A Learning-based Domain Decomposition Method

TL;DR

This paper introduces a learning-based domain decomposition method that leverages pre-trained neural operators to efficiently solve complex PDEs in intricate geometries, outperforming existing methods and demonstrating strong generalization.

Contribution

It proposes a novel approach combining neural operators with domain decomposition, enabling efficient and scalable solutions for complex PDEs with unseen microstructures.

Findings

Outperforms state-of-the-art methods on complex PDE problems

Demonstrates resolution-invariance and generalization to unseen microstructures

Provides theoretical guarantees on neural operator approximation within domain decomposition

Abstract

Recent developments in mechanical, aerospace, and structural engineering have driven a growing need for efficient ways to model and analyse structures at much larger and more complex scales than before. While established numerical methods like the Finite Element Method remain reliable, they often struggle with computational cost and scalability when dealing with large and geometrically intricate problems. In recent years, neural network-based methods have shown promise because of their ability to efficiently approximate nonlinear mappings. However, most existing neural approaches are still largely limited to simple domains, which makes it difficult to apply to real-world PDEs involving complex geometries. In this paper, we propose a learning-based domain decomposition method (L-DDM) that addresses this gap. Our approach uses a single, pre-trained neural operator-originally trained on simple domains-as a surrogate model within a domain decomposition scheme, allowing us to tackle large and complicated domains efficiently. We provide a general theoretical result on the existence of neural operator approximations in the context of domain decomposition solution of abstract PDEs. We then demonstrate our method by accurately approximating solutions to elliptic PDEs with discontinuous microstructures in complex geometries, using a physics-pretrained neural operator (PPNO). Our results show that this approach not only outperforms current state-of-the-art methods on these challenging problems, but also offers resolution-invariance and strong generalization to microstructural patterns unseen during training.