We propose a novel approach for training Physics-enhanced Neural ODEs (PeN-ODEs) by expressing the training process as a dynamic optimization problem. The full model, including neural components, is discretized using a high-order implicit Runge-Kutta method with flipped Legendre-Gauss-Radau points, resulting in a large-scale nonlinear program (NLP) efficiently solved by state-of-the-art NLP solvers such as Ipopt. This formulation enables simultaneous optimization of network parameters and state trajectories, addressing key limitations of ODE solver-based training in terms of stability, runtime, and accuracy. Extending on a recent direct collocation-based method for Neural ODEs, we generalize to PeN-ODEs, incorporate physical constraints, and present a custom, parallelized, open-source implementation. Benchmarks on a Quarter Vehicle Model and a Van-der-Pol oscillator demonstrate superior accuracy, speed, generalization with smaller networks compared to other training techniques. We also outline a planned integration into OpenModelica to enable accessible training of Neural DAEs.

The convergence failure of iterative Newton solvers during the initialization of Modelica models is a serious show-stopper, particularly for inexperienced users. This paper presents the implementation in the OpenModelica tool of methods presented by two of the authors in a previous paper, to help diagnosing and resolving these convergence failure by providing ranked lists of potentially critical start attributes that might need to be fixed in order to successfully achieve convergence. The method also provides library developers with useful information about critical nonlinear equations, that could be replaced by equivalent, less nonlinear ones, or approximated by homotopy for more robust initialization.