Learning to Solve PDEs on Neural Shape Representations

Teaser figure: [short description] — Figure 1: Our mesh-free, geometry-conditioned learned operator solves surface PDEs (heat equation in the teaser) *directly in the neural domain* on multiple modalities, without mesh extraction, or per-instance optimization. Trained once on a single exemplar (*SPIKE*, top-left), the learned operator generalizes across unseen geometries, topologies, and input modalities. See below for heat flow videos and also Poisson solves.

Abstract

Solving partial differential equations (PDEs) on shapes underpins many shape analysis and engineering tasks; yet, prevailing PDE solvers operate on polygonal/triangle meshes while modern 3D assets increasingly live as neural representations. This mismatch leaves no suitable method to solve surface PDEs directly within the neural domain, forcing explicit mesh extraction or per-instance residual training, preventing end-to-end workflows.

We present a mesh-free formulation that learns a local update operator conditioned on neural local shape attributes, enabling surface PDEs to be solved directly where the neural data lives. The operator integrates with prevalent neural surface representations, is trained once on a single representative shape, and generalizes across shape and topology variations, enabling accurate, fast inference without explicit meshing or per-instance optimization while preserving differentiability.

Across analytic benchmarks (heat equation and Poisson solve on a sphere) and real neural assets across different representations, our method slightly outperforms CPM while remaining reasonably close to FEM, and, to our knowledge, delivers the first end-to-end pipeline that solves surface PDEs on both neural and classical surface representations.

Method Overview

Pipeline figure — Figure 2: **Pipeline overview.**

From a surface \( \mathcal{S} \), we sample points and geometric features (normals, local features). Around an \( \varepsilon \)-narrow band around the shape, we gather Cartesian grids \( \mathcal{B}_{\mathcal{S}} \) to store an initial field \( U_0 \) extended from surface values and covered by overlapping, surface-centred patches \( \{ \mathcal{P}_i \} \). Each patch is reoriented to its local frame, yielding \( \{ \hat{\mathcal{B}}_i, \hat{\mathcal{F}}_i \} \), which are processed by our lightweight geometry-conditioned operator \( \mathcal{N}_\Theta \) to produce local updates (see Figure 3). The local updates are smoothly aggregated to form the global band update \( \tilde{U}_{t} \) and advanced with a standard grid PDE time step to get \( U_{t+1} \). Iterating this loop yields surface PDE solutions directly in the neural domain via grid-to-grid updates, without mesh extraction or extend–restrict shuttling.

Neural update operator (overview).

Given a query location \( q \), the local band \( \hat{\mathcal{B}}_i \) expressed in the local frame \( \mathcal{L}_i \) centered around \( p_i^c \) for patch \( \mathcal{P}_i \), with locally transformed surface features \( \hat{\mathcal{F}}_i \) (e.g., positions, normals), and current band values \( u^i \) at grid sites (time index \( t \) omitted for brevity; full notation \( u^i_t \)), our operator predicts an updated function value at location \( q \). Trainable components include compact MLP blocks \( \Phi_{\theta_1}, \Phi_{\theta_2}, \Phi_{\theta_3} \) and a scalar \( \lambda \). The full network \( \mathcal{N}_\Theta \) produces the updated band value at \( q \), yielding a single geometry–conditioned grid-to-grid step.

Results

Our precomputed results for different evaluations in the paper.

Generalization

Generalization across shapes

Different neural surfaces

Tests on different neural representations

Boundary handling

Dirichlet boundary conditions

Different neural surface representations

Our method operates natively on a wide spectrum of neural and classical surface representations without requiring meshing, re-meshing, or per-instance optimization—as long as local geometric features such as positions and normals can be sampled. To demonstrate this versatility, we evaluate the solver on neural SDFs, overfitted SDF, Spherical Neural Surfaces (SNS), Gaussian splatting, point clouds, and triangle meshes. Across all representations, the solver achieves stable and consistent accuracy despite large differences in geometric quality—for example, SNS provides clean differential estimates, whereas point clouds and GSplats often yield noisy normals. As shown in Figure 1 and Table 2 of the main paper, the operator adapts robustly to these variations: SNS achieves the strongest accuracy, while SDFs, meshes, and point clouds follow closely, demonstrating resilience to imperfect or approximate geometry. Finally, our method remains reasonably close to FEM in accuracy and, to our knowledge, provides the first end-to-end pipeline capable of solving surface PDEs directly on both neural and classical surface representations.

Generalization across shapes

To assess cross-shape generalization, we train our network once on a single shape and evaluate it on a diverse gallery of unseen and challenging geometries by solving both heat and Poisson equations. The test set includes smooth organic shapes, objects with sharp features, CAD models, and thin structures where the closest-point projection becomes ambiguous (i.e., cp is not unique). As shown in Figure 4 of the main article, the method closely matches SFEM reference solutions across this entire collection, with noticeable deviations only in inherently ambiguous regions such as sharp edges or thin components. These artifacts are expected and inherited from the underlying CPM formulation. Despite these natural limitations, errors remain consistently low and stable across shapes (Table 3), confirming a key strength of the approach: generalizing from a single training shape to a wide family of unseen geometries while maintaining reliable accuracy.Generalization examples: Apple, fastener, holey_human, jared, octopus, Sofa.

Boundary handling

We also evaluate our method on surfaces with Dirichlet boundary conditions applied on a cut version of the Max-Planck head (Figure 5 in the main article). We consider two regimes of increasing difficulty: a constant boundary condition, and a more challenging sinusoidal boundary condition. As in the Closest Point Method, boundary values are enforced directly on band nodes whose closest-point projections lie on the boundary. Our learned operator naturally inherits this behavior: in both cases, the solver remains stable, with only a mild increase in error for the oscillatory boundary condition (see Table 15 in the supplemental). These experiments confirm that the method handles Dirichlet boundaries reliably, leveraging CPM's boundary treatment while still producing accurate solutions on open surfaces.