George Em Karniadakis | Brown University | United States
PINNs is a fundamentally new approach for developing a data-driven, learning-based framework to predict outcomes of physical and biological systems and to discover hidden physics from noisy data. We will introduce a deep learning approach based on neural networks (NNs) and generative adversarial networks (GANs). Unlike other approaches that rely on big data, here we “learn” from small data by exploiting the information provided by the physical conservation laws, which are used to obtain informative priors or regularize the neural networks. We will present several variations of Bayesian PINNs, and we will make connections between uncertainty quantified by Gauss Process Regression and by Bayesian NNs and PINNs as well as other methods, e.g. Variational Inference and Dropout. We will demonstrate the power of PINNs for several inverse problems in fluid mechanics, solid mechanics and biomedicine, including wake flows, shock tube problems, material characterization, brain aneurysms, etc, where traditional methods fail due to lack of boundary and initial conditions or material properties. We will address the issue of total uncertainty quantification, namely the uncertainty associated with the network separately from the uncertainty in the parameters, data and models.