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

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.

Variational Bayesian Approximation of Inverse Problems using Sparse Precision Matrices

“…Firstly, taking advantage of fast implementations of sparse linear algebra routines would further improve the scalability of VB with the structured precision matrix, as proposed in our work. Secondly, casting the inverse problem in a multi-level setting and taking advantage of low-dimensional projections has potential to further improve computational efficiency (Nagel and Sudret, 2016;Ghattas and Willcox, 2021). Thirdly, the results provided in this paper use standard off-the-shelf optimisation routines; further computational improvements may be achieved using customised algorithms.…”

The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running

“…As an initial illustration of the method's practical relevance, we here present an incomplete list of recent applications. In the time-period of May-Sep 2021, we find applications of the SPDE-approach to Gaussian fields in astronomy (Levis et al, 2021), health (Mannseth et al, 2021;Scott, 2021;Moses et al, 2021;Bertozzi-Villa et al, 2021;Moraga et al, 2021;Asri and Benamirouche, 2021), engineering (Zhang et al, 2021), theory (Ghattas and Willcox, 2021;Sanz-Alonso and Yang, 2021a;Lang and Pereira, 2021;Bolin and Wallin, 2021), environmetrics Beloconi et al, 2021;Vandeskog et al, 2021a;Wang and Zuo, 2021;Wright et al, 2021;Gómez-Catasús et al, 2021;Valente and Laurini, 2021b;Bleuel et al, 2021;Florêncio et al, 2021;Valente and Laurini, 2021a;Hough et al, 2021), econometrics (Morales and Laurini, 2021;Maynou et al, 2021), agronomy (Borges da Silva et al, 2021), ecology (Martino et al, 2021;Sicacha-Parada et al, 2021;Williamson et al;Bell et al, 2021;Humphreys et al;Xi et al, 2021;Fecchio et al), urban planning (Li, 2021), imaging (Aquino et al, 2021), modelling of forest fires (Taylor et al; Lindenmayer et al), fisheries (Babyn et al, 2021;van Woesik and Cacciapaglia, 2021;Jarvis et al, 2021;…”

Non-intrusive reduced-order models for parametric partial differential equations via data-driven operator inference

“…Affine Operator Inference for Systems of PDEs. Systems of partial differential equations with affine polynomial structure admit low-dimensional representations similar to (2.7a)-(2.7b) [17]. Consider the system of d partial differential equations where each equation can be written as in (2.1a)-(2.1d):…”

“…Such parametric ROMs are critical for enabling outer-loop applications such as design, inverse problems, optimization, and uncertainty quantification. Furthermore, as highfidelity simulations become increasingly sophisticated and simulation data becomes more available, there is a growing need for non-intrusive model reduction methods, which aim to learn ROMs primarily from simulation data and/or outputs, as opposed to making a direct reduction of the underlying high-fidelity code that produced them [17]. Non-intrusive approaches combine data-driven learning with physics-based modeling in a way that enables both flexibility and robustness.…”