🤖 AI Summary
This work addresses the limited reusability and heavy reliance on problem-specific data in conventional global surrogate-based neural PDE solvers. The authors propose NEST, a novel framework that integrates local neural operators with classical domain decomposition techniques. By training reusable local solvers on 3×3×3 voxel patches and coupling them via overlapping subdomains, Schwarz iterations, and a partition of unity, NEST constructs globally consistent solutions without dependence on a fixed family of problems. The method is geometry-agnostic, scalable, and demonstrates strong generalization to complex three-dimensional nonlinear elasticity problems far exceeding the scale and configuration of its training data, enabling efficient solution transfer across scales, geometries, and boundary conditions.
📝 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.