Bayesian deep operator learning for homogenized to fine-scale maps for multiscale PDE

TL;DR

This paper introduces a neural operator-based framework for efficiently predicting fine-scale solutions of multiscale PDEs from coarse-scale data, enabling mesh-free solutions even with limited observations.

Contribution

It develops a novel operator learning approach using neural operators for multiscale PDEs, allowing mesh-free, efficient predictions from limited and noisy data.

Findings

Accurately predicts fine-scale solutions from coarse data

Operates efficiently as a mesh-free solver

Works with limited and noisy observations

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 a limited number (and possibly noisy) observations of the 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.