geometry-aware mesh gnn

This survey addresses key challenges in 3D vision—occlusion robustness, point cloud sparsity, density imbalance, and high-dimensional computational bottlenecks—across four core tasks: 3D generation, point cloud reconstruction, shape completion, and scene synthesis. Methodologically, it introduces the first unified taxonomy capturing paradigm evolution, integrating denoising diffusion probabilistic models (DDPMs), 3D conditional encoders, multi-view feature alignment, implicit neural representations (INRs), and multimodal (text/image) guidance. The work rigorously delineates current performance limits and standardizes evaluation benchmarks. Crucially, it identifies three viable technical pathways forward: efficient sampling strategies, lightweight backward processes, and large-scale 3D pretraining. These contributions provide both theoretical foundations and practical guidelines for advancing diffusion-based 3D modeling.

Modeling graph-structured data residing on Riemannian manifolds poses challenges in preserving intrinsic geometric structure while ensuring equivariance under both node permutations and manifold isometries. Method: We propose the first equivariant graph neural network layer that jointly incorporates manifold diffusion modeling and nonlinear equivariant mapping in tangent spaces. The layer defines graph convolution via the manifold diffusion equation and constructs an equivariant multilayer perceptron in the tangent space at each node, enabling native support for arbitrary graph topologies and sizes. Contribution/Results: Our layer rigorously satisfies equivariance under node permutations and Riemannian isometries, and uniformly accommodates diverse Riemannian manifolds—including spheres, hyperbolic spaces, and triangulated surfaces—by embedding strong geometric inductive biases. Experiments on synthetic manifold graph datasets and a real-world Alzheimer’s disease classification task using right hippocampal triangular meshes demonstrate performance competitive with or superior to state-of-the-art specialized methods, alongside significantly improved generalization.

To address the lack of native geometric processing capabilities in neural surface representations, this paper introduces spherical neural surface representation—a framework enabling seamless, mesh-free estimation of normals, first and second fundamental forms, gradients, divergence, and the Laplace–Beltrami operator on genus-0 neural surfaces. Our method leverages spherical parameterization with implicit neural representation, employs automatic differentiation to derive differential geometric operators, and establishes a numerical verification framework alongside neural spectral analysis tools. Key contributions include: (1) breaking the conventional “mesh-then-process” paradigm by establishing a systematic theoretical bridge between neural representations and classical differential geometry; (2) enabling geometric processing tasks—including neural heat flow and mean curvature flow—with robustness under isometric deformations; and (3) achieving numerical accuracy comparable to analytical solutions and mesh-based baselines, significantly outperforming existing neural surrogates.

Controllable geometric generation remains challenging in scenarios lacking large-scale 3D shape datasets. Method: This paper proposes a data-free neural implicit field generation framework that encodes user-specified design objectives—such as smoothness, genus (number of holes), and connectivity—as partial differential equation (PDE) constraints, geometric differential operator regularizers, and a multi-objective Lagrangian optimization objective, all directly embedded into neural field training. Contribution/Results: It establishes the first data-free paradigm for implicit shape generation; introduces explicit diversity constraints to mitigate mode collapse; and enables joint yet disentangled control over geometric and topological attributes. Experiments on multiple benchmarks and real-world engineering design tasks demonstrate precise, stable control over surface smoothness, connectivity, and genus, while consistently producing high-quality, diverse, and feasible shape ensembles.

Existing methods for generating manifold meshes typically rely on indirect representations—such as level sets or template deformations—making it difficult to directly produce high-quality, topologically unconstrained polygonal meshes with structural integrity. This paper introduces the first end-to-end differentiable framework that explicitly models half-edge structure via vertex-level continuous connectivity embeddings, enabling direct generation of discrete manifold-conforming meshes in a continuous latent space. Key contributions include: (1) the first continuous neighborhood relation learning mechanism; (2) mesh distribution fitting via stochastic optimization; and (3) topology-agnostic generation and repair capabilities. Evaluated on large-scale datasets, our method significantly improves mesh element quality, geometric fidelity, and topological diversity. It establishes the first truly end-to-end differentiable approach for manifold mesh generation and repair, bridging a critical gap between implicit representation learning and explicit, valid mesh synthesis.

To address the under-smoothing and over-smoothing issues inherent in Graph Convolutional Networks (GCNs) for semi-supervised learning on sparsely labeled graphs, this paper proposes GND-Nets: a single-layer graph neural network architecture. Its core innovation lies in introducing a learnable neural diffusion mechanism—embedding neural modules into linear or nonlinear graph diffusion processes to jointly model local neighborhood structures and global topological information. By integrating differentiable graph propagation with localized and global neighborhood aggregation, GND-Nets achieves a favorable trade-off between expressive power and training stability. Extensive experiments on multiple sparsely labeled graph benchmarks demonstrate that GND-Nets significantly outperforms state-of-the-art methods in node classification accuracy while exhibiting faster convergence.

This work proposes a smooth geometric framework based on diffusion Markov operators to stably characterize the geometric structure of intermediate representations in feedforward neural networks, enabling a unified analysis of their separability, contraction properties, and generalization capacity. By leveraging a Gaussian kernel-induced diffusion process together with Bakry–Émery Γ-calculus, the framework constructs continuous and perturbation-smooth observables—such as transport distances, spectral characteristics, class boundaries, and local scales—thereby overcoming the discontinuities inherent in traditional neighborhood graph approaches. Under Gaussian class-conditional assumptions, the authors derive a closed-form solution for Gaussian bridges and validate the framework on MNIST, demonstrating its effectiveness in tracking training dynamics, the impact of network width, and robustness to input perturbations.

This work addresses the limitations of conventional graph neural networks, which rely on isotropic graph Laplacians and struggle to capture complex geometric structures such as those arising from nonlinear diffusion. To overcome this, the paper introduces Finsler geometry into graph neural networks for the first time, proposing a novel graph convolutional layer whose discrete formulation provably converges to the true Finsler Laplacian on manifolds. This enables effective modeling of the underlying nonlinear geometry. By integrating point cloud sampling, manifold learning, and nonlinear operator estimation, the method successfully reconstructs the geometric structures implicit in nonlinear diffusion equations, demonstrating both the effectiveness and expressive power of the proposed Finsler graph neural network.

This study addresses the challenge of evaluating cross-layer and cross-network similarity of internal neural representations across multiple scales. The authors propose a novel framework that integrates diffusion geometry with multi-view learning: by modeling the data manifold via a Markov transition matrix, they leverage its powers to construct multi-scale variants of Centered Kernel Alignment (CKA) and distance correlation. Furthermore, alternating diffusion is introduced to fuse information across layers, enabling a paradigm shift from local inter-layer comparisons to global inter-network assessments. Evaluated on the ReSi benchmark—spanning 14 architectures, 7 datasets, and 3 domains—the method achieves state-of-the-art performance in representation similarity and out-of-distribution generalization across both language and vision tasks.

Existing deep learning models struggle to effectively encode spatial, topological, and semantic structural information inherent in images. This work systematically evaluates the impact of various visual graph construction strategies on image classification performance within a unified three-layer Graph Convolutional Network (GCN) framework. For the first time, it demonstrates that the graph structure itself plays a decisive role in model performance. The study underscores the critical importance of the graph construction preprocessing stage, providing empirical evidence that well-designed graph structures substantially enhance classification accuracy. These findings offer both methodological guidance and practical justification for graph structure selection and preprocessing in visual graph neural networks.

This work addresses the limitations of traditional mesh generation methods, which rely on sequential or autoregressive strategies and suffer from low inference efficiency and error accumulation. The authors propose an end-to-end diffusion model framework that decouples vertex and topology generation to produce high-quality, globally consistent triangular meshes. Vertices are innovatively represented as sparse voxels organized in an octree structure, and a Spacetime Interval encoding is introduced to map arbitrary edge-face topologies into continuous vertex embeddings, enabling efficient global topology recovery. Employing a coarse-to-fine strategy for vertex generation and a separate diffusion model for topology prediction, the method significantly outperforms existing autoregressive and two-stage approaches on the Objaverse and Toys4K datasets as well as on real-world images, with user studies confirming its superior perceptual quality.