In order to make data-driven models of physical systems interpretable and reliable, it is essential to include prior physical knowledge in the modeling framework. Hamiltonian Neural Networks (HNNs) implement Hamiltonian theory in deep learning and form a comprehensive framework for modeling autonomous energy-conservative systems. Despite being suitable to estimate a wide range of physical system behavior from data, classical HNNs are restricted to systems without inputs and require noiseless state measurements and information on the derivative of the state to be available. To address these challenges, this paper introduces an Output Error Hamiltonian Neural Network (OE-HNN) modeling approach to address the modeling of physical systems with inputs and noisy state measurements. Furthermore, it does not require the state derivatives to be known. Instead, the OE-HNN utilizes an ODE-solver embedded in the training process, which enables the OE-HNN to learn the dynamics from noisy state measurements. In addition, extending HNNs based on the generalized Hamiltonian theory enables to include external inputs into the framework which are important for engineering applications. We demonstrate via simulation examples that the proposed OE-HNNs results in superior modeling performance compared to classical HNNs.

翻译：为使物理系统的数据驱动模型具有可解释性和可靠性，在建模框架中融入先验物理知识至关重要。哈密顿神经网络将哈密顿理论融入深度学习，为自主能量守恒系统建模提供了综合性框架。尽管经典哈密顿神经网络适用于从数据中估计多种物理系统行为，但其局限于无输入系统，且要求状态测量值无噪声并需获取状态导数信息。为解决上述挑战，本文提出一种输出误差哈密顿神经网络建模方法，用于处理含输入及噪声状态测量的物理系统建模问题。该方法无需已知状态导数，而是在训练过程中嵌入常微分方程求解器，使OE-HNN能够从含噪声状态测量中学习系统动力学。此外，基于广义哈密顿理论扩展HNN框架，可将外部输入纳入其中，这对工程应用具有重要意义。仿真实验表明，与经典HNN相比，本文提出的OE-HNN具有更优的建模性能。