Data-driven modelling and scientific machine learning have been responsible for significant advances in determining suitable models to describe data. Within dynamical systems, neural ordinary differential equations (ODEs), where the system equations are set to be governed by a neural network, have become a popular tool for this challenge in recent years. However, less emphasis has been placed on systems that are only partially-observed. In this work, we employ a hybrid neural ODE structure, where the system equations are governed by a combination of a neural network and domain-specific knowledge, together with symbolic regression (SR), to learn governing equations of partially-observed dynamical systems. We test this approach on two case studies: A 3-dimensional model of the Lotka-Volterra system and a 5-dimensional model of the Lorenz system. We demonstrate that the method is capable of successfully learning the true underlying governing equations of unobserved states within these systems, with robustness to measurement noise.

翻译：数据驱动建模与科学机器学习在确定适合描述数据的模型方面取得了重大进展。在动力系统中，神经常微分方程（ODE）——其中系统方程由神经网络支配——近年来已成为应对这一挑战的流行工具。然而，对于仅部分可观测的系统，相关研究较少。本文采用混合神经ODE结构，将神经网络与领域特定知识相结合来支配系统方程，并借助符号回归（SR）学习部分可观测动力系统的支配方程。我们通过两个案例研究验证该方法：一个三维的洛特卡-沃尔泰拉系统模型和一个五维的洛伦兹系统模型。结果表明，该方法能够成功学习这些系统中未观测状态的真实潜在支配方程，并对测量噪声具有鲁棒性。