Abstract
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.
Highlights
We discuss its mathematical properties as a general framework for deriving surrogate models and present reduced models derived using proper orthogonal decomposition combined with Galerkin projection for time-dependent and parametrized systems
We present computation of the lowdimensional subspace using the proper orthogonal decomposition (POD) method
We first discuss the use of hyper-reduction methods such as discrete empirical interpolation method (DEIM) to approximate non-linear terms
Summary
Many PDE models of natural and engineered systems exhibit an intrinsically lowdimensional solution manifold. To address the challenges of learning models from data in the large-scale setting of PDEs, we must exploit the low-dimensional structure of the map from inputs to outputs. Here in Part 2 we pose the learning-from-data problem as an inverse problem governed by the forward (PDE) problem, and exploit the low-dimensionality of the parameter-to-observable map to efficiently and scalably recover the informed components of the model at a cost – measured in forward model solutions – that scales independent of the parameter or data dimensions. The statistical inverse problem is addressed in the Bayesian framework in Section 4; low-rank approximation of the Hessian of the log likelihood exploits the low-dimensionality, resulting in dimension-independence We begin Part 2 with a discussion of ill-posedness, and several model elliptic, parabolic and hyperbolic inverse problems intended to illustrate the underlying concepts
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