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Mathematics Colloquium

Guang Lin
Purdue University

Title:Bayesian Data-driven Discovery of Physical Laws in a Heterogeneous Environment from Noisy Data
Date: Friday, March 22, 2024
Place and Time: Love 101, 3:05-3:55 pm

Abstract. The discovery of Partial Differential Equations (PDEs) is an essential task for applied science and engineering. However, data-driven discovery of PDEs is generally challenging, primarily stemming from the sensitivity of the discovered equation to noise and the complexities of model selection. In this talk, I will introduce an advanced Bayesian sparse learning algorithm for PDE discovery with variable coefficients, predominantly when the coefficients are spatially or temporally dependent. Specifically, we apply threshold Bayesian group Lasso regression with a spike-and-slab prior and leverage a Gibbs sampler for Bayesian posterior estimation of PDE coefficients. This approach not only enhances the robustness of point estimation with valid uncertainty quantification. The capability of this method is illustrated by the discovery of several classical benchmark PDEs with spatially or temporally varying coefficients from solution data obtained from the reference simulations. In the experiments, we show that the proposed approaches are more robust than the baseline methods under noisy environments and provide better model selection criteria along the regularization path.