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. In many instances, this problem is further compounded by a lack of available data and only partial observations of the system state, e.g. forecasting fluid flow driven by unknown sources or predicting optical signal propagation without phase measurements. We propose a machine learning framework for discovering these governing equations 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 ODE and PDE systems.