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.

PoissonNet is the first method for learning on surfaces that simultaneously satisfies several key properties:

1. Full spectrum. Our network features retain their native frequency components without any spectral truncation, preserving high-frequency details while avoiding expensive precomputation of eigenbases.


2. Global receptive field. As an integral-like operator, our proposed network block efficiently propagates local feature updates across the entire surface.


3. Triangulation agnosticism. The core mechanism in our network approximates a well-defined object: the continuous Poisson's equation. This allows PoissonNet to produce near-identical predictions under changes in mesh discretization (subdivision, simplification, remeshing, corruption, etc.).


4. Efficiency. PoissonNet can operate on high-resolution meshes and forgo lengthy pre-computation before training and inference, which facilitates training on large datasets.

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, and blue), with color indicating signal intensity. The gradient features are transformed locally in each tangent basis (denoted by basis vectors u₁ and u₂) 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.