired by this, we introduce GyroBN, a principled Riemannian batch normalization framework for gyrogroups. We establish two necessary conditions, namely pseudo-reduction and gyroisometric gyrations, that guarantee GyroBN with theoretical control over sample statistics, and show that these conditions hold for all known gyrogroups in machine learning. Our framework also incorporates several existing Riemannian normalization methods as special cases. We further instantiate GyroBN on seven representative geometries, including the Grassmannian, five constant curvature spaces, and the correlation manifold, and derive novel gyro and Riemannian structures to enable these instantiations. Experiments across these geometries demonstrate the effectiveness of GyroBN. The code is available at https://github.com/GitZH-Chen/GyroBN.git.

Object detectors have achieved remarkable performance in many applications; however, these deep learning models are typically designed under the i.i.d. assumption, meaning they are trained and evaluated on data sampled from the same (source) distribution. In real-world deployment, however, target distributions often differ from source data, leading to substantial performance degradation. Domain Generalisation (DG) seeks to bridge this gap by enabling models to generalise to Out-Of-Distribution (OOD) data without access to target distributions during training, enhancing robustness to unseen conditions. In this work, we examine the generalisability and robustness of state-of-the-art object detectors under real-world distribution shifts, focusing particularly on spatial domain shifts. Despite the need, a standardised benchmark dataset specifically designed for assessing object detection under realistic DG scenarios is currently lacking. To address this, we introduce Real-World Distribution Shifts (RWDS), a suite of three novel DG benchmarking datasets that focus on humanitarian and climate change applications. These datasets enable the investigation of domain shifts across (i) climate zones and (ii) various disasters and geographic regions. To our knowledge, these are the first DG benchmarking datasets tailored for object detection in real-world, high-impact contexts. We aim for these datasets to serve as valuable resources for evaluating the robustness and generalisation of future object detection models. Our datasets and code are available at https://github.com/RWGAI/RWDS.

Domain Generation Algorithms (DGAs) are frequently used to generate numerous domains for use by botnets. These domains are often utilized as rendezvous points for servers that malware has command and control over. There are many algorithms that are used to generate domains, however many of these algorithms are simplistic and easily detected by traditional machine learning techniques. In this paper, three variants of Generative Adversarial Networks (GANs) are optimized to generate domains which have similar characteristics of benign domains, resulting in domains which greatly evade several state-of-the-art deep learning based DGA classifiers. We additionally provide a detailed analysis into offensive usability for each variant with respect to repeated and existing domain collisions. Finally, we fine-tune the state-of-the-art DGA classifiers by adding GAN generated samples to their original training datasets and analyze the changes in performance. Our results conclude that GAN based DGAs are superior in evading DGA classifiers in comparison to traditional DGAs, and of the variants, the Wasserstein GAN with Gradient Penalty (WGANGP) is the highest performing DGA for uses both offensively and defensively.

Recent advances in generative models highlight the power of geometry-aware modeling in manifold-constrained settings. Yet, for natural images, the field remains confined to Euclidean assumptions, failing to exploit the potential of intrinsic geometric structures within the data. In this work, we investigate the geometry of natural images and observe that semantic information is predominantly encoded in directional components, while norm components can be approximated by the global average. This property holds across both RGB and latent spaces, suggesting that natural images can be effectively modeled on a hypersphere. Building on this finding, we introduce Spherical Optimal Transport Flow Matching (SOT-CFM), which utilizes angular distance, and Spherical Flow Matching (SFM), which constrains dynamics directly on the manifold. Our experiments demonstrate that these geometry-aware methods achieve superior performance against Euclidean baselines. Ultimately, this work provides a novel perspective that bridges the gap between Riemannian manifold-based modeling and natural image generation.

In machine learning, there is a long history of trying to build neural networks that can learn from fewer example data by baking in strong geometric priors. However, it is not always clear a priori what geometric constraints are appropriate for a given task. Here, we explore the possibility that one can uncover useful geometric inductive biases by studying how training molds the Riemannian geometry induced by unconstrained neural network feature maps. We first show that at infinite width, neural networks with random parameters induce highly symmetric metrics on input space. This symmetry is broken by feature learning: networks trained to perform classification tasks learn to magnify local areas along decision boundaries. This holds in deep networks trained on high-dimensional image classification tasks, and even in self-supervised representation learning. These results begin to elucidate how training shapes the geometry induced by unconstrained neural network feature maps, laying the groundwork for an understanding of this richly nonlinear form of feature learning.

We present AlphaGeometry2, a significantly improved version of AlphaGeometry introduced in Trinh et al. (2024), which has now surpassed an average gold medalist in solving Olympiad geometry problems. To achieve this, we first extend the original AlphaGeometry language to tackle harder problems involving movements of objects, and problems containing linear equations of angles, ratios, and distances. This, together with other additions, has markedly improved the coverage rate of the AlphaGeometry language on International Math Olympiads (IMO) 2000-2024 geometry problems from 66% to 88%. The search process of AlphaGeometry2 has also been greatly improved through the use of Gemini architecture for better language modeling, and a novel knowledge-sharing mechanism that combines multiple search trees. Together with further enhancements to the symbolic engine and synthetic data generation, we have significantly boosted the overall solving rate of AlphaGeometry2 to 84% for all geometry problems over the last 25 years, compared to 54% previously. AlphaGeometry2 was also part of the system that achieved silver-medal standard at IMO 2024 https://dpmd.ai/imo-silver. Last but not least, we report progress towards using AlphaGeometry2 as a part of a fully automated system that reliably solves geometry problems directly from natural language input.

Large Multimodal Models (LMMs) often struggle with geometric reasoning due to visual hallucinations and a lack of mathematically precise Chain-of-Thought (CoT) data. To address this, we propose the GeoSym Engine, an automated and scalable neuro-symbolic framework. By leveraging a type-conditional grammar and an analytic SymGT Solver, it derives exact symbolic ground truths and seamlessly integrates with a robust rendering pipeline to produce high-precision geometric diagrams. Using this engine, we construct GeoSym127K, a difficulty-stratified dataset featuring 51K high-resolution images, 127K questions with symbolic ground truths, and 55K answer-verified CoT QA pairs. We also introduce GeoSym-Bench, an expert-curated suite of 511 complex samples for rigorous evaluation. Through extensive supervised fine-tuning (SFT), we demonstrate that GeoSym drives concentrated improvements specifically on diagram-dependent and multi-step geometry tasks. Our Qwen3-VL-8B model gains an absolute +22.21% on the MathVerse Vision-Only subset and reaches 61.52% (+6.19% improvement) on WeMath, mitigating long-horizon logic fragmentation and outperforming advanced closed-source models like Doubao-1.8. Furthermore, applying Reinforcement Learning with Verifiable Rewards (RLVR) via GRPO reveals that initializing from structural SFT checkpoints substantially elevates the performance ceiling over zero-shot RL. Driven by deterministic exact-match signals, this showcases the robust scaling potential of our verifiable reasoning synthesis. Datasets and code are available at https://huggingface.co/datasets/Tomie0506/GeoSym127K and https://github.com/Tomie56/GeoSym127K.

This work studies the generalization issue of face anti-spoofing (FAS) models on domain gaps, such as image resolution, blurriness and sensor variations. Most prior works regard domain-specific signals as a negative impact, and apply metric learning or adversarial losses to remove them from feature representation. Though learning a domain-invariant feature space is viable for the training data, we show that the feature shift still exists in an unseen test domain, which backfires on the generalizability of the classifier. In this work, instead of constructing a domain-invariant feature space, we encourage domain separability while aligning the live-to-spoof transition (i.e., the trajectory from live to spoof) to be the same for all domains. We formulate this FAS strategy of separability and alignment (SA-FAS) as a problem of invariant risk minimization (IRM), and learn domain-variant feature representation but domain-invariant classifier. We demonstrate the effectiveness of SA-FAS on challenging cross-domain FAS datasets and establish state-of-the-art performance.

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of effective species in an environment; when applied to a one-parameter family of metric spaces tX with scale parameter t, the magnitude captures much of the underlying geometry of the space. Prior work has mostly focussed on properties of magnitude in a global sense; in this paper we restrict the sets to finite subsets of Euclidean space and investigate its individual components. We give an explicit formula for the corrected inclusion-exclusion principle, and define a quantity associated with each point, called the moment which gives an intrinsic ordering to the points. We exploit this in order to form an algorithm which approximates the convex hull.

A triangular mesh is one of the most popular 3D data representations. As such, the deployment of deep neural networks for mesh processing is widely spread and is increasingly attracting more attention. However, neural networks are prone to adversarial attacks, where carefully crafted inputs impair the model's functionality. The need to explore these vulnerabilities is a fundamental factor in the future development of 3D-based applications. Recently, mesh attacks were studied on the semantic level, where classifiers are misled to produce wrong predictions. Nevertheless, mesh surfaces possess complex geometric attributes beyond their semantic meaning, and their analysis often includes the need to encode and reconstruct the geometry of the shape. We propose a novel framework for a geometric adversarial attack on a 3D mesh autoencoder. In this setting, an adversarial input mesh deceives the autoencoder by forcing it to reconstruct a different geometric shape at its output. The malicious input is produced by perturbing a clean shape in the spectral domain. Our method leverages the spectral decomposition of the mesh along with additional mesh-related properties to obtain visually credible results that consider the delicacy of surface distortions. Our code is publicly available at https://github.com/StolikTomer/SAGA.

Machine learning systems typically assume that the distributions of training and test sets match closely. However, a critical requirement of such systems in the real world is their ability to generalize to unseen domains. Here, we propose an inter-domain gradient matching objective that targets domain generalization by maximizing the inner product between gradients from different domains. Since direct optimization of the gradient inner product can be computationally prohibitive -- requires computation of second-order derivatives -- we derive a simpler first-order algorithm named Fish that approximates its optimization. We demonstrate the efficacy of Fish on 6 datasets from the Wilds benchmark, which captures distribution shift across a diverse range of modalities. Our method produces competitive results on these datasets and surpasses all baselines on 4 of them. We perform experiments on both the Wilds benchmark, which captures distribution shift in the real world, as well as datasets in DomainBed benchmark that focuses more on synthetic-to-real transfer. Our method produces competitive results on both benchmarks, demonstrating its effectiveness across a wide range of domain generalization tasks.

The creation of photorealistic virtual worlds requires the accurate modeling of 3D surface geometry for a wide range of objects. For this, meshes are appealing since they 1) enable fast physics-based rendering with realistic material and lighting, 2) support physical simulation, and 3) are memory-efficient for modern graphics pipelines. Recent work on reconstructing and statistically modeling 3D shape, however, has critiqued meshes as being topologically inflexible. To capture a wide range of object shapes, any 3D representation must be able to model solid, watertight, shapes as well as thin, open, surfaces. Recent work has focused on the former, and methods for reconstructing open surfaces do not support fast reconstruction with material and lighting or unconditional generative modelling. Inspired by the observation that open surfaces can be seen as islands floating on watertight surfaces, we parameterize open surfaces by defining a manifold signed distance field on watertight templates. With this parameterization, we further develop a grid-based and differentiable representation that parameterizes both watertight and non-watertight meshes of arbitrary topology. Our new representation, called Ghost-on-the-Shell (G-Shell), enables two important applications: differentiable rasterization-based reconstruction from multiview images and generative modelling of non-watertight meshes. We empirically demonstrate that G-Shell achieves state-of-the-art performance on non-watertight mesh reconstruction and generation tasks, while also performing effectively for watertight meshes.

We present a dataset of large-scale indoor spaces that provides a variety of mutually registered modalities from 2D, 2.5D and 3D domains, with instance-level semantic and geometric annotations. The dataset covers over 6,000m2 and contains over 70,000 RGB images, along with the corresponding depths, surface normals, semantic annotations, global XYZ images (all in forms of both regular and 360{\deg} equirectangular images) as well as camera information. It also includes registered raw and semantically annotated 3D meshes and point clouds. The dataset enables development of joint and cross-modal learning models and potentially unsupervised approaches utilizing the regularities present in large-scale indoor spaces. The dataset is available here: http://3Dsemantics.stanford.edu/

Conformal symmetries, i.e.\ coordinate transformations that preserve angles, play a key role in many fields, including physics, mathematics, computer vision and (geometric) machine learning. Here we build a neural network that is equivariant under general conformal transformations. To achieve this, we lift data from flat Euclidean space to Anti de Sitter (AdS) space. This allows us to exploit a known correspondence between conformal transformations of flat space and isometric transformations on the AdS space. We then build upon the fact that such isometric transformations have been extensively studied on general geometries in the geometric deep learning literature. We employ message-passing layers conditioned on the proper distance, yielding a computationally efficient framework. We validate our model on tasks from computer vision and statistical physics, demonstrating strong performance, improved generalization capacities, and the ability to extract conformal data such as scaling dimensions from the trained network.

Multi-task learning (MTL) is an inductive transfer mechanism designed to leverage useful information from multiple tasks to improve generalization performance compared to single-task learning. It has been extensively explored in traditional machine learning to address issues such as data sparsity and overfitting in neural networks. In this work, we apply MTL to problems in science and engineering governed by partial differential equations (PDEs). However, implementing MTL in this context is complex, as it requires task-specific modifications to accommodate various scenarios representing different physical processes. To this end, we present a multi-task deep operator network (MT-DeepONet) to learn solutions across various functional forms of source terms in a PDE and multiple geometries in a single concurrent training session. We introduce modifications in the branch network of the vanilla DeepONet to account for various functional forms of a parameterized coefficient in a PDE. Additionally, we handle parameterized geometries by introducing a binary mask in the branch network and incorporating it into the loss term to improve convergence and generalization to new geometry tasks. Our approach is demonstrated on three benchmark problems: (1) learning different functional forms of the source term in the Fisher equation; (2) learning multiple geometries in a 2D Darcy Flow problem and showcasing better transfer learning capabilities to new geometries; and (3) learning 3D parameterized geometries for a heat transfer problem and demonstrate the ability to predict on new but similar geometries. Our MT-DeepONet framework offers a novel approach to solving PDE problems in engineering and science under a unified umbrella based on synergistic learning that reduces the overall training cost for neural operators.

We propose Riemannian Flow Matching (RFM), a simple yet powerful framework for training continuous normalizing flows on manifolds. Existing methods for generative modeling on manifolds either require expensive simulation, are inherently unable to scale to high dimensions, or use approximations for limiting quantities that result in biased training objectives. Riemannian Flow Matching bypasses these limitations and offers several advantages over previous approaches: it is simulation-free on simple geometries, does not require divergence computation, and computes its target vector field in closed-form. The key ingredient behind RFM is the construction of a relatively simple premetric for defining target vector fields, which encompasses the existing Euclidean case. To extend to general geometries, we rely on the use of spectral decompositions to efficiently compute premetrics on the fly. Our method achieves state-of-the-art performance on many real-world non-Euclidean datasets, and we demonstrate tractable training on general geometries, including triangular meshes with highly non-trivial curvature and boundaries.

Graphs are able to model interconnected entities in many online services, supporting a wide range of applications on the Web. This raises an important question: How can we train a graph foundational model on multiple source domains and adapt to an unseen target domain? A major obstacle is that graphs from different domains often exhibit divergent characteristics. Some studies leverage large language models to align multiple domains based on textual descriptions associated with the graphs, limiting their applicability to text-attributed graphs. For text-free graphs, a few recent works attempt to align different feature distributions across domains, while generally neglecting structural differences. In this work, we propose a novel Structure Alignment framework for text-free Multi-domain Graph Pre-Training and cross-domain adaptation (SAMGPT). It is designed to learn multi-domain knowledge from graphs originating in multiple source domains, which can then be adapted to address applications in an unseen target domain. Specifically, we introduce a set of structure tokens to harmonize structure-based aggregation across source domains during the pre-training phase. Next, for cross-domain adaptation, we design dual prompts, namely, holistic prompts and specific prompts, which adapt unified multi-domain structural knowledge and fine-grained, domain-specific information, respectively, to a target domain. Finally, we conduct comprehensive experiments on seven public datasets to evaluate and analyze the effectiveness of SAMGPT.

Fast, geometry-generalizing surrogates for unsteady flow remain challenging. We present a time-dependent, geometry-aware Deep Operator Network that predicts velocity fields for moderate-Re flows around parametric and non-parametric shapes. The model encodes geometry via a signed distance field (SDF) trunk and flow history via a CNN branch, trained on 841 high-fidelity simulations. On held-out shapes, it attains sim 5% relative L2 single-step error and up to 1000X speedups over CFD. We provide physics-centric rollout diagnostics, including phase error at probes and divergence norms, to quantify long-horizon fidelity. These reveal accurate near-term transients but error accumulation in fine-scale wakes, most pronounced for sharp-cornered geometries. We analyze failure modes and outline practical mitigations. Code, splits, and scripts are openly released at: https://github.com/baskargroup/TimeDependent-DeepONet to support reproducibility and benchmarking.

AI-driven geometric problem solving is a complex vision-language task that requires accurate diagram interpretation, mathematical reasoning, and robust cross-modal grounding. A foundational yet underexplored capability for this task is the ability to identify and interpret geometric elements based on natural language queries. To address this, we introduce the task of Referring Expression Comprehension (REC) for geometric problems, which evaluates whether models can localize points, shapes, and spatial relations in diagrams in response to textual prompts. We present GeoRef, a benchmark dataset constructed from existing geometric problem corpora, featuring diverse, high-quality annotations and queries. Due to the lack of annotated data for this task, we generate a large-scale synthetic training dataset using a structured geometric formal language, enabling broad coverage of geometric concepts and facilitating model adaptation. We explore two fine-tuning approaches: Supervised Fine-Tuning (SFT) and Group Relative Policy Optimization (GRPO). Our results show that GRPO significantly outperforms SFT by better aligning model behavior with task-specific rewards. Furthermore, we propose a verify-and-regenerate mechanism that detects incorrect predictions and re-infers answers using contextual reasoning history, further boosting accuracy. Notably, even state-of-the-art Multimodal Large Language Models (MLLMs) struggle with this task, underscoring the necessity of explicitly evaluating and strengthening geometric grounding as a prerequisite for robust geometric problem solving. Moreover, models trained on GeoRef demonstrate measurable improvements on downstream geometric reasoning tasks, highlighting the broader value of REC as a foundation for multimodal mathematical understanding.

We present a system for generating indoor scenes in response to text prompts. The prompts are not limited to a fixed vocabulary of scene descriptions, and the objects in generated scenes are not restricted to a fixed set of object categories -- we call this setting indoor scene generation. Unlike most prior work on indoor scene generation, our system does not require a large training dataset of existing 3D scenes. Instead, it leverages the world knowledge encoded in pre-trained large language models (LLMs) to synthesize programs in a domain-specific layout language that describe objects and spatial relations between them. Executing such a program produces a specification of a constraint satisfaction problem, which the system solves using a gradient-based optimization scheme to produce object positions and orientations. To produce object geometry, the system retrieves 3D meshes from a database. Unlike prior work which uses databases of category-annotated, mutually-aligned meshes, we develop a pipeline using vision-language models (VLMs) to retrieve meshes from massive databases of un-annotated, inconsistently-aligned meshes. Experimental evaluations show that our system outperforms generative models trained on 3D data for traditional, closed-universe scene generation tasks; it also outperforms a recent LLM-based layout generation method on open-universe scene generation.

In real-world scenarios, achieving domain generalization (DG) presents significant challenges as models are required to generalize to unknown target distributions. Generalizing to unseen multi-modal distributions poses even greater difficulties due to the distinct properties exhibited by different modalities. To overcome the challenges of achieving domain generalization in multi-modal scenarios, we propose SimMMDG, a simple yet effective multi-modal DG framework. We argue that mapping features from different modalities into the same embedding space impedes model generalization. To address this, we propose splitting the features within each modality into modality-specific and modality-shared components. We employ supervised contrastive learning on the modality-shared features to ensure they possess joint properties and impose distance constraints on modality-specific features to promote diversity. In addition, we introduce a cross-modal translation module to regularize the learned features, which can also be used for missing-modality generalization. We demonstrate that our framework is theoretically well-supported and achieves strong performance in multi-modal DG on the EPIC-Kitchens dataset and the novel Human-Animal-Cartoon (HAC) dataset introduced in this paper. Our source code and HAC dataset are available at https://github.com/donghao51/SimMMDG.

We introduce a framework for automatically defining and learning deep generative models with problem-specific structure. We tackle problem domains that are more traditionally solved by algorithms such as sorting, constraint satisfaction for Sudoku, and matrix factorization. Concretely, we train diffusion models with an architecture tailored to the problem specification. This problem specification should contain a graphical model describing relationships between variables, and often benefits from explicit representation of subcomputations. Permutation invariances can also be exploited. Across a diverse set of experiments we improve the scaling relationship between problem dimension and our model's performance, in terms of both training time and final accuracy. Our code can be found at https://github.com/plai-group/gsdm.

We present a novel geometric perspective on the latent space of diffusion models. We first show that the standard pullback approach, utilizing the deterministic probability flow ODE decoder, is fundamentally flawed. It provably forces geodesics to decode as straight segments in data space, effectively ignoring any intrinsic data geometry beyond the ambient Euclidean space. Complementing this view, diffusion also admits a stochastic decoder via the reverse SDE, which enables an information geometric treatment with the Fisher-Rao metric. However, a choice of x_T as the latent representation collapses this metric due to memorylessness. We address this by introducing a latent spacetime z=(x_t,t) that indexes the family of denoising distributions p(x_0 | x_t) across all noise scales, yielding a nontrivial geometric structure. We prove these distributions form an exponential family and derive simulation-free estimators for curve lengths, enabling efficient geodesic computation. The resulting structure induces a principled Diffusion Edit Distance, where geodesics trace minimal sequences of noise and denoise edits between data. We also demonstrate benefits for transition path sampling in molecular systems, including constrained variants such as low-variance transitions and region avoidance. Code is available at: https://github.com/rafalkarczewski/spacetime-geometry.

Although polygon meshes have been a standard representation in geometry processing, their irregular and combinatorial nature hinders their suitability for learning-based applications. In this work, we introduce a novel learnable mesh representation through a set of local 3D sample Points and their associated Normals and Quadric error metrics (QEM) w.r.t. the underlying shape, which we denote PoNQ. A global mesh is directly derived from PoNQ by efficiently leveraging the knowledge of the local quadric errors. Besides marking the first use of QEM within a neural shape representation, our contribution guarantees both topological and geometrical properties by ensuring that a PoNQ mesh does not self-intersect and is always the boundary of a volume. Notably, our representation does not rely on a regular grid, is supervised directly by the target surface alone, and also handles open surfaces with boundaries and/or sharp features. We demonstrate the efficacy of PoNQ through a learning-based mesh prediction from SDF grids and show that our method surpasses recent state-of-the-art techniques in terms of both surface and edge-based metrics.

Neural representations carry rich geometric structure; but does that structure causally shape behavior? To address this question, we intervene along paths through activation space defined by different geometries, and measure the behavioral trajectories they induce. In particular, we test whether interventions that respect the geometry of activation space will yield behaviors close to those the model exhibits naturally. Concretely, we first fit an activation manifold M_h to representations and a behavior manifold M_y to output probability distributions. We then test the link M_h leftrightarrow M_y via interventions: we find that steering along M_h, which we term manifold steering, yields behavioral trajectories that follow M_y, while linear steering -- which assumes a Euclidean geometry -- cuts through off-manifold regions and hence produces unnatural outputs. Moreover, optimizing interventions in activation space to produce paths along M_y recovers activation trajectories that trace the curvature of M_h. We demonstrate this bidirectional relationship between the geometry of representation and behavior across tasks and modalities. In language models, we use reasoning tasks with cyclic and sequential geometries as well as in-context learning tasks with more complex graph geometries. In a video world model, we use a task with geometry corresponding to physical dynamics. Overall, our work shows that geometry in neural representation is not merely incidental, but is in fact the proper object for enabling principled control via intervention on internals. This recasts the core problem of steering from finding the right direction to finding the right geometry.

Domain generalization (DG) seeks to learn robust models that generalize well under unknown distribution shifts. As a critical aspect of DG, optimizer selection has not been explored in depth. Currently, most DG methods follow the widely used benchmark, DomainBed, and utilize Adam as the default optimizer for all datasets. However, we reveal that Adam is not necessarily the optimal choice for the majority of current DG methods and datasets. Based on the perspective of loss landscape flatness, we propose a novel approach, Flatness-Aware Minimization for Domain Generalization (FAD), which can efficiently optimize both zeroth-order and first-order flatness simultaneously for DG. We provide theoretical analyses of the FAD's out-of-distribution (OOD) generalization error and convergence. Our experimental results demonstrate the superiority of FAD on various DG datasets. Additionally, we confirm that FAD is capable of discovering flatter optima in comparison to other zeroth-order and first-order flatness-aware optimization methods.

Domain generalization is the task of learning models that generalize to unseen target domains. We propose a simple yet effective method for domain generalization, named cross-domain ensemble distillation (XDED), that learns domain-invariant features while encouraging the model to converge to flat minima, which recently turned out to be a sufficient condition for domain generalization. To this end, our method generates an ensemble of the output logits from training data with the same label but from different domains and then penalizes each output for the mismatch with the ensemble. Also, we present a de-stylization technique that standardizes features to encourage the model to produce style-consistent predictions even in an arbitrary target domain. Our method greatly improves generalization capability in public benchmarks for cross-domain image classification, cross-dataset person re-ID, and cross-dataset semantic segmentation. Moreover, we show that models learned by our method are robust against adversarial attacks and image corruptions.