Locally Subspace-Informed Neural Operators for Efficient Multiscale PDE Solving

TL;DR

The paper tackles high-contrast multiscale PDEs by introducing GMsFEM-NO, a hybrid method that trains neural operators to learn the subspace spanned by GMsFEM basis functions via a subspace-alignment loss. This approach yields large offline speedups while maintaining or improving accuracy over traditional GMsFEM and standalone NOs, and it demonstrates robustness to varying forcing terms and boundary conditions. The key contributions include the subspace-informed loss (SAL and SAL-PR), the division of local domains into specialized NOs, and substantial performance gains (roughly 60× faster basis generation and ~60% error reduction in some benchmarks). The method offers a practical, scalable solution for heterogeneous PDEs, with future directions including grid-agnostic architectures and time-dependent extensions to broaden applicability.

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.

Figures (4)

• Figure 1: Illustration of training (a) and inference (b) stages of the proposed GMsFEM-NO method. $\text{NO}$ is trained on heterogeneous fields $\kappa^{\omega_i}$ that defined on subdomain $\omega_i$ to predict subspace of basis functions $\{\psi^{\omega_i}_j\}_{j=1}^{N_c}$. During training the subspace-informed loss $\mathcal{L}$ is applied to align predicted subspace $\{\widetilde{\psi}^{\omega_i}_j\}_{j=1}^{N_c}$ with $\{\psi^{\omega_i}_j\}_{j=1}^{N_c}$. During inference stage (b), the predicted subspace forms the matrix $\widetilde{R}$ that projects matrix $A$ and vectors to the coarse space.
• Figure 2: Illustration of the $5 \times 5$ coarse grid $\mathcal{T}_H$ with local domains ($\omega_0$, $\omega_{20}$, $\omega_{34}$) that are denoted as corner ($1$ cell, e.g., $\omega_0$), half ($2$ cells, e.g., $\omega_{34}$), or full ($4$ cells, e.g., $\omega_{20}$) based on their coarse-cell count.
• Figure 3: Multiscale basis generation algorithm for three subdomain types $\omega_i$: (a)full, (b)half, (c)corner - using dedicated NOs per type with further extension to $\Omega$.
• Figure 4: Example of input coefficient and right-hand side.