Neural Network Localized Orthogonal Decomposition for Numerical Homogenization of Diffusion Operators with Random Coefficients

Abstract

This paper presents a neural network--enhanced surrogate modeling approach for diffusion problems with spatially varying random field coefficients. The method builds on numerical homogenization, which compresses fine-scale coefficients into coarse-scale surrogates without requiring periodicity. To overcome computational bottlenecks, we train a neural network to map fine-scale coefficient samples to effective coarse-scale information, enabling the construction of accurate surrogates at the target resolution. This framework allows for the fast and efficient compression of new coefficient realizations, thereby ensuring reliable coarse models and supporting scalable computations for large ensembles of random coefficients. We demonstrate the efficacy of our approach through systematic numerical experiments for two classes of coefficients, emphasizing the influence of coefficient contrast: (i) lognormal diffusion coefficients, a standard model for uncertain subsurface structures in geophysics, and (ii) hierarchical Gaussian random fields with random correlation lengths.