Uncertainty Quantification and Machine Learning of the Physical Laws Hidden Behind the Noisy Data

I will present a new data-driven paradigm on how to quantify the structural uncertainty (model-form uncertainty) and learn the physical laws hidden behind the noisy data in the complex systems governed by partial differential equations. The key idea is to identify the terms in the underlying equations and to approximate the coefficients of the terms with error bars using Bayesian machine learning algorithms using the available noisy measurement. In particular, Bayesian sparse feature selection and parameter estimation are performed. Numerical experiments show the robustness of the learning algorithms with respect to noisy data and size, and it ability to learn various candidate equations with error bars to represent the quantified uncertainty.