Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media

Abstract: We present an operator learning framework based on a coarse data-driven multiscale space for nonlinear flow in random heterogeneous porous media. The multiscale space is constructed from local representative fine-scale solution snapshots, yielding an accurate low-dimensional representation of the solution manifold. This multiscale basis serves as the trunk of a neural operator, while a branch network predicts the corresponding reduced coefficients from the input permeability field. Unlike Galerkin projection methods, the neural operator learns a global nonlinear mapping from permeability fields to solution coefficients, providing greater flexibility, improved accuracy, and eliminating the need for online nonlinear coarse-grid solves and coefficient evaluations. Numerical results show that the proposed approach achieves good accuracy and substantially lower computational cost than projection-based methods for nonlinear flow in high-contrast heterogeneous media.