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PoissonNet: A Local-Global Approach for Learning on Surfaces

Arman Maesumi, Tanish Makadia, Thibault Groueix, Vladimir G. Kim, Daniel Ritchie, Noam Aigerman

arXiv:2510.14146v1cs.GRcs.CVcs.LG

TL;DR

Existing mesh-learning architectures face trade-offs among high-frequency representation, receptive-field size, discretization robustness, and computational cost. PoissonNet alternates local learned gradient transformations with global Poisson-based scalar updates, achieving state-of-the-art performance across surface-learning applications while remaining efficient and scalable. Its main scope limitation is that disconnected components are handled independently, and very large meshes still do not support real-time use.

  • Problem

    Existing intrinsic mesh-learning methods repeatedly trade off receptive-field size, full-frequency representation, discretization robustness, and computational efficiency.

  • Method

    PoissonNet transforms feature gradients locally and solves Poisson’s equation to propagate global scalar feature updates across the surface.

  • Results

    PoissonNet achieves state-of-the-art performance across segmentation, deformation, and high-frequency surface-learning applications while remaining more efficient than comparable methods.

  • Takeaways & Limitations

    The local-global design supports efficient, scalable learning on detailed surfaces, including animation, semantic segmentation, and high-frequency geometry compression.

  • Takeaways & Limitations

    PoissonNet is not well suited to multi-component shapes, can become numerically unstable on extremely poor discretizations, and remains too slow for real-time use on very large meshes.

Abstract

from arXiv · show

Many network architectures exist for learning on meshes, yet their constructions entail delicate trade-offs between difficulty learning high-frequency features, insufficient receptive field, sensitivity to discretization, and inefficient computational overhead. Drawing from classic local-global approaches in mesh processing, we introduce PoissonNet, a novel neural architecture that overcomes all of these deficiencies by formulating a local-global learning scheme, which uses Poisson's equation as the primary mechanism for feature propagation. Our core network block is simple; we apply learned local feature transformations in the gradient domain of the mesh, then solve a Poisson system to propagate scalar feature updates across the surface globally. Our local-global learning framework preserves the features's full frequency spectrum and provides a truly global receptive field, while remaining agnostic to mesh triangulation. Our construction is efficient, requiring far less compute overhead than comparable methods, which enables scalability -- both in the size of our datasets, and the size of individual training samples. These qualities are validated on various experiments where, compared to previous intrinsic architectures, we attain state-of-the-art performance on semantic segmentation and parameterizing highly-detailed animated surfaces. Finally, as a central application of PoissonNet, we show its ability to learn deformations, significantly outperforming state-of-the-art architectures that learn on surfaces.

1 Introduction

PoissonNet addresses recurring trade-offs in intrinsic mesh learning by alternating local gradient-domain transformations with global Poisson-based feature propagation. It preserves full-frequency features, global support, triangulation agnosticism, and scalability, with state-of-the-art results across several applications.

  • Motivation: PoissonNet is designed to overcome limited receptive fields, spectral truncation, discretization sensitivity, and expensive computation in prior intrinsic methods.These deficiencies recur across existing differential-operator-based architectures.
  • Method: Each block transforms feature gradients locally, then solves Poisson’s equation to produce global scalar-domain feature updates.The architecture alternates between gradient and functional domains, treating gradients as first-class features.
  • Properties: PoissonNet simultaneously preserves the full frequency spectrum, provides global spatial support, remains triangulation agnostic, and scales to large meshes and datasets.Its Poisson formulation avoids spectral truncation and supports near-identical outputs under changes in mesh discretization.
  • Evaluation: Across shape segmentation, deformation, and high-frequency surface learning, PoissonNet achieves state-of-the-art performance while remaining more efficient than comparable methods.The authors report this pattern across the evaluated applications.

2 Related Work

Related surface-learning methods use point, convolutional, spectral, or differential-operator constructions, each with limitations in connectivity, frequency range, locality, or efficiency. PoissonNet instead uses Poisson solves to propagate learned features globally while retaining intrinsic geometric processing.

  • Surface learning: Point-cloud methods lack mesh connectivity, limiting their ability to encode highly detailed geometry and support intrinsic surface computations.Mesh-based representations explicitly encode topology and provide a discretization of the underlying manifold.
  • Intrinsic learning on meshes: Spectral approaches can require eigenbasis computation and restrict representations to lower frequencies, reducing expressivity and increasing preprocessing costs.DiffusionNet addresses PDE propagation spectrally, but its lower-frequency restriction loses detail.
  • Intrinsic learning on meshes: DeltaConv combines local differential operators, but its receptive field shrinks under refinement, requiring deeper networks and making behavior dependent on sampling density.This contrasts with PoissonNet’s global support per layer.
  • Deformation learning: NJF uses Poisson’s equation only in a deformation-prediction output layer, whereas PoissonNet solves it in every block to propagate learned features.PoissonNet is evaluated as a backbone together with an NJF final layer for deformation learning.
  • Deformation learning: PoissonNet’s deformation-learning application builds on gradient-domain methods developed for surface parameterization, registration, animation, and geometric modeling.Mesh deformation remains a primary benchmark and application for the network.

3 Preliminaries

The preliminaries represent mesh signals as piecewise-linear scalar functions whose facewise gradients are tangent vectors. Poisson’s variational formulation then recovers the scalar function whose gradient best matches a given tangent-vector field through a sparse linear system.

  • Tangent spaces and local coordinates: Each mesh triangle has a two-dimensional tangent space with an arbitrary orthonormal basis for local vector coordinates.Tangent vectors can be represented equivalently in the triangle’s local two-dimensional coordinate system.
  • Piecewise linear functions and their gradients: Vertex scalar values define an affine function on each triangle, so the resulting piecewise-linear signal has a constant tangent gradient on every face.The face gradient is obtained by applying the linear gradient operator to the triangle’s vertex values.
  • Poisson’s equation: Poisson’s variational formulation finds the scalar function whose gradient best matches a supplied tangent-vector field.On a mesh, facewise-constant gradients reduce the least-squares objective to a sparse linear system.

4 PoissonNet

PoissonNet alternates local learned transformations in the gradient domain with global scalar updates from Poisson solves. This design supports full-frequency, globally coupled, triangulation-agnostic feature propagation with efficient reusable sparse systems.

  • 4 PoissonNet: PoissonNet blocks compute gradients, apply learned per-face transformations, solve Poisson’s equation for global scalar updates, and combine those updates with the input features.The block alternates local gradient-domain processing with global propagation in the scalar domain.
  • 4 PoissonNet: The gradient representation uses tangent-plane complex features, whose learned transformations scale and rotate gradients while preserving coordinate-system equivariance.Nonlinearities are applied to gradient magnitudes, while directional components transform consistently with the tangent basis.
  • 4 PoissonNet: Scalar features modulate gradient phases and magnitudes through point-wise MLP outputs, providing intrinsic positional information for more discriminative local transformations.The modulation uses face-averaged scalar features to predict scale factors and angular rotations.
  • 4 PoissonNet: The Poisson solve converts transformed face gradients into globally coupled vertex updates using the cotangent Laplacian, mass matrix, and divergence operator.The solution is centered at zero, and coordinate-free divergence makes it invariant to tangent-basis choices.
  • 4 PoissonNet: PoissonNet avoids lossy spectral approximations while providing a truly global operator and a reusable sparse factorization across network blocks.The inverse Laplacian can be viewed as a geometry-dependent global attention kernel without materializing a quadratic attention matrix.

5 Implementation

The implementation constructs mesh operators on the fly with custom CUDA kernels and solves Poisson systems using shared sparse factorizations. These choices target scalable training, batching, and rapid experimentation.

  • 5 Implementation: Custom PyTorch CUDA extensions construct Laplacian, gradient, and mass operators on the fly, reducing precomputation and supporting flexible training pipelines.The operators are emitted directly in PyTorch COO format, with batching supported for meshes sharing connectivity.
  • 5 Implementation: Poisson systems reuse a shared Cholesky factorization of the Laplacian across network blocks and channels for parallel sparse solves.The implementation uses the CUDA-based Cholespy solver and zero Neumann boundary conditions.

6 Results and Experimentation

The paper evaluates PoissonNet on several mesh-learning applications against current state-of-the-art intrinsic methods. The supplied passage establishes the evaluation scope but reports no specific outcomes.

  • 6 Results and Experimentation: Experiments compare PoissonNet with current state-of-the-art intrinsic mesh-learning methods that use differential operators.The evaluation covers several applications, while the supplied passage does not provide individual metrics or winners.

6.1 Analysis of Full-Spectrum Learning

PoissonNet combines local gradient-domain transformations with global Poisson solves to retain high-frequency detail while reasoning across entire surfaces. Experiments show this design represents detailed signals, preserves deformation fidelity, supports segmentation, and remains efficient and robust across mesh conditions.

  • 6.1 Analysis of Full-Spectrum Learning: A 650K-parameter PoissonNet represents an animated 300K-face crumpling paper sequence using only 2% of the reference memory footprint.The model is conditioned on scalar time through each block’s MLP.
  • 6.1 Analysis of Full-Spectrum Learning: PoissonNet preserves high-frequency wrinkles that DiffusionNet loses when both models use the same parameter count, as confirmed by learned-feature power spectra.DiffusionNet’s performance also depends strongly on its eigenbasis size, while PoissonNet avoids that spectral restriction.
  • 6.2 Shape Deformation: PoissonNet accurately learns global humanoid reposing and retains fine surface details, whereas DiffusionNet distorts hands and faces; PoissonNet also converges faster and reaches lower loss.The deformation benchmark uses an NJF head that reconstructs deformations by solving Poisson’s equation from predicted Jacobians.
  • 6.3 Semantic Segmentation: 97.03% test accuracy surpasses DiffusionNet’s 96.12% and DeltaConv’s 88.2% on mesh semantic segmentation, while avoiding DiffusionNet’s 16-hour precomputation and 160GB memory overhead.Each method varies by less than 0.5% across runs.
  • 6.5 Analysis of Architectural Properties: PoissonNet remains stable under corruption, simplification, subdivision, and partial surfaces, and is less distorted around holes than DiffusionNet.The authors attribute this behavior to natural boundary conditions in the inhomogeneous Poisson equation.
  • 6.5 Analysis of Architectural Properties: Despite its purely global operator, PoissonNet learns heat kernel signatures across diverse held-out shapes, showing that global propagation can represent local multi-scale signals.The HKS outputs are independently normalized to [0, 1], so predictions are meaningful up to global scaling.

7 Conclusion

PoissonNet’s local-global approach is efficient, scalable, and robust to out-of-distribution geometry, supporting animation, segmentation, and high-frequency geometry compression. Its scope is limited by disconnected components, poor discretizations, non-real-time speed on very large meshes, and distance-decaying coupling.

  • PoissonNet enables animation of intricately detailed character models without rigs, semantic segmentation, and compression of high-frequency geometry.
  • PoissonNet is efficient, scalable, and robust to out-of-distribution geometry.
  • Limitations: Meshes with multiple connected components are poorly suited because the intrinsic Poisson equation operates independently on each component.Generalizing to multi-component meshes without ad-hoc KNN solutions remains unsolved.
  • Limitations: Sliver triangles may cause numerical instability, PoissonNet remains too slow for real-time use on very large meshes, and coupling diminishes with distance.

A Experimental and Implementation Details

The appendix provides implementation details for the paper’s experiments, including architectures, comparison methods, and training hyperparameters.

  • The appendix documents PoissonNet architectures, compared methods, and training hyperparameters for all experiments described in the main text.

A.1 Analysis of Full-Spectrum Learning

The full-spectrum analysis compares PoissonNet and DiffusionNet features in the crumpled-paper Laplace–Beltrami eigenbasis. PoissonNet retains more high-frequency power, including frequencies truncated by DiffusionNet’s spectral PDE solve.

  • Features from the last network block are projected into a Laplace–Beltrami eigenbasis with K=1024 to compute power spectra.
  • PoissonNet features retain more high-frequency power than DiffusionNet features, including frequencies truncated by DiffusionNet’s spectral PDE solve.
  • The appendix identifies Table 4 as a comparison of methods on Maron et al.’s [2017] human mesh segmentation task.

A.2 Shape Deformation

Shape-deformation experiments use matched NJF heads and compare PoissonNet with DiffusionNet and DeltaConv across detailed character-reposing and deformation settings. PoissonNet’s global support contrasts with DeltaConv’s local operators, whose deeper variants yielded only marginal or harmful improvements.

  • Shape Deformation: The deformation experiments use five-block, 128-width PoissonNet and equal-parameter DiffusionNet backbones, each with an NJF deformation head.
  • Shape Deformation: The experiments use 16k SMPL-X source-target pairs with MOYO poses and condition networks on 153 SMPL-X pose parameters.
  • NJF Head: NJF heads outperform direct target-vertex prediction across backbones, especially when generalizing to out-of-distribution geometries.
  • Shape Deformation: PoissonNet reposes diverse MeshyAI-generated characters using pose inputs from the MOYO dataset [Tripathi et al. 2023].
  • DeltaConv Baseline: PoissonNet’s blocks have fully global support, whereas DeltaConv uses local mesh operators; deeper DeltaConv variants only marginally improved performance and sometimes hurt it through over-parametrization.

A.3 Semantic Segmentation on MOYO Dataset

On the MOYO shape-deformation benchmark, PoissonNet converges to higher test accuracy than comparable methods while using substantially less compute.

  • PoissonNet converges to higher test accuracy than the compared methods while remaining far more compute efficient.The comparison uses a 128-width, three-block PoissonNet against parameter-matched DiffusionNet and DeltaConv baselines.
  • Vector feature modulation clearly improves performance on the shape-deformation benchmark, with the gain reported as representative across experiments.

C.1 MOYO Dataset

The MOYO-based experiments use sampled SMPL-X human bodies, while the separate crumpling-paper benchmark contains 118 frames of a 300k-face mesh.

  • The MOYO experiments use 32k training and 4k validation SMPL-X human bodies generated from yoga poses sampled from temporal motion captures.Near-identical MOYO frames are reduced using greedy farthest point sampling.
  • The simulated crumpling-paper sequence contains 118 frames, each sharing a canonical mesh topology of 300k faces.
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