Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems

Kurzfassung

As the repeated simulation and optimization of technical systems described by parametric partial differential equations (PDEs) is expensive, we combine neural networks, being known for their good approximation properties, with the classic finite element method (FEM) to obtain a cheap-to-evaluate surrogate model. However, in case of the saddle-point problems arising from the Stokes and Navier-Stokes flows considered in this work, the FEM residual used to train the neural network is highly ill-conditioned and optimizing the model parameters becomes hard. By analogy to the linear case, we propose preconditioning of the loss function and observe improved model accuracy with significantly less training effort, despite the nonlinearity introduced by the neural network. After training, we show the successful application of the FENN to a related inverse problem at moderate Reynolds numbers. Unfortunately, the used preconditioner becomes ineffective at higher Reynolds numbers. We identify the reasons for that, point out the properties a preconditioner should have to be optimal in our setting, and eventually state why it is hard to find such a preconditioner.