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. In addition, we discuss generalizability and computational cost of the learned mesh motion operators.

翻译：网格退化是流固耦合（FSI）模拟及基于映射方法的形状优化中的关键瓶颈。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依赖于启发式方法，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）。后者虽在大位移场景下表现出良好的数值性能，但计算成本高昂。此外，从连续视角看，网格运动技术的选择在一定程度上具有任意性，且不影响物理相关量。为此，我们探索了受机器学习启发的方案。提出了一种混合PDE-神经网络（NN）方法，其中神经网络用于参数化非线性二阶PDE中的系数，并通过网络架构设计确保非线性PDE解的存在性。同时，我们提出了另一种方法：利用神经网络修正调和延拓，保证边界位移不变。为避免有限元与机器学习软件耦合的技术难点，将整体FSI系统分解为三个子系统，使网格运动方程可在独立步骤求解。通过FSI基准问题评估所学习网格运动技术的性能，并进一步讨论了学习得到的网格运动算子的泛化能力与计算成本。