Deep Micro Solvers for Rough-Wall Stokes Flow in a Heterogeneous Multiscale Method

TL;DR

This paper introduces a neural network-based precomputation method for the heterogeneous multiscale method applied to rough-wall Stokes flow, significantly reducing computational costs while maintaining accuracy.

Contribution

It develops a Fourier neural operator to approximate local flow averages, enabling efficient and boundary-condition-independent precomputation in multiscale simulations.

Findings

The learned precomputation is stable across different roughness scales.

The method achieves accuracy comparable to classical micro problem solutions.

Computational cost is significantly reduced without sacrificing solution quality.

Abstract

We propose a learned precomputation for the heterogeneous multiscale method (HMM) for rough-wall Stokes flow. A Fourier neural operator is used to approximate local averages over microscopic subsets of the flow, which allows to compute an effective slip length of the fluid away from the roughness. The network is designed to map from the local wall geometry to the Riesz representors for the corresponding local flow averages. With such a parameterisation, the network only depends on the local wall geometry and as such can be trained independent of boundary conditions. We perform a detailed theoretical analysis of the statistical error propagation, and prove that under suitable regularity and scaling assumptions, a bounded training loss leads to a bounded error in the resulting macroscopic flow. We then demonstrate on a family of test problems that the learned precomputation performs stably with respect to the scale of the roughness. The accuracy in the HMM solution for the macroscopic flow is comparable to when the local (micro) problems are solved using a classical approach, while the computational cost of solving the micro problems is significantly reduced.