Adaptive Uncertainty-Penalized Model Selection for Data-Driven PDE Discovery

We propose a new parameter-adaptive uncertainty-penalized Bayesian information criterion (UBIC) to discover the stable governing partial differential equation (PDE) composed of a few important terms. Since the naive use of the BIC for model selection yields an overfitted PDE, the UBIC penalizes the found PDE not only by its complexity but also by its quantified uncertainty. Representing the PDE as the best subset of a few candidate terms, we use Bayesian regression to compute the coefficient of variation (CV) of the posterior PDE coefficients. The PDE uncertainty is then derived from the obtained CV. The UBIC follows the premise that the true PDE shows relatively lower uncertainty when compared with overfitted PDEs. Thus, the quantified uncertainty is an effective indicator for identifying the true PDE. We also introduce physics-informed neural network learning as a simulation-based approach to further validate the UBIC-selected PDE against the other potential PDE. Numerical results confirm the successful application of the UBIC for data-driven PDE discovery from noisy spatio-temporal data. Additionally, we reveal a positive effect of denoising the observed data on improving the trade-off between the BIC score and model complexity.