Operator Learning with Domain Decomposition for Geometry Generalization in PDE Solving

TL;DR

This work tackles the geometry-generalization and data-efficiency challenges of neural operators for PDEs by introducing a local-to-global framework that marries operator learning with overlapping domain decomposition. Local neural operators are trained on simple polygonal building blocks and then assembled on arbitrary geometries via Schwarz Neural Inference (SNI), an additive Schwarz-inspired iterative scheme with symmetry-based normalization and data augmentation. The authors provide convergence and error-bounded guarantees for SNI and demonstrate substantial improvements in geometry generalization and data efficiency across multiple PDEs, including time-dependent problems. The approach offers a scalable path to accurate PDE solving on complex domains with limited training data, with potential impact across engineering and scientific computing.

Abstract

Neural operators have become increasingly popular in solving \textit{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 \textit{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.

Figures (9)

• Figure 1: An illustration of Operator Learning with Domain Decomposition Framework. (a) During training stage, the goal is to ensure that the neural operator can effectively model the local solution operator on various building blocks of shapes. These building blocks are selected and generated based on specific criteria, allowing for a more efficient and targeted learning process. Proper boundary conditions are then imposed to generate local solutions which serve as training data for neural operator. (b) During inference, for an arbitrary given domain, an automated decomposition algorithm is employed to decompose the domain into subdomains. By leveraging the trained local operator and Schwarz Neural Inference (SNI), global solution can be obtained by stitching local solutions on subdomains.
• Figure 2: Illustration of experiment domain A, B, C from left to right respectively.
• Figure 3: Comparison between the $l_2$ relative errors from SNI (blue), GNOT direct inference (orange) and validation (red) on two different PDEs (Laplace2d-Dirichlet and Darcy2d) upon three domains (A, B and C) with different numbers of training samples. The results of SNI and GNOT direct inference are presented based on 100 inferences with different boundary conditions. The best validation errors during training are also provided as a reference.
• Figure 4: Comparison between convergence rate of SNI on Laplace2d-Dirichlet domain A.
• Figure 5: Illustration of mixed Dirichlet and Neumann boundaries for domain A, B, C in Laplace2d-Mixed.

Theorems & Definitions (4)

• Theorem 1
• Theorem
• proof
• Corollary 1