Locally Subspace-Informed Neural Operators for Efficient Multiscale PDE Solving

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.