Locally Subspace-Informed Neural Operators for Efficient Multiscale PDE Solving

TL;DR

This paper introduces a hybrid neural operator approach that learns subspaces for multiscale PDEs, significantly improving accuracy and efficiency over traditional methods and standalone neural operators.

Contribution

It proposes a novel subspace-informed neural operator framework that reduces basis construction costs and enhances multiscale PDE solving accuracy.

Findings

Reduces solution error by ~60% compared to standalone NOs.

Cuts basis-construction time by ~60 times relative to classical GMsFEM.

Achieves efficient and accurate multiscale PDE solutions independent of forcing and boundary conditions.

Abstract

Neural operators (NOs) struggle with high-contrast multiscale partial differential equations (PDEs), where fine-scale heterogeneities cause large errors. To address this, we use the Generalized Multiscale Finite Element Method (GMsFEM) that constructs localized spectral basis functions on coarse grids. This approach efficiently captures dominant multiscale features while solving heterogeneous PDEs accurately at reduced computational cost. However, computing these basis functions is computationally expensive. This gap motivates our core idea: to use a NO to learn the subspace itself - rather than individual basis functions - by employing a subspace-informed loss. On standard multiscale benchmarks - namely a linear elliptic diffusion problem and the nonlinear, steady-state Richards equation - our hybrid method cuts solution error by approximately $60\%$ compared with standalone NOs and reduces basis-construction time by about $60$ times relative to classical GMsFEM, while remaining independent of forcing terms and boundary conditions. The result fuses multiscale finite-element robustness with NO speed, yielding a practical solver for heterogeneous PDEs.