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

arXiv: https://arxiv.org/abs/2605.12343 · HTML: https://mlsanity.com/paper/2605.12343/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 34 references that are in the map.
- First posted on arXiv 12 May 2026.

## Builds on (most cited first, 23 of 23)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior](https://mlsanity.com/paper/2306.00258/) (2023)
- [Transolver++: An Accurate Neural Solver for PDEs on Million-Scale Geometries](https://mlsanity.com/paper/2502.02414/) (2025)
- [HAMLET: Graph Transformer Neural Operator for Partial Differential Equations](https://mlsanity.com/paper/2402.03541/) (2024)
- [Geometric Operator Learning with Optimal Transport](https://mlsanity.com/paper/2507.20065/) (2025)
- [SimJEB: Simulated Jet Engine Bracket Dataset](https://mlsanity.com/paper/2105.03534/) (2021)
- [Learning Interface Conditions in Domain Decomposition Solvers](https://mlsanity.com/paper/2205.09833/) (2022)
- [On the feasibility of foundational models for the simulation of physical phenomena](https://mlsanity.com/paper/2410.14645/) (2024)
- [Operator Learning with Domain Decomposition for Geometry Generalization in PDE Solving](https://mlsanity.com/paper/2504.00510/) (2025)
- [A Learning-based Domain Decomposition Method](https://mlsanity.com/paper/2507.17328/) (2025)
- [Data-Efficient Discovery of Hyperelastic TPMS Metamaterials with Extreme Energy Dissipation](https://mlsanity.com/paper/2405.19507/) (2024)
- [Generalizing PDE Emulation with Equation-Aware Neural Operators](https://mlsanity.com/paper/2511.09729/) (2025)
- [Towards Universal Neural Operators through Multiphysics Pretraining](https://mlsanity.com/paper/2511.10829/) (2025)
- [Learning Data-Efficient and Generalizable Neural Operators via Fundamental Physics Knowledge](https://mlsanity.com/paper/2602.15184/) (2026)

## Built on by (most cited first, 2 of 2)

- [Convex Neural Energy Elements: Monolithic Finite-Element Assembly of Geometry-Parameterized Neural Operators with Stability and Error Guarantees](https://mlsanity.com/paper/2608.02036/) (2026)
- [Learnable composition for neural operators](https://mlsanity.com/paper/2609.03069/) (2026)

## Abstract (arXiv)

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 NEST (Neural-Schwarz Tiling), 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 × 3 × 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.