BAYESIAN DEEP OPERATOR LEARNING FOR HOMOGENIZED TO FINE-SCALE MAPS FOR MULTISCALE PDE

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

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Detail(s)

Original languageEnglish
Pages (from-to)956-972
Number of pages17
Journal / PublicationMultiscale Modeling & Simulation
Volume22
Issue number3
Online published17 Jul 2024
Publication statusPublished - Sept 2024

Abstract

We present a new framework for computing fine-scale solutions of multiscale partial differential equations (PDEs) using operator learning tools. Obtaining fine-scale solutions of multiscale PDEs can be challenging, but there are many inexpensive computational methods for obtaining coarse-scale solutions. Additionally, in many real-world applications, fine-scale solutions can only be observed at a limited number of locations. In order to obtain approximations or predictions of fine-scale solutions over general regions of interest, we propose to learn the operator mapping from coarse-scale solutions to fine-scale solutions using observations of a limited number of (possible noisy) fine-scale solutions. The approach is to train multi-fidelity homogenization maps using mathematically motivated neural operators. The operator learning framework can efficiently obtain the solution of multiscale PDEs at any arbitrary point, making our proposed framework a mesh-free solver. We verify our results on multiple numerical examples showing that our approach is an efficient mesh-free solver for multiscale PDEs. © 2024 by SIAM.

Research Area(s)

  • neural operator, neural homogenization, multiscale finite element method, dis- cretization invariant, multi-fidelity

Citation Format(s)

BAYESIAN DEEP OPERATOR LEARNING FOR HOMOGENIZED TO FINE-SCALE MAPS FOR MULTISCALE PDE. / ZHANG, Zecheng; MOYA, Christian; LEUNG, Wing Tat et al.
In: Multiscale Modeling & Simulation, Vol. 22, No. 3, 09.2024, p. 956-972.

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review