PoissonNet: A Local-Global Approach for Learning on Surfaces

2025-10-15

AI Summary

Existing mesh learning networks suffer from inherent trade-offs: difficulty modeling high-frequency features, limited receptive fields, sensitivity to discretization, and excessive computational overhead. To address these challenges, we propose PoissonNet—the first deep learning framework to integrate the Poisson equation as a core feature propagation mechanism. PoissonNet establishes a local–global collaborative learning paradigm: it extracts fine-grained details via learnable gradient-domain local transformations and performs truly global, triangulation-invariant feature updates by efficiently solving a sparse linear system derived from the Poisson equation. This design inherently transcends rigid discrete mesh structural constraints, substantially enhancing representational capacity and generalization. Experiments demonstrate that PoissonNet achieves state-of-the-art performance on semantic segmentation, high-fidelity surface parameterization, and surface deformation learning—while maintaining computational efficiency and scalability.

Technology Category

• Computer Vision: Representation Learning for Vision
• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

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.

Problem

• Overcomes trade-offs in mesh learning architectures
• Provides global receptive field with local feature transformations
• Enables efficient deformation learning on surface meshes

Innovation

• Local-global learning scheme using Poisson's equation
• Learned local feature transformations in gradient domain
• Efficient global feature propagation via Poisson system