用非线性规划方法高效训练含物理约束的神经微分方程，提升精度与速度。
Efficient Training of Physics-enhanced Neural ODEs via Direct Collocation and Nonlinear Programming
- 将训练过程转化为非线性规划问题，直接优化网络参数与状态轨迹
- 在车辆模型和范德波尔振子上实现更优精度与更快训练速度
- 支持小网络高泛化，适合需物理一致性建模的工程场景
我们提出一种新方法训练含物理约束的神经微分方程（PeN-ODEs），将训练过程表述为动态优化问题。通过高阶隐式龙格-库塔法结合翻转勒让德-高斯-拉多-欧点进行全模型离散，得到大规模非线性规划（NLP）问题，由Ipopt等先进求解器高效求解。该框架可同时优化网络参数与状态轨迹，克服传统基于微分方程求解器训练在稳定性、运行时间与精度上的局限。相较于近期基于直接配点的神经微分方程方法，本工作拓展至PeN-ODEs，引入物理约束，并提供定制化、并行化的开源实现。在四分之一车辆模型与范德波尔振子上的基准测试表明，该方法在更小网络下仍具更高精度、更快训练速度与更强泛化能力。此外，计划集成至OpenModelica，以支持神经微分代数方程的便捷训练。
原文摘要 · Abstract (English)
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.
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