Science and engineering in the aerospace sector often requires the simulation and/or optimization of partial differential equations (PDEs), e.g., of flow or transport problems, and often in parametric/multi-query scenarios. Physics-informed machine learning techniques, such as, e.g., physics-informed neural networks (PINNs), have proven highly useful for various problems in this context over the recent years. In this talk we address a physics-informed approach that is slightly different from classical PINNs: Finite element-informed neural networks (FENNs) combine the best from two worlds, the finite element method (FEM) and neural networks (NNs)—the governing equations are discretized a priori by the theoretically well-understood FEM, and a NN is trained to approximate the parameter-to-solution map by minimizing the residual of the discrete equations, enabling data-free, physics-informed training while retaining the geometric and boundary-condition flexibility of FEM. It turns out that preconditioning the FEM-residuals in the loss function substantially accelerates, or even enables at all, convergence of quasi-Newton training for FENNs, in particular for aerospace-relevant problems with saddle-point or nonlinear structure such as 2D stationary Navier-Stokes equations. We will present and discuss results and examples from our recent paper as well as corresponding ongoing work on further applications at DLR.