Group sparse bayesian learning for data-driven discovery of explicit model forms with multiple parametric datasets

Extracting the explicit governing equations of a dynamic system from limited data has attracted increasing attention in the data-driven modeling community. Compared to black-box learning approaches, the sparse regression-based learning method enables discovering an analytical model form from data, which is more appealing due to its white-box nature. However, distilling explicit equations from real-world measurements with data uncertainty is challenging, where many existing methods are less robust. Moreover, it is unclear how to efficiently learn a parametric system from multiple data sets with different parameters. This paper presents a group sparse Bayesian learning approaches to uncover the explicit model forms of a parametric dynamical system with estimated uncertainties. A deep neural network is constructed to improve the calculation of derivatives from noisy measurements. Group sparsity is leveraged to enable synchronous learning from a group of parametric datasets governed by the equations with the same functional form but different parameter settings. The proposed approach has been studied over a few linear/nonlinear ODE systems in explicit and implicit settings. In particular, a simplified parametric model of intracranial dynamics was identified from multiple synthetic datasets with different patient-specific parameters. The numerical results demonstrated the effectiveness of the proposed approach and the merit of synchronous learning from multiple datasets in a group sparsifying Bayesian setting.