Data-driven system identification for piecewise dynamical systems via Sparse Identification Neural Ordinary Differential Equations

Traditional approaches to data-driven system identification typically focus on global information while neglecting the influence of local information. This oversight poses challenges to the model identification for piecewise dynamical systems. This paper introduces Sparse Identification Neural Ordinary Differential Equations (SI-NODEs), a novel framework for data-driven identification for piecewise dynamical systems. Neural Ordinary Differential Equations (NODEs) are initially employed to extract localized information. Utilizing reverse thinking, implicit NODEs are applied to identify points of imperfect fit which correspond to the piecewise points. Subsequently, the sparse identification or explicit NODEs are used to identify the differential equations for each data segment, thereby obtaining global information. This framework synergizes the fitting capabilities of implicit NODEs with the sparse regression or the flexible neural modeling of explicit NODEs. The uniqueness and convergence are rigorously proved about piecewise point identification. Compared to classical approaches such as Sparse Identification of Nonlinear Dynamics (SINDy) and Neural Ordinary Differential Equations (NODEs), the effectiveness and superiority of the proposed SI-NODEs framework are demonstrated by piecewise linear system, piecewise Lorenz system, and piecewise nonlinear system contains two-stroke and four-stroke oscillators, respectively. Robustness under noisy conditions is also verified using data with different noise levels.