Locally Subspace-Informed Neural Operators for Efficient Multiscale PDE Solving

We propose GMsFEM-NO, a novel hybrid framework that combines the robustness of the Generalized Multiscale Finite Element Method (GMsFEM) with the computational speed of neural operators (NOs) to create an efficient method for solving heterogeneous partial differential equations (PDEs). GMsFEM builds localized spectral basis functions on coarse grids, allowing it to capture important multiscale features and solve PDEs accurately with less computational effort. However, computing these basis functions is costly. While NOs offer a fast alternative by learning the solution operator directly from data, they can lack robustness. Our approach trains a NO to instantly predict the GMsFEM basis by using a novel subspace-informed loss that learns the entire relevant subspace, not just individual functions. This strategy significantly accelerates the costly offline stage of GMsFEM while retaining its foundation in rigorous numerical analysis, resulting in a solution that is both fast and reliable. On standard multiscale benchmarks—including a linear elliptic diffusion problem and the nonlinear, steady-state Richards equation—our GMsFEM-NO method achieves a reduction in solution error compared to standalone NOs and other hybrid methods. The framework demonstrates effective performance for both 2D and 3D problems. A key advantage is its discretization flexibility: the NO can be trained on a small computational grid and evaluated on a larger one with minimal loss of accuracy, ensuring easy scalability. Furthermore, the resulting solver remains independent of forcing terms, preserving the generalization capabilities of the original GMsFEM approach. Our results prove that combining NO with GMsFEM creates a powerful new type of solver that is both fast and accurate.