# ERE Seminar: Nicholas Zabaras, PhD, University of Notre Dame β Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data

- When:
- Monday, Feb 11, 2019 12:30 PM
- Where:
- Room 104, Green Earth Sciences Building, 367 Panama Street, Stanford
- More Info:
- ERE Seminar: Nicholas Zabaras, PhD, University of Notre Dame β Physics-Constβ¦
- Audience:
- General Public, Faculty/Staff, Students, Alumni/Friends, Members
- Sponsor:
- Energy Resources Engineering

**Title**Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data

**Nicholas Zabaras, PhD**

Viola D. Hank Professor of Computational Science and Engineering**Abstract**Surrogate modeling and uncertainty quantification tasks for systems governed by PDEs are most often considered as supervised learning problems where input π and output π data pairs are used for training. A wide array of deterministic and Bayesian regression models have been proposed and, more recently, Deep Learning approaches have also been employed. The construction of such emulators is by definition a Small Data problem which poses challenges to modern tools such as Deep Learning approaches that have been developed to operate in a 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 without labeled data (i.e. 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 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. The methodology is posed as a minimization problem of the reverse Kullback-Leibler (KL) divergence π·πΎπΏ(π(π)ππ½(π|π)||π(π)ππ½(π|π)) between the approximate joint density π(π)ππ½(π|π) and the reference joint density π(π)ππ½(π|π). The generalization capability of these models for predictions beyond those corresponding to the input training data is considered. Quantification and interpretation of the predictive uncertainty is provided for a number of problems that include flows in random heterogeneous media

**Bio**Prof. Nicholas Zabaras joined Notre Dame in 2016 as the Viola D. Hank Professor of Computational Science and Engineering after serving as Uncertainty Quantification Chair and founding director of the βWarwick Centre for Predictive Modeling (WCPM)β at the University of Warwick. He is the Director of the interdisciplinary University of Notre Dame βCenter for Informatics and Computational Science (CICS)β that aims to bridge the areas of data-sciences, scientific computing and uncertainty quantification for complex multiscale/multiphysics problems in science and engineering. He was also the Hans Fisher Senior Fellow with the Institute for Advanced Study at the Technical University of Munich where he is currently serving as "TUM Ambassador". He is also an Honorary Professor at the Dept. of Mathematics at the University of Hong Kong. Prior to this, he spent 23 years serving in all academic ranks of the faculty at Cornell University where he was the director of the βMaterials Process Design and Control Laboratory (MPDC)β. He received his Ph.D. in Theoretical and Applied Mechanics from Cornell, after which he started his academic career at the faculty of the University of Minnesota. Professor Zabaras' research focuses on the integration of computational mathematics, statistics, and scientific computing for the predictive modeling of complex systems.