2025-09-09 –, Audi-Midi
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.
Linus Langenkamp is a research associate in the Department of Engineering Sciences and Mathematics at Hochschule Bielefeld (Bielefeld University of Applied Sciences). He earned his Master of Science degree from Hochschule Bielefeld and joined the department in January 2025. His work on the OpenModelica project focuses on dynamic optimization, efficient numerical methods for differential equation systems, and physics-enhanced neural ordinary differential equations.
- Professor of Mathematics and Technical Applications, Bielefeld University of Applied Sciences (HSBI, since 1999)
- Research focus: numerical mathematics, nonlinear optimization, symbolic and numerical methods for large hybrid differential-algebraic systems
- Founding member of the Modelica Association (1996) and Open Source Modelica Consortium (2004)
- Key contributions to the BackEnd and C-Runtime of the OpenModelica Compiler
- Co-Author of the Modelica Petri Net Library and Modelica Neural Network Library
- Research stays at ABB Research Center (Switzerland, USA, Sweden), Linköping University (Sweden), and Politecnico di Milano (Italy)
- Member, Promotionskolleg NRW (since 2022)
- Founding board member, Institute for Data Science Solutions (IDaS), HSBI (since 2022)