Learning Governing Equations of Unobserved States in Dynamical Systems

TL;DR

This work tackles learning the governing equations for unobserved states in dynamical systems by marrying hybrid neural ODEs with symbolic regression. It demonstrates that replacing unknown terms with neural networks and then extracting explicit equations via SR can recover the true dynamics of unobserved variables in partially observed Lotka-Volterra and Lorenz systems, even under measurement noise. The method leverages an ensemble of models to stabilize predictions and uses SR to obtain interpretable, partially learned models that extend beyond the training range, though extrapolation is limited by short training windows and chaotic behavior. Overall, the approach provides a robust pathway to derive interpretable governing equations from partial observations, with potential future enhancements like Bayesian networks to improve efficiency and uncertainty quantification.

Abstract

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.

Figures (18)

• Figure 1: Temporal evolution of the 3D Lotka-Volterra system with initial condition $[x_0,y_0,z_0]=[0.5,1.0,2.0]$. This synthetic data is used for training and validation.
• Figure 2: Temporal evolution of the 5D Lorenz system with initial condition $[x_0,y_0,z_0,v_0,w_0]=[-8.0,8.0,27.0,0.4,0.5]$. This synthetic data is used for training and validation.
• Figure 3: Prediction of the hybrid neural ODE (shown by the solid curves) against the ground truth data (shown by the scatter points), both before (Figure \ref{['fig:LV3Untrained']}) and after (Figure \ref{['fig:LV3Trained']}) the training process. The ground truth data for the state variable $y$ is faint as this data is not available during training.
• Figure 4: Extrapolation of trained hybrid neural ODE. The predictions are shown by the solid curves and the ground truth is shown by the scatter points. The ground truth for the state variable $y$ is faint as this data is assumed to be unavailable. The blue region to the left of each plot represents the training range.
• Figure 5: Sliding window errors of hybrid neural ODE extrapolations, using three different training ranges. The training ranges of 2.0, 3.25 and 4.5 units of time are abbreviated to TR1, TR2 and TR3, respectively. Only the unobserved state variable $y$ is shown. This comparison is done at each level of noise.