Geometric Deep Learning

Recognizing hand-drawn geometric shapes is a foundational sub-problem of sketch recognition, with applications in education, human-computer interaction, and diagram digitization. This paper presents the design, implementation, and evaluation of a desktop application that recognizes four basic hand-drawn geometric shapes, circle, square, rectangle, and triangle using a compact Convolutional Neural Network (CNN). A dataset of 2,000 labeled 28x28-pixel shape images was collected independently and released publicly. The classifier consists of three convolutional blocks (16, 32, and 64 filters) with max-pooling, an in-model data-augmentation stage (random horizontal flip, rotation, and zoom), a dropout-regularized dense layer of 128 units, and a 4-way linear output layer, totaling 97{,}956 trainable parameters. The network is trained with the Adam optimizer on a sparse categorical cross-entropy objective computed directly on logits. On an 80/20 train-validation split, the model achieves 94.80% training accuracy and 96.01% validation accuracy with a validation loss of 0.1437. A Tkinter-based graphical interface allows a user to draw a shape with the mouse and receive an immediate class prediction with a confidence score. We situate this system within the broader sketch and shape-recognition literature, compare its accuracy against related hand-drawn shape classification studies, and discuss the limitations inherent to a small, single-contributor dataset. The complete source code, trained model, and per-class datasets are released publicly to support reproducibility.

Rotational symmetry is one of the most important structural principles in machine learning on 3D data. In applications ranging from physics and materials science to 3D computer vision, predictions should not depend on an arbitrary choice of coordinate frame. Rotational equivariance captures this requirement mathematically by enforcing that a rotation of the input induces a corresponding transformation of the model output. This tutorial provides a comprehensive introduction to rotational equivariance, starting from the physical and geometric intuition behind coordinate independence and building up the necessary machinery from geometric deep learning, group theory, and representation theory. We introduce message passing on Euclidean graphs, group actions and representations, spherical harmonics, Wigner matrices, tensor products, and Clebsch-Gordan decomposition, and explain how these ingredients give rise to modern equivariant architectures. We then survey the principal strategies for incorporating rotational equivariance in deep learning, including group convolutions, internal tensorial representations, and canonicalization-based methods, and discuss their practical strengths and limitations. The tutorial aims to lower the barrier to the subject by connecting the underlying mathematics to practical model design, by unifying ideas that are often expressed in different formal languages, and by helping practitioners choose among competing approaches through a clear discussion of their trade-offs.

Deep learning systems perform mainly within the 2D for a single image domain and take the face as a single-dimension representation, losing sight of the 3D anatomy of sheep and cross-landmark spatial relationships that are intrinsic to the clinically proven Sheep Pain Facial Expression Scale (SPFES). This paper presents the \textbf{3D Sheep Pain Facial Expression System (3D-SPFES)}, a novel, monocular depth-aware geometric graph neural network system that integrates each SPFES facial landmark, such as the ears, eyes, and nose, into 3D Euclidean space estimated from a single RGB camera by using VideoDepthAnything, thus preventing the need for specialized depth hardware. Each landmark node includes a feature vector containing its 3D spatial coordinates, estimated surface normal, and facial attribute class embedding. Edges linked to nodes are assigned weights based on an aggregate metric that combines both Euclidean distance and surface co-planarity in a 3D space. A Weighted Geometric Graph Neural Network (WG-GNN) studies this graph using $\mathcal{K} = 3$ geometry-aware message-passing layers enhanced by a scaled dot-product attention method that selectively enhances anatomically relevant inter-landmark messages. The resultant node embeddings are combined into $\mathcal{O} = 3$ pain-level clusters and integrated into a Normalized Pain Score (NPS) within the range of $[0, 100$ a confidence-weighted, SPFES-derived scoring method.

We introduce cut-cell skinning, a geometric prior designed to augment data-driven skinning weight generation. While data-driven methods show promise in producing high-quality skinning weights, they often lack the generalizability of classic geometric approaches. To bridge this gap, we propose a geometric prior that can be robustly computed for in-the-wild meshes and is efficient for large-scale machine learning workflows. The key idea of our cut-cell skinning is a fast graph-based approximation of the volumetric geodesics distances, motivated by their importance in classic skinning weight computation. Our method achieves orders of magnitude speedup compared to optimization-based solvers and remains resilient to topological artifacts common in cage- or voxel-based alternatives. We demonstrate the efficacy of the cut-cell skinning prior by integrating it into recent neural skinning models, showing consistent improvements across existing methods and achieving state-of-the-art results. Project page: https://wenchao-m.github.io/CutCell.github.io/

Blood-brain barrier permeability (BBBP) prediction is a critical screening task in central nervous system drug discovery, where candidate molecules must be assessed for whether they can cross, or should be prevented from crossing, the blood-brain barrier. However, this task remains challenging because of limited, class-imbalanced datasets and sensitivity to molecular structure. Recent advances in deep learning have established graph neural networks (GNNs) as a powerful approach for molecular representation learning, while pre-trained molecular GNNs provide transferable knowledge for downstream tasks. However, full fine-tuning is often parameter-inefficient and prone to overfitting, whereas existing parameter-efficient fine-tuning (PEFT) methods mainly adapt node features or the two-dimensional covalent graph, limiting their ability to capture three-dimensional geometry and second-order interactions. To address these limitations, we propose BBBP-GeoPEFT, a geometry-informed PEFT framework for pre-trained molecular GNNs. BBBP-GeoPEFT constructs distance-based graphs at multiple cutoffs and their corresponding line graphs from molecular conformers to capture spatial atom and second-order edge interactions. Lightweight auxiliary geometric graph encoders generate cutoff-specific representations, which are incorporated into each pre-trained layer through node-wise cutoff attention and gated residual connections. This design preserves pre-trained knowledge while incorporating permeability-relevant geometric information with a small trainable-parameter budget. Experiments on a curated BBBP dataset show that BBBP-GeoPEFT achieves competitive performance compared with full fine-tuning and representative PEFT baselines. Under both random and scaffold splitting, BBBP-GeoPEFT achieves competitive or improved ROC-AUC and accuracy in most experiments while updating only 10.1% of the model parameters.

Brain tissue microstructure estimation with machine learning provides higher computational efficiency than conventional fitting. However, machine learning still presents important limitations that hamper its clinical utility. Specifically, current models typically lack generalisation across diffusion MRI acquisition protocols and require retraining whenever b-vectors or b-values change. Moreover, the recent machine learning methods that were developed to address protocol generalisation lack rotational equivariance. Particularly suitable for dMRI parameter estimation is a geometric deep learning model known as spherical convolutional neural network (SCNN), which guarantees rotational equivariance and b-vector generalisation. However, this architecture currently does not account for b-values. Therefore, obtaining a model that combines protocol generalisation and rotational equivariance remains an open challenge. In this paper, we directly address this issue by incorporating explicit b-value dependence into an SCNN architecture via a hypernetwork. This new approach is illustrated using NODDI as an example forward model for estimating brain tissue microstructure. To evaluate b-value generalisation, the original and newly proposed SCNN architectures are trained on synthetic data and tested on both synthetic and real data across different b-value pairs. Results demonstrate that the proposed method achieves reduced RMSE and bias on synthetic data, as well as higher agreement with conventional NODDI fitting on real data, indicating improved robustness to unseen b-values and a reduced need for retraining. By combining generalisation across b-values with generalisation across b-vectors and rotational equivariance, the proposed framework enhances the applicability of deep learning to clinical diffusion MRI parameter estimation. Code available at https://github.com/aerdnairo/arXiv\_generalisedSCNN.

Polarimetric Synthetic Aperture Radar (PolSAR) classification underpins all-weather Earth observation. Conventional Wishart methods depend on rigid handcrafted operators with limited adaptability, while mainstream deep networks ignore PolSAR native Wishart scattering statistics. Additionally, fixed convolution windows fail to capture multi-scale, multi-directional terrain patterns, harming boundary detection and small-object characterization. To mitigate these drawbacks, we propose WGDNet, a Wishart-guided geometric-aware deep network. It integrates three core designs: (1) learnable Wishart convolutions with directional kernels for multi-scale statistical edge feature extraction; (2) an orientation-prior aggregation module that estimates dominant local directions and confidences to refine directional Wishart outputs adaptively; (3) GAnet, a scale-direction adaptive geometric-aware convolution that dynamically reshapes sampling grids to model anisotropic terrain and retain fine details. Our contributions lie in learnable Wishart statistical modeling, orientation-prior feature aggregation, and geometry-adaptive convolution. Evaluations across four real PolSAR datasets verify WGDNet surpasses existing state-of-the-art approaches in classification accuracy and boundary fidelity.

We present a privacy-preserving framework for synthetic lung CT slice generation developed for the Image-CLEFmed GANs 2026 challenge. The approach combines Optimal Transport Conditional Flow Matching with privacy-oriented training and a post-generation "Supervisor" pipeline that filters generated candidates in learned geometric latent spaces using autoencoder embeddings, Determinantal Point Processes, and Stein Kernel Thinning. Official results show a strong realism-privacy trade-off, with the best-performing model achieving a Privacy Preservation Score of 0.549 and competitive visual fidelity with an FID of 0.3290. While the proposed geometric filtering substantially reduces nearest-neighbor memorization and membership-inference leakage, persistent patient re-identification scores indicate that preventing direct image copying is not sufficient to remove deeper patient-specific anatomical identity, highlighting an important frontier for future privacy-preserving medical image generation.

We present a novel isogeometric deep learning method, termed SplineNet, for the seamless design and analysis of shell structures with complex geometries. The proposed approach is built upon watertight spline representations, e.g., analysis-suitable unstructured T-splines, and features exact geometric descriptions of Computer-Aided Design (CAD) models in neural networks. Bézier extraction is used to build the network architecture, where Bernstein polynomials serve as the nonlinear activation functions. SplineNet can be applied in a data-free or data-driven way. In the data-free case, energy-based formulations can be naturally incorporated as loss terms, which fulfill the need of Computer-Aided Engineering (CAE) and can be accurately calculated. In particular, the Kirchhoff--Love (KL) model is adopted to solve for the mechanical behaviors of shell structures. This way, CAD and CAE can be tightly integrated in a deep neural network without the time-consuming model/data exchange process. In the data-driven case, SplineNet can be used as the trunk net of Deep Operator Networks (DeepONet) to provide interpretability. Given such a trained network and unseen input data, results can be immediately obtained without retraining the network or repeatedly performing the traditional workflow for analysis. In the end, a variety of numerical examples are studied to demonstrate the effectiveness of the proposed method, especially when real-world complex geometries are involved.

Scientists often seek to draw causal inferences from structured data that is not independently and identically distributed, such as spatial data, network data, or molecular data. We develop geometric causal models (GCMs), a framework for causal inference from dependent data that exploits underlying symmetries of the data generating process. For example, in spatial data, we consider processes that are symmetric under translations, or in graph data, symmetric under permutations of the nodes. We show how symmetries, formalized with group theory, can enable causal identification and estimation. We deploy ergodic theory for amenable groups to establish identification, and combine geometric deep learning with scalable Bayesian inference for estimation. We recover i.i.d. causal models and do-calculus when the data is a sequence and the symmetry is permutation equivariance, and find novel types of causal models when we use alternate structures and symmetries. As an example, we construct a causal model that satisfies the symmetries of DNA. This GCM enables new estimators for the effects of genetic variation, combining deep functional genomics models to describe outcomes and DNA language models to describe propensities. We illustrate on semisynthetic data.

Symmetry is everywhere in nature and society. Geometric deep learning exploits symmetries in data to improve the performance and efficiency of deep learning systems. In this paper, we extend geometric deep learning to utilize richer symmetry structures. Specifically, we develop order-equivariant neural networks (OENN), which generalize standard graph message passing and sheaf neural networks via the theory of equivariant bundles over face posets (face categories). We (i) characterize all linear order-equivariant maps, (ii) build OENN layers, and (iii) prove universal approximation theorems (UATs) for continuous order-equivariant maps, which are new results even when restricted to sheaf neural networks (for which no UAT was known before). We illustrate the framework on graph and sheaf models. Our results can also be seen as extending the known UAT for graph neural networks to a more general setting that subsumes sheaf neural networks as well. In addition, we show that OENN can be extended further to CENN, Category-Equivariant Neural Network, which gives the general form of equivariant neural networks as well as of equivariant universal approximation theorems, allowing us to leverage categorical symmetry in data (e.g., non-invertible symmetries on multiple objects with compositional relations on those symmetries).

This paper presents OCD SLAM, a dynamic stereo visual SLAM framework that extends ORB-SLAM2 by jointly addressing dynamic objects and dynamic features in the scene. Usual visual SLAM systems operating in dynamic environments often fail in the presence of moving objects, due to the static-world assumption used in pose estimation and mapping. To address this predicament, we introduce a novel geometric approach based on the discrepancy between disparity and a newly proposed notion called ``cross disparity'', which exploits both temporal and stereo inconsistency to identify dynamic feature points. Complementary to this feature-level motion analysis, OCD SLAM integrates a 3D object detection module (SMOKE) with Kalman filter-based object tracking to perform object-level motion classification, enabling robust separation of static and dynamic scene elements for accurate pose estimation. The proposed approach has been evaluated on various sequences from the KITTI Odometry and KITTI Raw datasets. Results demonstrate that OCD SLAM achieves significant improvement in trajectory accuracy compared to ORB-SLAM2 and several state-of-the-art dynamic SLAM methods. Ablation studies further demonstrate the effectiveness of the cross disparity module in the KITTI Raw dataset and show that this method is able to detect dynamic features that are missed by the 3D object detection scheme alone.

Real-time texture transfer expands the creative horizon for interactive applications, enabling seamless detail projection in scenarios that range from digital character cosmetics to procedural automotive texturing. Yet, its practical application is governed by inherent trade-offs between processing speed and suppression of artifacts. Low-latency transfer methods frequently fail to suppress artifacts, and robust alternatives rely on large-scale models that are costly in training and memory. Our proposed method bridges the gap between efficiency and robustness by using a cage-based geometric filtering method to identify Non-Cosmetic Zones (NCZs) for artifact suppression. While other models are resource-intensive and require multiple days of training on manually annotated datasets, we are able to successfully suppress artifacts and achieve immediate deployment on consumer-grade hardware. Our framework achieved highly efficient runtimes of ~70ms on mobile devices for a ~4.8k triangle mesh.

Deep neural networks transform input data into latent representations that support a wide range of downstream tasks. These representations can be characterized along information-theoretic and geometric dimensions, but their relationship remains poorly understood. A central open question is whether low mutual information (MI) between inputs and representations necessarily implies geometrically compressed latent spaces and vice versa. We investigate this question using class-wise clustering as a measure of geometric compression and theoretically sound MI estimation in conditional entropy bottleneck (CEB) networks and continuous dropout networks. We evaluate the interplay between MI, geometric compression, and generalization on classification tasks under controlled noise injection schemes. Our findings show that low MI does not reliably correspond to geometric compression, and that the connection between the two is more nuanced than often assumed. Indeed, our experiments reveal a negative and nonlinear relationship that can reverse when varying training setup. Our results put forward a hypothesis that generalization acts as a potential confounder in this connection rather than being their direct consequence.

The concept of geodesic-like curves was introduced by Chen in 2010 as a method for estimating shortest paths (geodesics) on parametric surfaces, with its convergence established theoretically. However, an efficient numerical computational framework has not yet been developed. In this paper, we propose an elegant and efficient approach for computing geodesic-like curves by leveraging deep learning and Physics-Informed Neural Networks (PINNs). Under the proposed framework, not only can single parametric surfaces be handled efficiently, but a broad class of complex parametric surfaces including multi-surface systems with $C^0$ or higher continuity and surfaces of revolution can also be robustly addressed.

Accurate prediction of hydrodynamic performance is central to ship design, yet high-fidelity computational fluid dynamics remains prohibitively expensive for large-scale parametric exploration. This motivates the development of data-driven surrogate models that provide rapid approximations to hydrodynamic predictions at substantially reduced cost. We present ShipNet, a geometric deep-learning surrogate that predicts both hull-surface pressure distributions and far-field free-surface wave patterns directly from hull geometry and speed. The network employs a regularized dynamic graph convolutional backbone on hull point clouds, with a multi-head decoder for simultaneous near-body pressure and free-surface elevation outputs. Training data consist of 420 inviscid free-surface simulations generated using a potential-flow panel method for two parent yacht hulls, each parameterized into 70 variants and evaluated at three speeds. ShipNet predicts per-point pressure coefficient and two-dimensional wave elevation map using a composite loss that combines point-wise regression and image-structure terms. On a geometry-held-out test set, ShipNet achieves R^2=0.98 for hull pressure and R^2=0.91 for wave fields. Inference requires approximately 0.15s per case, yielding over a 550x speedup relative to the potential-flow solver on conventional hardware. Limitations include the restricted geometry and speed ranges and the inviscid training data, while future work will extend the model to high-fidelity viscous simulations with physics-informed regularization.

Accurate prediction of protein-ligand binding affinity is essential for structure-based drug discovery. Recent geometric deep learning methods have achieved promising performance by representing protein-ligand complexes as three-dimensional graphs. However, most existing approaches mainly rely on static interaction geometry from a single bound conformation, while neglecting molecular flexibility and binding-induced conformational changes. To address this limitation, we propose a curvature-informed potential energy surface (CPES) graph neural network for protein-ligand binding affinity prediction, which incorporates physics-informed curvature representations to model conformational flexibility. CPES first derives curvature spectral descriptors from the Hessian of the potential energy surface evaluated at equilibrium configurations, whose eigenvalues define the local principal curvatures of the potential energy surface. It then uses spectral cross-attention to compare the unbound ligand and protein with the bound complex, thereby capturing binding-induced changes in conformational dynamics. In parallel, hierarchical protein-ligand interaction representations are learned from static structural features through geometry-aware message passing, soft clustering, and bidirectional cross-attention. Finally, CPES fuses the curvature-informed dynamic representations with static interaction representations for affinity regression. Extensive evaluations on multiple benchmark datasets demonstrate that CPES achieves improved predictive performance and offers physical interpretability.

This research introduces a framework for incorporating Concept Bottleneck Models (CBMs) into 3D generative architectures to address the inherent 'semantic gap' in deep geometric learning. As deep models become central to 3D content creation, explainability shifts from a peripheral feature to a fundamental requirement for trust and accountability in safety-critical domains such as healthcare and manufacturing. CBMs provide an intrinsic interpretability solution by constraining latent representations to align with human-defined concepts, yet their application to unstructured 3D data remains largely unexplored. We design, implement, and validate a formal 3D-CBM architecture that maps raw geometric inputs, including point clouds and meshes, into a multi-tiered taxonomy of interpretable primitives and functional attributes. The framework further identifies strategic datasets, such as PartNet and ShapeNet, specialized for concept-based supervision. Experimental results from a 3D part-manipulation proof-of-concept experiment demonstrate the framework's efficacy, achieving a concept prediction accuracy of 88.8% and a Chamfer Distance of 0.0115. Critically, the model enables precise test-time intervention, allowing for the interactive correction of structural errors. This work establishes a foundation for semantically-steerable 3D generation and invites further exploration into collaborative human-in-the-loop design systems.

Finite element analysis (FEA) is essential for structural design but remains computationally expensive, particularly when evaluating multiple design iterations or load scenarios. Machine learning surrogate models offer a promising alternative, yet most approaches struggle with a critical limitation: generalizing across varying geometries. This work presents a mesh graph network (MGN) for predicting von Mises stress fields in 2D structural components with arbitrary hole geometries. Unlike traditional machine learning approaches that use absolute node coordinates as features, the proposed model builds on existing MGN frameworks that encode node types (e.g., fixed boundary, free surface, hole edge), relative edge features (distance between neighbors), and global features (applied load). This architecture is inherently translation- and rotation-invariant, enabling generalization to unseen geometries without retraining. The MGN was trained on 11 plate geometries under 20 load conditions and evaluated on 7 unseen geometries and 3 unseen loads. In the most favorable case, the model achieves $R^2 \geq 0.97$ on an unseen geometry and unseen load, compared to $R^2 \approx 0.01$--$0.86$ for conventional models (Random Forest, Gradient Boosting , K-Nearest Neighbors) trained on identical data. However, even in less favorable cases, the MGN model still outperforms conventional models. This work extends the mesh-based simulation framework of Pfaff et al. (arXiv:2010.03409) to structural mechanics, demonstrating that graph neural networks can serve as efficient surrogates for finite element analysis across varying geometries.

High-fidelity computational fluid dynamics (CFD) is crucial to vehicle aerodynamic analysis, but its cost still constrains early-stage design exploration. Machine-learning-based surface-field prediction offers a faster alternative if the model can efficiently capture both global flow context and local geometric detail. This work proposes a machine-learning-based method, named the geometry-aware triplane field network (GTF-Net), for vehicle aerodynamic pressure and wall shear stress prediction. GTF-Net constructs triplane features directly from sampled surface points through a shared multilayer perceptron (MLP) and smooth bilinear rasterization. The planes are then processed by a dual-stream backbone that combines adaptive Fourier neural operator (AFNO) spectral mixing with convolutional neural network (CNN) refinement, so long-range aerodynamic coupling and local geometry-induced variations are modeled in the same representation. At query stage, sampled triplane features are combined with vehicle-aligned directional coordinates, normal-projection features, and a voxel-based curvature proxy. GTF-Net is compared with Transolver, geometry-informed neural operator (GINO), and TripNet, a triplane-based surrogate model. GTF-Net improves the relative L2 error from the strongest baseline value of 0.157 to 0.145 for pressure prediction and from 0.237 to 0.226 for wall shear stress prediction. Ablation results show that AFNO mixing, local CNN refinement, and query-side geometric encoding each contribute to accuracy, supporting the proposed mechanism of combining structured triplane representation with explicit aerodynamic geometry cues.

3D Gaussian Splatting (3DGS) is a recent approach for scene rendering. Although primarily designed for view synthesis, its potential for scene understanding tasks remains underexplored. In this work, we conduct a comparative evaluation of various geometric deep learning architectures for the classification of 3D scenes represented using Gaussian Splatting. We benchmark point-based and graph-based models across both traditional point cloud datasets and dedicated Gaussian Splatting datasets. Scenes are embedded into latent representations, which are evaluated through end-to-end classification, linear probing, and clustering analysis. Our study provides insight into the suitability of different geometry-aware architectures and input feature configurations for learning effective 3D Gaussian Splat representations. The results highlight consistent differences between architectural families and reveal the impact of Gaussian-specific attributes on the quality of representation.

Geometric deep learning organises neural architectures around the symmetries of their data domain, with the choice of symmetry group serving as a geometric prior that determines what representations can be learned. Metric-Aware Principal Component Analysis (MAPCA) parameterises principal component analysis by a positive-definite metric matrix, with a canonical subfamily interpolating between standard PCA and output whitening and a diagonal-metric point recovering Invariant PCA (IPCA). This paper positions MAPCA within the geometric deep learning framework. The metric is read as the geometric prior; the orthogonal group preserving it is the symmetry group it induces; MAPCA solutions are equivariant under this group with the resulting spectrum invariant; and MAPCA's defining constraint is the linear analogue of the Schur-type weight constraints used in equivariant networks. Across six axes - domain, symmetry group, equivariance, invariance, architectural primitive, and geometric prior - we construct a precise dictionary between MAPCA and geometric deep learning. The technical anchor is a uniqueness theorem characterising IPCA as the unique linear data-derived metric in the MAPCA family that is equivariant under arbitrary diagonal rescaling and projects onto the fixed-point set of the action, equivalent under normalisation to the variance-maximisation criterion in its precise form. The paper closes with three bridges: kernel PCA as the nonlinear extension, spectral graph methods as MAPCA on graphs, and a deep MAPCA construction extending the positioning into deep equivariant networks

Nucleic acids are increasingly recognized as therapeutic targets beyond conventional protein-centered drug discovery, yet accurate and efficient docking of small molecules to nucleic acid structures remains challenging. Physics-based docking methods often show limited accuracy and efficiency, whereas deep learning approaches are constrained by the scarcity of experimentally resolved nucleic acid-ligand complexes. Here, we present NucleoDock, a deep learning framework for nucleic acid-small molecule docking. To address data scarcity, NucleoDock combines physics-guided large-scale pretraining on millions of docking-generated synthetic complexes with fine-tuning on curated experimental co-crystal structures. It further integrates sequence- and structure-informed nucleotide representations with atomistic three-dimensional features to capture both biological context and binding-site geometry. A mixture density network-based geometric scoring head is used to model conditional interaction-distance distributions for pose ranking. On an external benchmark of 125 nucleic acid-ligand complexes, NucleoDock achieved a top-1 success rate of 56 percent at an RMSD cutoff of 2.0 Angstrom, outperforming rDock with 29 percent, while generating 100 poses in approximately 5 seconds per complex. Retrospective virtual screening on the ROBIN benchmark further showed improved early enrichment. NucleoDock represents a step toward bridging the methodological gap between protein- and nucleic acid-directed computational drug discovery.

Foundational Models pretrained on huge amount of data learn representations that evolve across depth, forming a hierarchy of embeddings with distinct semantic content and geometric structure. Contrary to the widespread practice of using only the final layer or shallow mixtures, we show that task-relevant information is distributed non-monotonically across layers and cannot be recovered by naïve aggregation. Through a geometric and empirical study across multiple modalities, we show that effective transfer depends on identifying which layers encode task-discriminative structure and how their embeddings are geometrically organized. We introduce Layer-wise Optimal Embedding Selection (LOES), a constructive spectral method that identifies task-discriminative subspaces by minimizing residual error under orthogonality and isotropy constraints. To align fine-tuning with this selection principle, we further propose Geometric Regularization Loss (GeoReg), which enforces a simplicial structure on class manifolds and stabilizes representation geometry during fine-tuning. Across a wide range of architectures, depths, modalities, and data regimes, LOES consistently outperforms standard baselines, with gains that grow as model depth increases. Beyond accuracy, our method reveals how semantic factors are distributed across layers, thereby enabling cross-lingual and cross-modal interpretability analyses. Together, our results provide strong evidence that layerwise embedding geometry is not incidental but central to how deep models represent and transfer knowledge.

While deep neural networks have achieved remarkable success across a wide range of domains, their underlying mechanisms remain poorly understood, and they are often regarded as black boxes. This gap between empirical performance and theoretical understanding poses a challenge analogous to the pre-axiomatic stage of classical geometry. In this work, we introduce the Pursuit of Subspaces (PoS) hypothesis, an axiomatic framework that formulates neural network behavior through a set of geometric postulates. These axioms, together with their derived consequences, provide a unified perspective on representation, computation, and generalization in both shallow and deep architectures. We show that this framework yields geometric explanations for fundamental questions in deep learning, including representation structure, architectural mechanisms, and generalization behavior, offering a principled step toward a coherent theoretical foundation.

Predicting protein-ligand binding affinity remains intractable for multi-domain proteins, where inter-domain dynamics govern molecular recognition. Existing geometric deep learning methods typically treat proteins as monolithic static graphs, suffering from rigid-body assumptions and aleatoric noise in flexible regions. To address this, we introduced HCLBind, a self-supervised framework that decouples geometric representation learning from affinity regression. HCLBind leverages a general-to-specific pre-training paradigm on the Q-BioLiP database to learn a robust physical grammar of binding. We propose a novel hierarchical decoy strategy: the model learns local physicochemical constraints through protein coordinate perturbation in single-domain proteins and global conformational geometry through inter-domain rotation in multi-domain complexes. Our hybrid architecture integrates a domain-gated graph attention network and cross-modal attention to explicitly prioritize domain interfaces. Furthermore, we employ LoRA on protein and ligand foundation models, ensuring efficient optimization while preserving evolutionary knowledge. Experiments on PDBBind demonstrate that HCLBind effectively learns discriminative interface features and provides robust uncertainty estimation, overcoming the limitations of standard supervised learning. The code is available at https://github.com/jiankliu/HCLBind.

Group-convolutional neural networks (GCNNs) are among the most important methods for introducing symmetry as an inductive bias in deep learning: In each linear layer, GCNNs sample a transformation group $G$ densely and correlate data and filters in different poses (with suitable anti-aliasing for steerable GCNNs) to maintain equivariance with respect to $G$. Unfortunately, applying filters to many data items resulting from this sampling is expensive (even for translations alone, i.e., in ordinary CNNs), and costs grow exponentially with increasing degrees of freedom (such as translations and rotations in 3D), which often hinders practical applications. In this paper, we propose sampling in feature space, i.e., replacing geometrically dense samples with representative samples selected by feature similarity. This decouples geometric resolution from memory and processing costs during training and inference, providing a novel way to trade off computational effort and accuracy. Our main empirical finding is that a coarse feature-space sampling already preserves classification accuracy remarkably well, which permits precomputation based on geometric similarity, accelerating the training of equivariant 3D classifiers substantially.

Modern deep learning treats neural networks primarily as endpoint functions from inputs to outputs. Inspired by the shift from force to geometry in physics, we ask whether a network should instead be understood through the geometry of its internal propagation. We define a neural propagation field as the collection of hidden-state trajectories and local Jacobian operators across depth. Endpoint losses constrain only the boundary behavior of this field, leaving its interior geometry underdetermined. We show that endpoint-equivalent models can differ by orders of magnitude in trajectory and Jacobian structure, and introduce observable field metrics such as path sensitivity, solver consistency, and trajectory/Jacobian retention. In controlled teacher-flow and PDE systems, endpoint fitting fails to recover the underlying propagation law. In real multi-path tasks, field-aware objectives improve unseen-path generalization, OOD robustness, and calibration when aligned with the observation structure, but can collapse when over-constrained. In continual learning, field-preservation regularization complements replay and distillation: on Split CIFAR-100, DER++ with field preservation improves average accuracy, backward transfer, and field-retention metrics. These results identify propagation-field quality as a measurable and trainable property of neural networks beyond endpoint performance.

Modern deep learning architectures increasingly contend with sophisticated signals that are natively infinite-dimensional, such as time series, probability distributions, or operators, and are defined over irregular domains. Yet, a unified learning theory for these settings has been lacking. To start addressing this gap, we introduce a novel convolutional learning framework for possibly infinite-dimensional signals supported on a manifold. Namely, we use the connection Laplacian associated with a Hilbert bundle as a convolutional operator, and we derive filters and neural networks, dubbed as \textit{HilbNets}. We make HilbNets and, more generally, the convolution operation, implementable via a two-stage sampling procedure. First, we show that sampling the manifold induces a Hilbert Cellular Sheaf, a generalized graph structure with Hilbert feature spaces and edge-wise coupling rules, and we prove that its sheaf Laplacian converges in probability to the underlying connection Laplacian as the sampling density increases. Notably, this result is a generalization to the infinite-dimensional bundle setting of the Belkin & Niyogi \cite{BELKIN20081289} convergence result for the graph Laplacian to the manifold Laplacian, a theoretical cornerstone of geometric learning methods. Second, we discretize the signals and prove that the discretized (implementable) HilbNets converge to the underlying continuous architectures and are transferable across different samplings of the same bundle, providing consistency for learning. Finally, we validate our framework on synthetic and real-world tasks. Overall, our results broaden the scope of geometric learning as a whole by lifting classical Laplacian-based frameworks to settings where the signal at each point lives in its own Hilbert space.

Neural operators perform well on structured domains, yet their behaviour on irregular geometries remains poorly understood. We show that this limitation is not merely an encoding issue, but a depth-wise failure mode inherent to deep operator architectures. We formalise the Geometric Forgetting Hypothesis: due to the Markovian structure of operator layers and their reliance on global mixing mechanisms, neural operators progressively lose access to domain geometry as depth increases. Using layer-wise geometric probing, we demonstrate that both spectral and attention-based operators systematically lose geometric fidelity. We show that this geometric forgetting degrades accuracy, stability, and generalisation. To counteract it, we introduce a lightweight geometry memory injection mechanism that restores geometric constraints at intermediate depths with minimal architectural overhead. This simple intervention consistently mitigates forgetting and exposes a geometric shortcut instability in transformer-based operators, revealing that geometric retention is a structural requirement rather than a design choice.

Machine Learning surrogates for Computational Fluid Dynamics (CFD), particularly Graph Neural Networks (GNNs) and Transformers, have become a new important approach for accelerating physics simulations. However, we identify a critical bottleneck in the field: while architectures have advanced significantly, the common underlying training paradigms remain bound to naive assumptions, such as node-wise supervision and explicit Euler time-stepping. These legacy choices ignore the stiff dynamics and local flux continuity inherent to numerous partial differential equations resolution methods, such as Finite Element, Difference, or Volume (FEM). In this work, we propose a unified framework to bridge the gap between geometric deep learning and rigorous numerical analysis. We introduce three key innovations: (1) Multi Node Prediction, a stencil-level objective that predicts field values for a node's full local topology, enforcing spatial derivative consistency; (2) Temporal Correction, replacing unstable explicit schemes with a predictor-corrector via temporal Cross-Attention; and (3) Geometric Inductive Biases, leveraging 3D Rotary Positional Embeddings (RoPE) to robustly capture rotational symmetries in unstructured meshes. We evaluate this framework across three architectures (MeshGraphNet, Transolver, and a Transformer) on diverse physics datasets. Our approach yields consistent improvements in accuracy and stability, particularly in long-horizon rollouts, while producing latent representations that generalize to unseen subtasks such as Wall Shear Stress or Pressure prediction. Code is available at https://github.com/DonsetPG/graph-physics.

Safety-critical applications are required to perform as expected in normal operations. Image processing functions are often required to be insensitive to small geometric perturbations such as rotation, scaling, shearing or translation. This paper addresses the formal verification of neural networks against geometric perturbations on their image dataset. Our method Super-DeepG improves the reasoning used in linear relaxation techniques and Lipschitz optimization, and provides an implementation that leverages GPU hardware. By doing so, Super-DeepG achieves both precision and computational efficiency of robustness certification, to an extent that outperforms prior work. Super-DeepG is shared as an open-source tool on GitHub.

Monocular depth estimation (MDE) has witnessed remarkable progress driven by Convolutional Neural Networks and transformer-based architectures. However, these approaches typically treat the problem as a generic image-to-image regression on Euclidean grids, thereby overlooking the intrinsic algebraic and geometric structures induced by perspective projection. To address this limitation, we propose LAGRNet, a novel framework that fundamentally grounds MDE in algebraic geometry by explicitly embedding learnable group, ring, and sheaf structures into the deep learning pipeline. Modeling feature maps as sections of a sheaf over an approximated image manifold, our method first establishes a Group-defined Feature Manifold (GFM) parameterized by a learned algebraic group action to enforce projective equivariance and robustness against view changes. To facilitate algebraically consistent cross-scale interactions, we subsequently introduce a Ring Convolution Layer (RCL) that formulates feature fusion as a graded ring homomorphism. Furthermore, to ensure global topological consistency, a Sheaf-based Module (SM) aggregates local depth cues via Čech nerve on the image topology. Extensive zero-shot evaluations across the KITTI, NYU-Depth V2, and ETH3D benchmarks demonstrate that LAGRNet significantly outperforms state-of-the-art methods in both accuracy and generalization capabilities.

Depth is widely viewed as a central contributor to the success of deep neural networks, whereas standard neural network approximation theory typically provides guarantees only for the final output and leaves the role of intermediate layers largely unclear. We address this gap by developing a quantitative framework in which depth admits a precise scale-dependent interpretation. Specifically, we design a single shared mixed-activation architecture of fixed width $2dN+d+2$ and any prescribed finite depth such that each intermediate readout $Φ_\ell$ is itself an approximant to the target function $f$. For $f\in L^p([0,1]^d)$ with $p\in [1,\infty)$, the approximation error of $Φ_\ell$ is controlled by $(2d+1)$ times the $L^p$ modulus of continuity at the geometric scale $N^{-\ell}$ for all $\ell$. The estimate reduces to the geometric rate $(2d+1)N^{-\ell}$ if $f$ is $1$-Lipschitz. Our network design is inspired by multigrade deep learning, where depth serves as a progressive refinement mechanism. For every prescribed terminal depth, the construction yields a finite nested family of prefix readouts whose earlier correction terms remain embedded in later readouts. Thus the approximation may be truncated within the prescribed depth range once the desired certified accuracy is reached.

Symmetric objects are common in daily life and industry, yet their inherent orientation ambiguities that impede the training of deep learning networks for pose estimation are rarely discussed in the literature. To cope with these ambiguities, existing solutions typically require the design of specific loss functions and network architectures or resort to symmetry-invariant evaluation metrics. In contrast, we focus on the numeric representation of the rotation itself, modifying trigonometric identities with the degrees of symmetry derived from the objects' shapes. We use our representation, SARR, to obtain canonic (symmetry-resolved) poses for the symmetric objects in two popular 6D pose estimation datasets, T-LESS and ITODD, where SARR is unique and continuous w.r.t. the visual appearance. This allows us to use a standard CNN for 3D orientation estimation whose performance is evaluated with the symmetry-sensitive cosine distance $\text{AR}_{\text{C}}$. Our networks outperform the state of the art using $\text{AR}_{\text{C}}$ and achieve satisfactory performance when using conventional symmetry-invariant measures. Our method does not require any 3D models but only depth, or, as part of an additional experiment, texture-less RGB/grayscale images as input. We also show that networks trained on SARR outperform the same networks trained on rotation matrices, Euler angles, quaternions, standard trigonometrics or the recently popular 6d representation -- even in inference scenarios where no prior knowledge of the objects' symmetry properties is available. Code and a visualization toolkit are available at https://github.com/akriegler/SARR .

In many real-world applications of regression, conditional probability estimation, and uncertainty quantification, exploiting symmetries rooted in physics or geometry can dramatically improve generalization and sample efficiency. While geometric deep learning has made empirical advances by incorporating symmetry and geometry priors, less attention has been given to statistical learning guarantees. In this paper, we introduce an equivariant representation learning framework that simultaneously addresses regression, conditional probability estimation, and uncertainty quantification while providing first-of-its-kind non-asymptotic statistical learning guarantees. Grounded in operator and group representation theory, our framework approximates the spectral decomposition of the conditional expectation operator, building representations that are both equivariant and disentangled along independent symmetry quotient groups. Empirical evaluations on synthetic datasets and real-world robotics applications confirm the potential of our approach, matching or outperforming existing equivariant baselines in regression while providing well-calibrated uncertainty estimates.

The geometric morphology of deposited filaments can significantly influence the structural performance and stability of 3D concrete-printed (3DCP) structures. However, most finite element (FEM)-based approaches for buildability assessment represent printed layers as simplified rectangles, potentially limiting predictive accuracy. This study proposes a geometry-informed modelling framework that integrates the deep-learning-based filament shape prediction tool ShapeGen3DCP with a layer-activation FEM approach to investigate the effect of realistic filament geometries on buildability. The framework generates geometry-aware numerical models directly from material and process parameters, eliminating the need for experimental filament characterization or computationally intensive fluid-flow simulations. Validation against experimental data and a parametric study of rectilinear walls demonstrate that extrusion parameters and the resulting filament geometry can significantly influence buildability predictions. Realistic filament representations are particularly important for free-flow deposition, whereas layer-pressing strategies are less sensitive to geometric simplifications. Among the investigated representations, an elliptical approximation provides an effective balance between geometric fidelity and modelling simplicity. When rectangular representations are preferred to enable regular computational meshes for faster simulations, defining their dimensions based on volume conservation improves prediction reliability compared with calibrating them using either the maximum filament width or the interlayer contact width. Overall, the proposed methodology demonstrates the importance of incorporating filament geometry into 3DCP simulations and provides practical guidance for selecting efficient and accurate geometric representations for buildability assessment.