Neural-Schwarz Tiling for Geometry-Universal PDE Solving at Scale

TL;DR

NEST introduces a local-to-global neural PDE solver framework that learns local physics on small patches and assembles solutions for complex 3D domains, enabling scalable and generalizable PDE solving.

Contribution

The paper presents NEST, a novel local-to-global neural PDE solver that shifts learning from full-domain operators to reusable local physics models with iterative global assembly.

Findings

NEST generalizes across domain size, shape, and boundary conditions.

Local neural blocks combined with Schwarz iteration effectively solve complex 3D PDEs.

The approach outperforms traditional global surrogate models in scalability and flexibility.

Abstract

Most learned PDE solvers follow a global-surrogate paradigm: a neural operator is trained to map full problem descriptions to full solution fields for a prescribed distribution of geometries, boundary conditions, and coefficients. This has enabled fast inference within fixed problem families, but limits reuse across new domains and makes large-scale deployment dependent on expensive problem-specific data generation. We introduce $\textbf{NEST}$ ($\textbf{Ne}$ural-$\textbf{S}$chwarz $\textbf{T}$iling), a local-to-global framework that shifts learning from full-domain solution operators to reusable local physical solvers. The central premise is that, although global PDE solutions depend on geometry, scale, and boundary conditions, the physical response on small neighborhoods can be learned locally and composed into global solutions through classical domain decomposition. NEST learns a neural operator on minimal voxel patches ($3 \times 3 \times 3$) with diverse local geometries and boundary/interface data. At inference time, an unseen voxelized domain is tiled into overlapping patches, the learned local solver is applied patchwise, and global consistency is enforced through iterative Schwarz coupling with partition-of-unity assembly. In this way, generalization is shifted from a monolithic neural model to the combination of local physics learning and algorithmic global assembly. We instantiate NEST on nonlinear static equilibrium in compressible neo-Hookean solids and evaluate it on large, geometrically complex 3D domains far outside the scale of the training patches. Our results show that local neural building blocks, coupled through Schwarz iteration, offer a reusable local-training path toward scalable learned PDE solvers that generalize across domain size, shape, and boundary-condition configurations.