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

TL;DR

This paper introduces a Parsimony Enhanced Sparse Bayesian Learning (PeSBL) method for robustly discovering partial differential equations from noisy data, emphasizing model simplicity and sparsity, and extends it to stochastic PDE learning with uncertainty quantification.

Contribution

The paper proposes a novel PeSBL method that promotes model parsimony alongside sparsity, improving PDE discovery accuracy from noisy data and extending to stochastic PDEs with hierarchical Bayesian inference.

Findings

Successfully identified PDEs from data with up to 50% noise.

Extended framework for stochastic PDE learning with uncertainty quantification.

Demonstrated system response prediction and anomaly diagnosis.

Abstract

Robust physics discovery is of great interest for many scientific and engineering fields. Inspired by the principle that a representative model is the one simplest possible, a new model selection criteria 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, the model parameters are updated through Bayesian inference with the 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 where all parameters and modeling error 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 diagnosis. Codes of all demonstrated examples in this study are available on the website: https://github.com/ymlasu.