  We propose a new parameter-adaptive uncertainty-penalized Bayesian
information criterion (UBIC) to prioritize the parsimonious partial
differential equation (PDE) that sufficiently governs noisy spatial-temporal
observed data with few reliable terms. Since the naive use of the BIC for model
selection has been known to yield an undesirable overfitted PDE, the UBIC
penalizes the found PDE not only by its complexity but also the quantified
uncertainty, derived from the model supports' coefficient of variation in a
probabilistic view. We also introduce physics-informed neural network learning
as a simulation-based approach to further validate the selected PDE flexibly
against the other discovered PDE. Numerical results affirm the successful
application of the UBIC in identifying the true governing PDE. Additionally, we
reveal an interesting effect of denoising the observed data on improving the
trade-off between the BIC score and model complexity. Code is available at
https://github.com/Pongpisit-Thanasutives/UBIC.
