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

TL;DR

This work tackles the challenge of reliably discovering governing PDEs from noisy spatio-temporal data by embedding uncertainty into model selection. It introduces the parameter-adaptive uncertainty-penalized Bayesian information criterion (UBIC), which augments the traditional BIC with an uncertainty-based penalty derived from the posterior coefficient covariance, and it uses denoising (regularized KSVD) to improve data quality prior to candidate formation. The approach pairs a weak-form PDE library with best-subset regression and adaptive UBIC tuning, then optionally validates the selected PDE via simulation-based model comparison using physics-informed neural networks. Across Burgers', KdV, KS, NS, RD, and GS PDEs, UBIC consistently recovers the true governing equations, with denoising substantially strengthening the BIC-uncertainty trade-off and PINN validation providing a practical check on predictive fidelity. The work also outlines extensions (G-UBIC, UWAIC) and supplementary tools (spike-and-slab priors, thresholded Bayesian methods) to broaden uncertainty-guided PDE discovery under diverse conditions and datasets.

Abstract

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.

Figures (15)

• Figure 1: Schematic diagram of the Burgers' PDE discovery example incorporating the UBIC for the model selection
• Figure 2: We plot the BIC, uncertainty $\mathrm{U}^{k}$ and UBIC with tuned $\lambda_{\mathrm{U}} = 10^{\lambda}$ for the model selection in the Burgers, KdV, KS, and NS examples, arranged from left to right. "$\checkmark$" indicates that the UBIC selects the true PDE form.
• Figure 3: Robust adaptive model selection by the UBIC under the extremely noisy scenarios.
• Figure 4: Model selection results for the RD and GS PDEs.
• Figure 5: Successful model selection results by G-BIC with $\Gamma = 1$. In each subfigure, the (re)tuned $\lambda$(s) is/are annotated. It should be noted all the $\mathrm{U}^{k}$s are already given in Figure (\ref{['fig:UBIC_res1']}) and Figure (\ref{['fig:UBIC_res2']}) in the main text.