Learning governing equations of unobserved states in dynamical systems

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. • A hybrid neural ODE structure together with symbolic regression is used to recover governing equations of unobserved state variables. • Ensembles of the framework are employed to increase robustness to measurement noise. • Applicability is demonstrated in ecological (Lotka–Volterra) and chaotic (Lorenz) systems. • The framework is adapted to handle additional system uncertainty and to quantify confidence in the learned models.