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

arXiv: https://arxiv.org/abs/2505.03552 · HTML: https://mlsanity.com/paper/2505.03552/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 38 references that are in the map.
- First posted on arXiv 6 May 2025.

## Builds on (most cited first, 7 of 7)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Maximizing information from chemical engineering data sets: Applications to machine learning](https://mlsanity.com/paper/2201.10035/) (2022)
- [Accelerating Neural ODEs Using Model Order Reduction](https://mlsanity.com/paper/2105.14070/) (2021)
- [A Simultaneous Approach for Training Neural Differential-Algebraic Systems of Equations](https://mlsanity.com/paper/2504.04665/) (2025)
- [NeuralFMU: Presenting a workflow for integrating hybrid NeuralODEs into real world applications](https://mlsanity.com/paper/2209.03933/) (2022)
- [Training Neural ODEs Using Fully Discretized Simultaneous Optimization](https://mlsanity.com/paper/2502.15642/) (2025)

## Abstract (arXiv)

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.