Learning to Solve PDEs on Neural Shape Representations

TL;DR

This work introduces a mesh-free, neural PDE solver that directly operates in neural shape domains to solve surface PDEs such as the heat and Poisson equations. The core idea is a lightweight, geometry-conditioned neural update operator that learns the narrow-band extension around a surface, enabling grid-to-grid updates without meshing or per-instance optimization. Trained once on a single exemplar, the operator generalizes across unseen shapes, topologies, and modalities, maintaining differentiability and robustness across representations like SNS, SDFs, point clouds, and splatting. Empirically, the method achieves competitive accuracy with zero meshing overhead, offers stability under remeshing, and provides an end-to-end neural PDE layer that complements classical solvers for broad geometric modeling tasks on neural shapes. This bridges neural representations and PDE-based shape analysis, enabling end-to-end training pipelines for editing, reconstruction, and analysis that leverage surface PDE priors in a neural context.

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 novel, 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 naturally 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 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. Code will be released on acceptance.

Figures (5)

• 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 supplemental for heat flow videos and also Poisson solves.
• 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 operators $\mathcal{N}_\Theta$ to produce local updates (see \ref{['fig:SUN_network']}). 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.
• Figure 3: 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 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.
• Figure 4: Comparison to SFEM on diverse shapes. For each object, left shows SFEM and right shows ours; the small inset below visualizes the pointwise error (ours vs. SFEM) with a hot–cold colormap. See color bar for error scale and the supplemental for per-shape statistics. (Error colormaps are normalized per instance.)
• Figure 5: Dirichlet boundaries on an open surface. Heat diffusion on the Max Planck head cut at the neck (left to right) with boundary values clamped on the cut. See supplemental.