Neural ODEs are a widely used, powerful machine learning technique in particular for physics. However, not every solution is physical in that it is an Euler-Lagrange equation. We present Helmholtz metrics to quantify this resemblance for a given ODE and demonstrate their capabilities on several fundamental systems with noise. We combine them with a second order neural ODE to form a Lagrangian neural ODE, which allows to learn Euler-Lagrange equations in a direct fashion and with zero additional inference cost. We demonstrate that, using only positional data, they can distinguish Lagrangian and non-Lagrangian systems and improve the neural ODE solutions.

翻译：神经常微分方程是一种广泛应用于物理领域的强大机器学习技术。然而，并非所有解都具有物理意义，即满足欧拉-拉格朗日方程。本文提出亥姆霍兹度量方法，用于量化给定常微分方程与拉格朗日体系的相似度，并在多个含噪声基础系统中验证其有效性。通过将亥姆霍兹度量与二阶神经常微分方程结合，构建出拉格朗日神经常微分方程，该模型能以零额外推理成本直接学习欧拉-拉格朗日方程。实验表明，仅使用位置数据时，该模型能有效区分拉格朗日与非拉格朗日系统，并显著提升神经常微分方程的求解精度。