  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.
