Nicholas Zabaras, Center for Informatics and Computational Science, University of Notre Dame
601 Pao Yue-Kong Library
We are interested in the development of surrogate models for uncertainty quantification and propagation in problems governed by stochastic PDEs using a deep convolutional encoder-decoder network in a similar fashion to approaches used in deep learning for image to image regression tasks. To evaluate the performance of this approach, we consider standard uncertainty quantification benchmark problems including flow in heterogeneous media defined in terms of limited data-driven permeability realizations. The performance of the surrogate model developed is surprisingly very good even though there is no underlying structure shared between the input (permeability) and output (flow/pressure) fields as is often the case in the image-to-image regression models used in computer vision problems. Since normal neural networks are data intensive and cannot provide predictive uncertainty, we propose a Bayesian approach to convolutional neural nets. A recently introduced variational gradient descent algorithm based on Stein’s method is scaled to deep convolutional networks to perform approximate Bayesian inference on millions of uncertain network parameters. This approach achieves state of the art performance in terms of predictive accuracy and uncertainty quantification in comparison to other approaches in Bayesian neural networks as well as techniques such as Gaussian processes and ensemble methods even when the training data size is relatively small. Uncertainty propagation tasks are considered and the predictive output Bayesian statistics are compared to those obtained with Monte Carlo estimates.
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 “Warwick Centre for Predictive Modeling (WCPM)” at the University of Warwick. He was recently appointed 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. He is also serving as the Hans Fisher Senior Fellow with the Institute for Advanced Study at the Technical University of Munich where recently was appointed “TUM Ambassador”. 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 “Materials Process Design and Control Laboratory (MPDC)”. 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. He has been honored with the Wolfson Research Merit Award from the Royal Society, the Michael Tien ‘72 Excellence in Teaching Prize from Cornell University, and the Presidential Young Investigator Award from the National Science Foundation.