Bayesian Deep Operator Learning for Homogenized to Fine-Scale Maps for Multiscale PDE

Multiscale Modeling &Simulation, Volume 22, Issue 3, Page 956-972, September 2024.
Abstract. We present a new framework for computing fine-scale solutions of multiscale partial differential equations (PDEs) using operator learning tools. Obtaining fine-scale solutions of multiscale PDEs can be challenging, but there are many inexpensive computational methods for obtaining coarse-scale solutions. Additionally, in many real-world applications, fine-scale solutions can only be observed at a limited number of locations. In order to obtain approximations or predictions of fine-scale solutions over general regions of interest, we propose to learn the operator mapping from coarse-scale solutions to fine-scale solutions using observations of a limited number of (possible noisy) fine-scale solutions. The approach is to train multi-fidelity homogenization maps using mathematically motivated neural operators. The operator learning framework can efficiently obtain the solution of multiscale PDEs at any arbitrary point, making our proposed framework a mesh-free solver. We verify our results on multiple numerical examples showing that our approach is an efficient mesh-free solver for multiscale PDEs.