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

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

URL: http://arxiv.org/abs/2507.13902v1
Date: Fri, 18 Jul 2025 13:29:24 GMT
Title: Deep Micro Solvers for Rough-Wall Stokes Flow in a Heterogeneous Multiscale Method
Authors: Emanuel Ström, Anna-Karin Tornberg, Ozan Öktem,
Abstract summary: We propose a learned precomputation for rough-wall Stokes flow.<n>A network is designed to map from the local wall geometry to the Riesz representors for the corresponding local flow averages.<n>The accuracy in the HMM solution for the macroscopic flow is comparable to when the local (micro) problems are solved using a classical approach.
Score: 1.747820331822631
License: http://creativecommons.org/licenses/by/4.0/
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.

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