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Learning physics-based models from data: perspectives from inverse problems and model reduction

Published online by Cambridge University Press:  04 August 2021

Omar Ghattas
Affiliation:
Oden Institute for Computational Engineering & Sciences, Departments of Geological Sciences and Mechanical Engineering, The University of Texas at Austin, TX78712, USA E-mail: omar@oden.utexas.edu
Karen Willcox
Affiliation:
Oden Institute for Computational Engineering & Sciences, Department of Aerospace Engineering & Engineering Mechanics, The University of Texas at Austin, TX78712, USA E-mail: kwillcox@oden.utexas.edu
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Abstract

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This article addresses the inference of physics models from data, from the perspectives of inverse problems and model reduction. These fields develop formulations that integrate data into physics-based models while exploiting the fact that many mathematical models of natural and engineered systems exhibit an intrinsically low-dimensional solution manifold. In inverse problems, we seek to infer uncertain components of the inputs from observations of the outputs, while in model reduction we seek low-dimensional models that explicitly capture the salient features of the input–output map through approximation in a low-dimensional subspace. In both cases, the result is a predictive model that reflects data-driven learning yet deeply embeds the underlying physics, and thus can be used for design, control and decision-making, often with quantified uncertainties. We highlight recent developments in scalable and efficient algorithms for inverse problems and model reduction governed by large-scale models in the form of partial differential equations. Several illustrative applications to large-scale complex problems across different domains of science and engineering are provided.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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