Parsimony-Enhanced Sparse Bayesian Learning for Robust Discovery of Partial Differential Equations

Robust physics discovery is of great interest for many scientific and engineering fields. Inspired by the principle that a representative model is the simplest one among all possible models, a new model selection criterion considering both model's Parsimony and Sparsity is proposed. A Parsimony-Enhanced Sparse Bayesian Learning (PESBL) method is developed for discovering the governing Partial Differential Equations (PDEs) of nonlinear dynamical systems. Compared with the conventional Sparse Bayesian Learning (SBL) method, the PESBL method promotes parsimony of the learned model in addition to its sparsity. In this method, the parsimony of model terms is evaluated using their locations in the prescribed candidate library, for the first time, considering the increased complexity with the power of polynomials and the order of spatial derivatives. Subsequently, model parameters are updated through Bayesian inference with raw data. This procedure aims to reduce the error associated with the possible loss of information in data preprocessing and numerical differentiation prior to sparse regression. Results of numerical case studies indicate that the governing PDEs of many canonical dynamical systems can be correctly identified using the proposed PESBL method from highly noisy data (up to 50% in the current study). Next, the proposed methodology is extended for stochastic PDE learning in which both model parameters and modeling errors are considered as random variables. Hierarchical Bayesian Inference (HBI) is integrated with the proposed framework for stochastic PDE learning from a population of observations. Finally, the proposed PESBL is demonstrated for system response prediction with uncertainties and anomaly detection. Codes of all demonstrated examples in this study are available on the website: https://github.com/ymlasu.