Bayesian model selection for variable-coefficient partial differential equation discovery

Data-driven discovery of partial differential equations (PDEs) has emerged as a promising approach for identifying underlying physics when domain knowledge about observed data is limited. Despite recent progress, the identification of governing equations with variable coefficients using conventional information criteria remains challenging, as the criteria tend to select overly complex PDEs, especially in noisy situations. In this paper, we introduce an extension of the uncertainty-penalized Bayesian information criterion (UBIC) to solve variable-coefficient PDE discovery problems efficiently, requiring no computationally expensive PDE simulations. Our extended UBIC uses quantified PDE uncertainty, accumulated across temporal or spatial points, to overcome the overfitting problem in model selection. We also extend the exact Bayesian model evidence to quickly reflect the best model most likely to represent the true governing PDE. Numerical experiments on canonical PDEs reveal that our extensions to the UBIC and Bayesian model evidence correctly identify the governing PDEs, along with their varying coefficients, even in the presence of noise. • Bayesian model selection enables data-driven discovery of variable-coefficient partial differential equations (PDEs). • We extend the uncertainty-penalized BIC (UBIC) and Bayesian model evidence for variable-coefficient PDE discovery. • Extended UBIC and Bayesian model evidence enable PDE discovery by uncertainty quantification and prior tuning, respectively.