Mesh-Invariant Neural Network

• Mesh-Invariant Neural Networks are deep learning models that decouple intrinsic shape properties from varying mesh triangulations and connectivities.
• They employ techniques such as graph-based embeddings, spectral projections, and autoencoder pretraining to achieve topology-agnostic performance.
• These networks enhance reliability in physics simulation, segmentation, and generative modeling by ensuring consistent outputs across diverse mesh representations.

A mesh-invariant neural network is a class of deep learning architecture designed to process, analyze, or simulate mesh-based geometric data while being insensitive to variations in mesh topology, connectivity, parameterization, or sampling. This property is essential for robust physics simulation, geometric learning, segmentation, generative modeling, and registration, where the same underlying shape can be instantiated through diverse mesh representations. Mesh-invariant methodologies rigorously decouple the notion of shape or physical property from the specifics of a mesh’s triangulation, vertex count, and edge structure, thereby offering reliability and generalization in downstream tasks on arbitrary or “wild” meshes (Vaska et al., 16 Jan 2025).

1. Problem Formulation and Motivation

Most high-fidelity physics simulators (e.g., in radar, aerodynamics, optical rendering) represent objects as triangular meshes $M=(V_M, F_M)$. Neural simulators $g_\theta$ attempt to predict physical response $\hat{R}$ given $M$, targeting agreement with ground truth $R$ (Vaska et al., 16 Jan 2025). However, mesh-invariance is a critical challenge: topologically equivalent meshes—representing the same object shape but with different triangle counts, connectivity, or face arrangements—often lead neural networks to produce widely different outputs, undermining the reliability of learned surrogates.

Mesh-invariant neural networks address this challenge by ensuring that the learned mapping $M \mapsto \hat{R}$ or shape embedding does not depend on how the mesh is triangulated, subdivided, or sampled. This property is quantified by evaluating metrics such as “Variation MSE”:

$$\mathrm{Variation\ MSE} = \frac{1}{|C|} \sum_{c \in C} \frac{1}{n} \sum_{i=1}^n (\hat{R}_{s,i} - \hat{R}_{c,i})^2$$

where $C$ indexes a set of complex mesh variants and $s$ is a reference simple topology. High Variation MSE indicates poor mesh-invariance (Vaska et al., 16 Jan 2025).

2. Architectures and Mathematical Strategies

Mesh-invariant models employ several foundational strategies to decouple learning from mesh topology:

• Graph-Based Face or Node Embedding: Mesh elements (faces, edges, vertices) are encoded using message passing or graph convolution on the mesh adjacency graph $G=(F_M, E)$, refining local geometric features:

$$h_v^{(k+1)} = \sigma \left( W_1 h_v^{(k)} + \sum_{u \in \mathcal{N}(v)} W_2 h_u^{(k)} + b \right)$$

These embeddings are pooled via set encoders or transformers to form global shape codes (Vaska et al., 16 Jan 2025).

• Autoencoder Pretraining: Mesh-invariant encoders are pretrained on large auxiliary corpora (e.g., ShapeNet) to reconstruct local geometric features (positions, normals) from embeddings, enforcing invariance to mesh connectivity via:

$$\mathcal L_{AE} = \frac{1}{N} \sum_{v=1}^N \|x_v - \hat{x}_v\|^2 + \lambda \sum_{v=1}^N \|z_v\|^2$$

This initialization substantially reduces topology-induced variance in downstream simulation (Vaska et al., 16 Jan 2025).

• Operator-intrinsic Projections and Poisson Solves: Neural Jacobian Fields (NJF) (Aigerman et al., 2022) predict a continuous field of matrix-valued deformations across shape, which are then intrinsically projected to the mesh tangent spaces and solved via a mesh-specific but non-learned Poisson equation. The neural network is blind to connectivity, embedding only local geometric information, yielding provably mesh-invariant mappings.
• Spectral Embedding: The Laplacian2Mesh framework (Dong et al., 2022) projects mesh signals into the Laplace–Beltrami eigenbasis. Classification, segmentation, and convolutional operations are performed on spectral coefficients, not directly on mesh vertices or edges. This makes all learned functions independent of triangulation, robust to mesh irregularities, and insensitive to noise or defects.
• Edge-centric Fundamental Form Encodings: MeshCNN Fundamentals (Barda et al., 2021) utilize the first and second fundamental forms—edge length and dihedral angle—guaranteeing rigid-motion invariance and reconstructability up to global rotation/translation.
• Caging and Barycentric Feature Transfer: CageNet (Edelstein et al., 24 May 2025) envelops any wild, non-manifold mesh within a single-component manifold “cage.” All features and learning occur on the cage. Per-vertex outputs are projected back to the original mesh using generalized barycentric coordinates, decoupling the network from mesh defects.
• Geometric Measure Matching Losses: Mesh-invariant generative models (Besnier et al., 2023) compare shapes using kernel-based metrics over currents or varifolds (measures in $\mathbb{R}^3 \times S^2$), eliminating dependence on mesh correspondence or parameterization. Loss functions based on varifold norms are provably robust to resampling and reparameterization.

3. Empirical Results and Performance

Mesh-invariant neural networks consistently outperform topology-sensitive baselines across multiple domains:

Architecture	Task/Dataset	Mesh-Invariant Property	Performance
Graph + autoencoder (Vaska et al., 16 Jan 2025)	Physics sim (Basic Shapes)	Topology equivariance (↓42% Var-MSE)	Simple MSE=58.2; Complex MSE=164.1; Var-MSE=125.6
NJF (Aigerman et al., 2022)	Morphing, re-posing	Triangulation agnostic & detail-preserving	$L_2$ pos error $2 \times 10^{-2}$, normal error $4.4^\circ$
Laplacian2Mesh (Dong et al., 2022)	Classification (SHREC)	Connectivity/irregularity invariance	100% accuracy, robust to mesh noise
MeshCNN Fundamentals (Barda et al., 2021)	Classification, segmentation	Rigid-motion & reconstructability	91–100% accuracy, denoising MSE=0.0096
CageNet (Edelstein et al., 24 May 2025)	Segmentation, skinning	Invariance to defects, multi-component, non-manifold meshes	$\approx$91.7% accuracy, best L$_1$ error 0.124
Geometric measure AE (Besnier et al., 2023)	Generative modeling (faces)	Mesh resampling invariance	Chamfer $\approx$0.088, varifold $\approx$0.011

These results demonstrate both quantitative and qualitative reliability under mesh variation. For example, CageNet achieves identical segmentation accuracy on clean and artificially “broken” meshes, while standard networks suffer catastrophic failure (accuracy drops to $\sim$50%) (Edelstein et al., 24 May 2025). NJF achieves detail-preserving mappings regardless of triangulation, with applications to UV parameterization and deformation transfer (Aigerman et al., 2022). Laplacian2Mesh offers top-tier accuracy and robustness to Gaussian noise and defects without retriangulation (Dong et al., 2022). Geometric-measure autoencoders remain accurate under mesh subdivision and parameterization changes, with difference in output $\leq$2% in Chamfer distance (Besnier et al., 2023).

4. Theoretical Guarantees and Invariance Principles

Most mesh-invariant designs employ hardwired invariance principles rooted in geometry and topology:

• Subdivision invariance via topological statistics (Euler characteristic curves) and translation/O(3)-invariant pooling (Paik, 2023).
• Isometry invariance by decoupling features from absolute Euclidean position and encoding only relational or intrinsic shape descriptors (Barda et al., 2021, Dong et al., 2022).
• Permutation invariance through message aggregation (sum or mean), ensuring predictions are independent of mesh vertex/face ordering (Liu et al., 22 Sep 2025).
• Resampling invariance by converting meshes to geometric measures (currents, varifolds), with theoretical bounds on RKHS loss variation under mesh refinement (Besnier et al., 2023).

No ad-hoc or regularization-based invariance enforcement is required; these properties hold by construction. For example, the mapping of Euler curve statistics through translation-invariant neural modules and O(3)-equivariant GNNs gives subdivision- and rotation-invariance (Paik, 2023).

5. Applications and Impact

Mesh-invariant neural networks have accelerated progress in several mesh-based learning and simulation domains:

• Physics Simulation: Surrogates that generalize to unseen mesh topologies (e.g., radar, elastic plates) provide stable long-term predictive accuracy and substantial speedup over conventional solvers (Liu et al., 22 Sep 2025, Vaska et al., 16 Jan 2025).
• Shape Understanding: Shape classification, segmentation, morphing, and UV parameterization tasks benefit from invariance to mesh quality, orientation, and sampling (Dong et al., 2022, Barda et al., 2021).
• Generative Modeling and Registration: Generative autoencoders and latent-space manipulators achieve mesh-independence during training and prediction, facilitating shape synthesis, interpolation, and expression transfer (Besnier et al., 2023).
• Robust Feature Transfer: CageNet enables learning on data with disrupted connectivity, multi-component structure, or severe defects, generalizing across a diversity of real-world mesh datasets (Edelstein et al., 24 May 2025).
• Topological Learning: Sufficient statistics built from mesh topology guarantee subdivision-invariant representations and robust clustering/classification even with minimal training data and under arbitrary isometries (Paik, 2023).

6. Methodological Innovations and Future Directions

Key advances in mesh-invariant modeling include:

• Scaling autoencoder pretraining to corpora of millions of meshes, as successful in vision/NLP, to further strengthen topology-insensitive feature extraction (Vaska et al., 16 Jan 2025).
• Introducing contrastive or consistency terms at the loss level to explicitly enforce alignment of embeddings from topologically distinct but shape-equivalent meshes (Vaska et al., 16 Jan 2025).
• Developing specialized GNN layers with explicit equivariance to rotations/reflections, advancing invariance guarantees (Vaska et al., 16 Jan 2025).
• Hybridizing spectral and spatial graph convolutions to decouple shape from connectivity (Vaska et al., 16 Jan 2025).
• Employing multi-scale varifold losses to match shapes across diverse mesh resolutions (Besnier et al., 2023).
• Combining topological and geometric statistics in mesh representations for subdivision-robust embeddings (Paik, 2023).

A plausible implication is that further integrating mesh-invariant design into large-scale geometric learning pipelines will yield next-generation surrogates and generative models, with robust generalization to real-world, “wild” mesh data.