A new certified hierarchical and adaptive RB-ML-ROM surrogate model for parametrized PDEs

Haasdonk B, Kleikamp H, Ohlberger M, Schindler F, Wenzel T

Research article (journal) | Peer reviewed

Abstract

We present a new surrogate modeling technique for efficient approximation of input-output maps governed by parametrized PDEs. The model is hierarchical as it is built on a full order model (FOM), reduced order model (ROM) and machine-learning (ML) model chain. The model is adaptive in the sense that the ROM and ML model are adapted on-the-fly during a sequence of parametric requests to the model. To allow for a certification of the model hierarchy, as well as to control the adaptation process, we employ rigorous a posteriori error estimates for the ROM and ML models. In particular, we provide an example of an ML-based model that allows for rigorous analytical quality statements. We demonstrate the efficiency of the modeling chain on a Monte Carlo and a parameter-optimization example. Here, the ROM is instantiated by Reduced Basis Methods and the ML model is given by a neural network or a VKOGA kernel model.

Details about the publication

Journal: SIAM Journal on Scientific Computing (SIAM J. Sci. Comput.)

Volume: 45

Issue: 3

Page range: A1039-1065

Status: Published

Release year: 2023 (11/05/2023)

Language in which the publication is written: English

DOI: 10.1137/22M1493318

Link to the full text: https://epubs.siam.org/doi/10.1137/22M1493318

Keywords: model order reduction; machine learning; reduced basis methods; error estimation; neural networks; kernel methods

Authors from the University of Münster

Kleikamp, Hendrik	Professorship of Applied Mathematics, especially Numerics (Prof. Ohlberger)
Ohlberger, Mario	Professorship of Applied Mathematics, especially Numerics (Prof. Ohlberger) Center for Data Science and Complexity (CDSC) Center for Multiscale Theory and Computation (CMTC)
Schindler, Felix Tobias	Professorship of Applied Mathematics, especially Numerics (Prof. Ohlberger)

Projects the publication originates from

ML-MORE: Machine learning and model order reduction to predict the efficiency of catalytic filters. Subproject 1: Model Order Reduction (ML-MORE) Duration: 01/04/2020 - 31/12/2023 Funded by: Federal Ministry of Research, Technology and Space Type of project: Participation in federally funded joint project
Cluster of Excellence 2044 - Mathematics Münster: Dynamics – Geometry – Structure (EXC 2044) Duration: 01/01/2019 - 31/12/2025 \| 1st Funding period Funded by: DFG - Cluster of Excellence Type of project: Main DFG-project hosted at University of Münster
EXC 2044 - C4: Geometry-based modelling, approximation, and reduction Duration: 01/01/2019 - 31/12/2025 \| 1st Funding period Funded by: DFG - Cluster of Excellence Type of project: Subproject in DFG-joint project hosted at University of Münster

Talks on the publication

A certified and adaptive RB-ML-ROM surrogate approach for parametrized PDEs Kleikamp, Hendrik (21/07/2022) YMMOR - Young Mathematicians in Model Order Reduction, Münster Type of talk: scientific Talk
A certified and adaptive RB-ML-ROM surrogate approach for parametrized PDEs Kleikamp, Hendrik (16/06/2022) HCM Workshop: Synergies between Data Science and PDE Analysis, Bonn Type of talk: scientific Talk

Doctorates the publication originates from

Parametrized optimal control and transport-dominated problems: Reduced basis methods, nonlinear reduction strategies and data driven surrogates
Candidate: Kleikamp, Hendrik | Supervisors: Ohlberger, Mario | Reviewers: Ohlberger, Mario; Breiten, Tobias
Period of time: 01/01/2021 - 19/12/2024
Doctoral examination procedure finished at: Doctoral examination procedure at University of Münster