Physics-Constrained Learning of Multiscale and Dynamical PDE Systems


Nicholas Zabaras, University of Notre Dame


2019.07.02 16:00-17:00


Room 306, No. 5, Science Building


Surrogate modeling and uncertainty quantification tasks for systems governed by PDEs are most often considered as supervised learning problems where input and output data pairs are used for training. The construction of such emulators is by definition a Small Data problem which poses challenges in Deep Learning approaches that have been developed to operate in the Big Data regime. Even in cases where such models have been shown to have good predictive capability in high-input dimensions, they fail to address constraints in the data implied by the PDE model. We will present a methodology in this direction that incorporates the governing equations of the physical model in the loss/likelihood functions. The resulting physics-constrained, deep learning models operate by employing only training input data and provide predictive responses that obey the constraints of the problem at hand just as well as typical deterministic models. This work employs a convolutional encoder-decoder convolutional neural network approach as well as a conditional flow-based generative model for the solution of PDEs, surrogate model construction for PDEs, and uncertainty quantification tasks. We will highlight how these techniques can be extended to learn Dynamics of complex PDE systems including turbulence models.

The above methods go beyond PDE systems and are directly applicable to discrete atomistic models in Chemistry, Biology and elsewhere. We will briefly discuss a data-free approach to coarse-graining in atomistic simulations using generative models based on variational auto-encoders.


Prof. Nicholas Zabaras joined Notre Dame in 2016 as the Viola D. Hank Professor of Computational Science and Engineering. He is the Director of the interdisciplinary University of Notre Dame “Center for Informatics and Computational Science (CICS)” that aims to bridge the areas of data-sciences, scientific computing and uncertainty quantification for complex multiscale/multiphysics problems in science and engineering. Among the various appointments, Prof. Zabaras was until recently the Hans Fisher Senior Fellow with the Institute for Advanced Study at the Technical University of Munich where he currently holds the position of “TUM Ambassador”. He served for nearly 23 years at all academic ranks on the faculty of Cornell University.