Mesh degeneration is a bottleneck for fluid-structure interaction (FSI) simulations and for shape optimization via the method of mappings. In both cases, an appropriate mesh motion technique is required. The choice is typically based on heuristics, e.g., the solution operators of partial differential equations (PDE), such as the Laplace or biharmonic equation. Especially the latter, which shows good numerical performance for large displacements, is expensive. Moreover, from a continuous perspective, choosing the mesh motion technique is to a certain extent arbitrary and has no influence on the physically relevant quantities. Therefore, we consider approaches inspired by machine learning. We present a hybrid PDE-NN approach, where the neural network (NN) serves as parameterization of a coefficient in a second order nonlinear PDE. We ensure existence of solutions for the nonlinear PDE by the choice of the neural network architecture. Moreover, we present an approach where a neural network corrects the harmonic extension such that the boundary displacement is not changed. In order to avoid technical difficulties in coupling finite element and machine learning software, we work with a splitting of the monolithic FSI system into three smaller subsystems. This allows to solve the mesh motion equation in a separate step. We assess the quality of the learned mesh motion technique by applying it to a FSI benchmark problem.

翻译：网格退化是流固耦合（FSI）模拟及通过映射方法进行形状优化的瓶颈。在这两种情况下，均需采用合适的网格运动技术，而此类技术的选择通常基于启发式方法，例如偏微分方程（PDE）的解算子（如拉普拉斯方程或双调和方程）。尤其是双调和方程在处理大位移时虽数值性能优异，但计算成本高昂。此外，从连续视角出发，网格运动技术的选择在某种程度上具有任意性，且不影响物理相关量。因此，我们考虑受机器学习启发的途径。我们提出一种混合PDE-NN方法，其中神经网络（NN）用于参数化二阶非线性偏微分方程中的系数。通过神经网络架构的设计，我们确保该非线性PDE解的存在性。同时，我们提出另一种方法：让神经网络对调和延拓进行修正，且不改变边界位移。为避免有限元与机器学习软件耦合的技术困难，我们将整体FSI系统分解为三个较小的子系统，从而可在独立步骤中求解网格运动方程。通过将所学的网格运动技术应用于FSI基准测试问题，我们评估了其质量。