标题： 粗糙壁面斯托克斯流的深度微分求解器在异质多尺度方法中的应用 显示英文标题

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

摘要： 我们提出了一种用于粗糙壁面Stokes流的异构多尺度方法（HMM）的学习预计算。使用傅里叶神经算子来近似流体微观子集上的局部平均值，从而能够在远离粗糙度的地方计算流体的有效滑移长度。该网络设计为从局部壁面几何结构映射到相应局部流平均值的Riesz表示。通过这种参数化，网络仅依赖于局部壁面几何结构，因此可以独立于边界条件进行训练。我们对统计误差传播进行了详细的理论分析，并证明在适当的正则性和缩放假设下，有界的训练损失会导致宏观流动结果中的有界误差。然后我们在一组测试问题上展示了学习的预计算在粗糙度尺度上的稳定性。HMM求解宏观流动的准确性与使用经典方法求解局部（微观）问题时相当，而求解微观问题的计算成本显著降低。 显示英文摘要

摘要： 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.