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

Abstract. We propose a learned precomputation for the heterogeneous multiscale method (HMM) applied to rough-wall Stokes flow. HMM captures the effect of the roughness on the macroscopic flow without needing to fully resolve the geometry, by posing a Navier slip condition on a smooth approximation of the wall. The slip amount is estimated as the ratio between locally averaged shear and streamwise velocity rates, which can in turn be computed by linearizing the flow in microscopic subsets of the boundary layer (micro problem). We formulate an adjoint flow problem based on boundary integral methods that maps from the wall geometry of such micro problems to the Riesz representors of the averaging functionals. For problems whose roughness distribution is known, we propose to parameterize the mapping with a Fourier neural operator. The network can be trained independently of boundary conditions and macroscopic geometry on data generated by the boundary integral method. We perform a detailed probabilistic analysis of the 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—without the need for retraining. The accuracy in the HMM solution for the macroscopic flow is comparable to when the local problems are solved using a classical approach, while the computational cost of solving the micro problems is significantly reduced.