Data-driven sparse identification of nonlinear dynamical systems using linear multistep methods

Linear multistep methods (LMMs) are popular time discretization schemes for solving the forward problem on differential equations. Recently, LMMs together with deep neural networks have been shown to successfully discover dynamical systems from data. In this work, we propose a class of LMM-based sparse regression approaches for the discovery of nonlinear dynamical systems. The work builds on the sparse identification of nonlinear dynamics (SINDy) framework presented in Brunton et al. (Proc Natl Acad Sci USA 113: 3932-3937, 2016), allowing closed form expression for the governing equations and therefore the resulting data-driven model can give insights into the underlying physics. Compared to the standard SINDy algorithm, the proposed LMM-based SINDy approach allows for more accurate and robust model recovery from data with a wide range of noise levels, without requiring pointwise derivative approximations and conventional noise filtering. Numerical results are presented to demonstrate the effectiveness of the proposed method.