Bi-level Identification of Governing Equations for Nonlinear Physical Systems

The identification of governing equations from data for nonlinear physical systems has been a long-standing challenge in scientific discovery. However, the inherently ill-posed nature of this inverse problem often leads to false discoveries that overfit certain datasets without capturing the true dynamics of the studied system. To overcome the challenge, we propose a Bi-level Identification of Equations (BILLIE) framework in this work to simultaneously perform equation discovery and validation in two hierarchical objectives of a bi-level optimization. BILLIE is effectively solved using policy gradient techniques from reinforcement learning and hence offers the possibility of identifying very challenging physical systems where the existing approaches fail. Our experimental results on the Navier-Stokes equation, the Burgers' equation, and the three-body system demonstrate BILLIE's dominant performance over existing methods. Moreover, we apply BILLIE to the task of discovering the RNA and protein velocity equations from real-world single-cell sequencing. For the first time, we can quantitatively discover these prominent velocity equations in a data-driven manner, which marks a departure from previous reliance on experts' empirical intuitions to hypothesize these equations. Notably, the equations identified by BILLIE surpass empirical equations in accurately characterizing the future states of differentiated cells. This demonstrates BILLIE's strong potential to contribute to the discovery of fundamental physical rules underlying diverse scientific phenomena.