  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.
