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 extension operators.

翻译：网格退化是流固耦合（FSI）模拟和基于映射方法的形状优化中的瓶颈问题。在这两种情况下，都需要合适的网格运动技术。通常，这类选择基于启发式方法，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。特别是双调和方程，虽然在大位移情形下表现出良好的数值性能，但其计算成本高昂。此外，从连续视角来看，网格运动技术的选择在一定程度上具有任意性，且不会对物理相关量产生影响。因此，我们考虑受机器学习启发的方法。我们提出了一种混合PDE-NN方法，其中神经网络（NN）作为二阶非线性PDE中系数的参数化。通过选择神经网络架构，我们确保非线性PDE解的存在性。此外，我们还提出了一种方法，即用神经网络校正调和延拓，使得边界位移保持不变。为了避免耦合有限元和机器学习软件的技术困难，我们将整体FSI系统拆分为三个较小的子系统。这样便可以在单独步骤中求解网格运动方程。我们通过将其应用于FSI基准问题来评估所学习的网格运动技术的质量。此外，我们还讨论了学习得到的延拓算子的泛化能力和计算成本。