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.

Direct collocation-based dynamic optimization plays an important role in the optimization of equation-based models. With this approach, continuous problems are transcribed into sparse nonlinear programs (NLPs) that can be solved efficiently. The open-source Modelica environment OpenModelica provides an implementation using Radau IIA collocation, but has major limitations, such as the lack of parameter optimization, no adaptive mesh refinement, and no support for higher-order integration schemes. This paper presents (1) a comprehensive reimplementation that addresses these limitations and (2) a novel $h$-method mesh refinement algorithm. Implemented in the custom Python / C++ optimization framework GDOPT, the approach demonstrates significant performance improvements, solving typical problems 2 to 3 times faster than OpenModelica under equivalent conditions. Using the proposed mesh refinement algorithm, the framework correctly identifies non-smooth regions and increases resolution accordingly, requiring only a small increase in computation time. The implementation lays the foundation for a future integration into the OpenModelica toolchain.