Data-driven reduced order modeling for time-dependent problems

By Mengwu Guo, Jan S. Hesthaven. Computer Methods in Applied Mechanics and Engineering, 2019

258 citations recorded by the source corpus.

Abstract A data-driven reduced basis (RB) method for parametrized time-dependent problems is proposed. This method requires the offline preparation of a database comprising the time history of the full-order solutions at parameter locations. Based on the full-order data, a reduced basis is constructed by the proper orthogonal decomposition (POD), and the maps between the time/parameter values and the projection coefficients onto the RB are approximated as a regression model. With a natural tensor grid between the time and the parameters in the database, a singular-value decomposition (SVD) is used to extract the principal components in the data of projection coefficients. The regression functions are represented as the linear combinations of several tensor products of two Gaussian processes, one of time and the other of parameters. During the online stage, the solutions at new time/parameter locations in the domain of interest can be recovered rapidly as outputs from the regression models. Featuring a non-intrusive nature and the complete decoupling of the offline and online stages, the proposed approach provides a reliable and efficient tool for approximating parametrized time-dependent problems, and its effectiveness is illustrated by non-trivial numerical examples.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/fec02dab4f12cdd6133f08710c3911bdf3e6719a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition (2021, methodology): “…in a data-driven and less intrusive way (avoiding, e.g., the costs entailed by the projection stage of projection-based ROMs); for instance, in [25, 26, 27, 28, 29] the use of ANNs or Gaussian Processes (GPs) regression models has been proposed to approximate the mapping from the input parameters…”
Integrating Physics-Based Modeling with Machine Learning: A Survey (2020, unknown): “Reduced Order Models [244] [179] [101] [273] [225] [286] [173] [188] [10] [148] [125] [262] [171] [74] [188] [190] [125] [223] [224] [257] [101] [190] [57]”
Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems (2020, unknown): “4) [209] [16] [167] [138] [86] [210] [138] [241] [242] [278] [115] [210] [67]”
A Comprehensive Deep Learning-Based Approach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs (2020, methodology): “For instance, in [26,27,29,59] the solution of a (nonlinear, timedependent) ROM for any new parameter value has been replaced by the evaluation of ANNbased regression models; similar ideas can be found, e.”
PDEBENCH: An Extensive Benchmark for Scientific Machine Learning (2022, methodology): “While complex methods such as Bayesian optimisation [38, 50, 42] or reduced order modelling [16] are in part an attempt to circumvent this lack of differentiability, gradients for neural networks are readily available and efficient.”