Parsimonious structure-exploiting deep neural network surrogates for Bayesian inverse problems

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Advanced Numerical Methods for Scientific Computing
Omar Ghattas
Oden Institute for Computational Science & Engineering, The University of Texas at Austin (USA)
Thursday 12th November 2020
Online seminar:
In an inverse problem, one seeks to infer unknown parameters or parameter fields from measurements or observations of the state of a natural or engineered system. Such problems are fundamental to many fields of science and engineering: often available models possess unknown or uncertain input parameters that must be inferred from experimental or observational data. The Bayesian framework for inverse problems accounts for uncertainty in the inferred parameters stemming from uncertainties in the observational data, the model, and any prior knowledge. Bayesian inverse problems (BIPs) governed by large-scale complex models in high parameter dimensions (such as nonlinear PDEs with uncertain infinite dimensional parameter fields) quickly become prohibitive, since the forward model must be solved numerous times---as many as millions---to characterize the uncertainty in the parameters. Efficient evaluation of the parameter-to-observable (p2o) map, definedby solution of the forward model, is the key to making BIPs tractable. Surrogate approximations of p2o maps have the potential to greatly accelerate BIP, provided that the p2o map can be accurately approximated using (far) fewer forward model solves than would be required for solving the BIP using the full p2o map. Unfortunately, constructing such surrogates presents significant challenges when the parameter dimension is high and the forward model is expensive. Deep neural networks (DNNs) have emerged as leading contenders for overcoming these challenges. We demonstrate that black box application of DNNs for problems with infinite dimensional parameter fields leads to poor results, particularly in the common situation when training data are limited due to the expense of the model. However, by constructing a network architecture that is adapted to the geometry and intrinsic low-dimensionality of the p2o map as revealed through adjoint PDEs, one can construct a "parsimonious" DNN surrogate with superior approximation properties with only limited training data. Application to an inverse problem in Antarctic ice sheet flow is discussed. This work is joint with Tom O'Leary-Roseberry, Peng Chen, Umberto Villa, and Nick Alger. Contact:
Dr. Omar Ghattas is a Professor of Geological Sciences and Mechanical Engineering at the University of Texas at Austin. He is also the Director of the Center for Computational Geosciences and Optimization in the Oden Institute for Computational Engineering and Sciences and holds the John A. and Katherine G. Jackson Chair in Computational Geosciences. He is a member of the faculty in the Computational Science, Engineering, and Mathematics (CSEM) interdisciplinary PhD program in the Oden Institute, and holds courtesy appointments in Computer Science and Biomedical Engineering. He has general research interests in forward and inverse modeling, optimization, and uncertainty quantification of large-scale complex mechanical, geological, and biological systems. With collaborators, he received the ACM Gordon Bell Prize in 2003 (for Special Achievement) and again in 2015 (for Scalability), and was a finalist for the 2008, 2010, and 2012 Bell Prizes. He received the 2019 SIAM Computational Science & Engineering Best Paper Prize, and the 2019 SIAM Geosciences Career Prize. He is a Fellow of the Society for Industrial and Applied Mathematics (SIAM).