A machine learning approach for efficient uncertainty quantification using multiscale methods

TL;DR

This paper presents a neural network-based method to efficiently estimate multiscale basis functions, significantly reducing computational costs in uncertainty quantification for elliptic problems.

Contribution

It introduces a data-driven approach that learns to generate basis functions, replacing local problem solving in multiscale methods, enhancing efficiency in uncertainty quantification tasks.

Findings

Neural network predictor accurately estimates basis functions.

Method reduces computational cost compared to traditional local problem solving.

Promising results demonstrated on elliptic problems.

Abstract

Several multiscale methods account for sub-grid scale features using coarse scale basis functions. For example, in the Multiscale Finite Volume method the coarse scale basis functions are obtained by solving a set of local problems over dual-grid cells. We introduce a data-driven approach for the estimation of these coarse scale basis functions. Specifically, we employ a neural network predictor fitted using a set of solution samples from which it learns to generate subsequent basis functions at a lower computational cost than solving the local problems. The computational advantage of this approach is realized for uncertainty quantification tasks where a large number of realizations has to be evaluated. We attribute the ability to learn these basis functions to the modularity of the local problems and the redundancy of the permeability patches between samples. The proposed method is evaluated on elliptic problems yielding very promising results.