PoissonNet: A Local-Global Approach for Learning on Surfaces

TL;DR

PoissonNet tackles learning on triangle meshes by marrying a local gradient-domain transform with a global Poisson solve, enabling full-spectrum feature propagation across surfaces. This local-global architecture preserves high-frequency geometry, achieves a truly global receptive field, and remains triangulation-agnostic while avoiding costly spectral bases. The approach delivers state-of-the-art results in semantic segmentation and deformation learning, with strong efficiency and scalability on large, detailed meshes. Its practical impact lies in providing a robust, flexible backbone for learning detailed geometry without heavy precomputation, applicable to animation, segmentation, and geometric analysis.

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.

Figures (20)

• Figure 1: We illustrate our PoissonNet block on a surface patch of a triangle mesh. Our block begins by computing spatial gradients of the incoming scalar features---for demonstrative purposes we depict the signal as having three channels (shown as green, orange, blue) with color indicating signal intensity. The gradient features are transformed locally in each tangent basis (denoted by basis vectors $u_1,u_2$) via a Vector MLP, which induces a linear combination of rotated and scaled gradients on each face. We then solve a global Poisson system using the transformed vector fields, producing new scalar features on vertices that are updated locally using a scalar (per-vertex) MLP. This process is repeated for $N$ blocks, thereby producing a final feature representation on the shape.
• Figure 2: PoissonNet stacks identical blocks that transform features locally in the gradient domain and globally in the functional domain.
• Figure 3: Comparison of architectures for representing highly-detailed signals. The networks are used to parametrize the evolution of a crumpling paper through time w.r.t. a rest configuration (see supplemental video). The mesh has 300k triangles and is available on TurboSquid crumplingPaperBall.
• Figure 4: Poisson and heat solutions on an input scalar field. The Poisson equation (inhomogeneous) retains structure; whereas the heat equation (homogeneous) converges to a global mean, losing all structure over long-time heat flow. Right plot: solution values along a cross-section at $y=0.5$ (dotted lines).
• Figure 5: We demonstrate that PoissonNet surpasses state-of-the-art intrinsic learning backbones for representing source-to-target deformations of humanoid characters. Here, we compare our deformations to that of DiffusionNet diffusionnet, using diverse poses from the MOYO dataset moyo_tripathi_2023. PoissonNet is able to faithfully match target poses while retaining intricate surface details from the source geometry, even for models that are substantially out of distribution (e.g. the top left mutant, and the Buck Bunny). We observe that DiffusionNet is unable to retain surface details, causing many body parts to become distorted (see close-ups). We additionally show in-distribution examples in the bottom half of the figure---of which these distortions remain noticeable.