Discovering sparse interpretable dynamics from partial observations

Identifying the governing equations of a nonlinear dynamical system is key to both understanding the physical features of the system and constructing an accurate model of the dynamics that generalizes well beyond the available data. Achieving this kind of interpretable system identification is even more difficult for partially observed systems. We propose a machine learning framework for discovering the governing equations of a dynamical system using only partial observations, combining an encoder for state reconstruction with a sparse symbolic model. The entire architecture is trained end-to-end by matching the higher-order symbolic time derivatives of the sparse symbolic model with finite difference estimates from the data. Our tests show that this method can successfully reconstruct the full system state and identify the equations of motion governing the underlying dynamics for a variety of ordinary differential equation (ODE) and partial differential equation (PDE) systems. Nonlinear dynamical systems are ubiquitous in nature and play an essential role in science, from providing models for the weather forecast to describing the chaotic behavior of plasma in nuclear reactors. This paper introduces an artificial intelligence framework that can learn the correct equations of motion for nonlinear systems from incomplete data, and opens up the door to applying interpretable machine learning techniques on a wide range of applications in the field of nonlinear dynamics.