Sparse coefficient distribution identification of control equations based on high-order integration

Identifying the governing equations of physical systems from datasets has long been a persistent challenge across various scientific disciplines. The sparse identification of nonlinear dynamics method typically relies on accurate derivative estimation, which makes it prone to failure in data-scarce and noisy environments. To enhance the robustness of identification, this paper proposes a novel data-driven approach, DIRK-BSINDy, that combines high-order integration (Gauss implicit Runge-Kutta method) with Bayesian sparse identification, and leverages deep neural networks to assist in solving the stage values of high-order integration, thereby improving computational efficiency. The proposed method obtains the posterior distribution of sparse coefficients in the governing equations via Bayesian inference and constructs a loss function by integrating forward and backward predictions to enhance stability. Furthermore, owing to the weak step-size constraint inherent in high-order integration, DIRK-BSINDy exhibits remarkable robustness even in scenarios with scarce data and strong noise interference, demonstrating distinct advantages over traditional methods.