Discovering Interpretable Dynamics by Sparsity Promotion on Energy and the Lagrangian

Data-driven modeling frameworks that adopt sparse regression techniques, such as sparse identification of nonlinear dynamics (SINDy) and its modifications, are developed to resolve difficulties in extracting underlying dynamics from experimental data. In contrast to neural-network-based methods, these methods are designed to obtain white-box analytical models. In this work, we incorporate the concept of SINDy and knowledge in the field of classical mechanics to identify interpretable and sparse expressions of total energy and the Lagrangian that shelters the hidden dynamics. Moreover, our method (hereafter referred as Lagrangian-SINDy) is developed to use knowledge of simple systems that form the system being analyzed to ensure the likelihood of correct results and to improve the learning pace. Lagrangian-SINDy is highly accurate in discovering interpretable dynamics via energy-related physical quantities. Its performance is validated with three popular multi-DOF nonlinear dynamical systems, namely the spherical pendulum, double pendulum and cart-pendulum system. Comparisons with other SINDy-based methods are made and Lagrangian-SINDy is found to provide the most compact analytical models.