Efficient Training of Physics-enhanced Neural ODEs via Direct Collocation and Nonlinear Programming

We propose a novel approach for training Physics-enhanced Neural ODEs (PeNODEs) 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 PeNODEs, 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, and generalization with smaller networks compared to other training techniques. We also outline a planned integration into OpenModelica to enable accessible training of Neural DAEs.

@article{langenkamp2025_2505.03552,
  title={ Efficient Training of Physics-enhanced Neural ODEs via Direct Collocation and Nonlinear Programming },
  author={ Linus Langenkamp and Philip Hannebohm and Bernhard Bachmann },
  journal={arXiv preprint arXiv:2505.03552},
  year={ 2025 }
}