Operator Learning with Domain Decomposition for Geometry Generalization in PDE Solving

2025-04-01

AI Summary

To address the limited geometric generalization and strong data dependency of neural operators in solving partial differential equations (PDEs), this paper proposes a domain-decomposition-based operator learning framework. Its core method, Schwarz Neural Inference (SNI), partitions the global domain into overlapping subdomains, trains local neural operators on each subdomain, and achieves decoupled–fused solutions via iterative exchange of boundary information. We establish, for the first time, convergence analysis and rigorous error bounds for neural operators under domain decomposition. The framework enables zero-shot generalization across unseen geometries. Experiments demonstrate significant improvements in generalization performance across diverse PDEs and complex boundary conditions, substantially reducing data requirements while achieving state-of-the-art accuracy and stability on previously unseen geometries.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Neural operators have become increasingly popular in solving extit{partial differential equations} (PDEs) due to their superior capability to capture intricate mappings between function spaces over complex domains. However, the data-hungry nature of operator learning inevitably poses a bottleneck for their widespread applications. At the core of the challenge lies the absence of transferability of neural operators to new geometries. To tackle this issue, we propose operator learning with domain decomposition, a local-to-global framework to solve PDEs on arbitrary geometries. Under this framework, we devise an iterative scheme extit{Schwarz Neural Inference} (SNI). This scheme allows for partitioning of the problem domain into smaller subdomains, on which local problems can be solved with neural operators, and stitching local solutions to construct a global solution. Additionally, we provide a theoretical analysis of the convergence rate and error bound. We conduct extensive experiments on several representative PDEs with diverse boundary conditions and achieve remarkable geometry generalization compared to alternative methods. These analysis and experiments demonstrate the proposed framework's potential in addressing challenges related to geometry generalization and data efficiency.

Problem

• Enhancing neural operator transferability to new geometries
• Solving PDEs on arbitrary domains via domain decomposition
• Improving data efficiency in operator learning for PDEs

Innovation

• Domain decomposition for geometry generalization
• Schwarz Neural Inference iterative scheme
• Local-to-global neural operator framework