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

2025-07-18

AI Summary

To address the prohibitively high computational cost of evaluating effective slip lengths for Stokes flow over rough walls within the Heterogeneous Multiscale Method (HMM), this work proposes a Fourier Neural Operator (FNO)-based precomputation framework. The method learns a mapping from local wall geometry to Riesz representers—statistical descriptors of the microscale flow field—enabling efficient, boundary-condition-agnostic slip-length prediction. Its key innovation lies in the first integration of neural operators into HMM’s local representation step, coupled with a theoretical proof establishing bounded propagation of microscale prediction error to the macroscale solution. Numerical experiments demonstrate strong robustness across multiscale roughness patterns: macroscopic flow fields achieve accuracy comparable to full microscale resolution, while microscale computational overhead is drastically reduced.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Social Networks and Social Media: Computational social science
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Estimating effective slip length in rough-wall Stokes flow
• Mapping local wall geometry to flow averages via neural operator
• Reducing computational cost in heterogeneous multiscale method

Innovation

• Fourier neural operator approximates local flow averages
• Network maps wall geometry to Riesz representors
• Learned precomputation reduces micro problems cost