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A Parsimonious Separated Representation Empowering PINN–PGD-Based Solutions for Parametrized Partial Differential Equations

Article dans une revue avec comité de lecture
Author
ccGHNATIOS, Chady
86289 Laboratoire Procédés et Ingénierie en Mécanique et Matériaux [PIMM]
ccCHINESTA SORIA, Francisco
86289 Laboratoire Procédés et Ingénierie en Mécanique et Matériaux [PIMM]
1166977 CNRS@CREATE Ltd.

URI
http://hdl.handle.net/10985/25753
DOI
10.3390/math12152365
Date
2024-07
Journal
Mathematics

Abstract

The efficient solution (fast and accurate) of parametric partial differential equations (pPDE) is of major interest in many domains of science and engineering, enabling evaluations of the quantities of interest, optimization, control, and uncertainty propagation—all them under stringent real-time constraints. Different methodologies have been proposed in the past within the model order reduction (MOR) community, based on the use of reduced bases (RB) or the separated representation at the heart of the so-called proper generalized decompositions (PGD). In PGD, an alternate-direction strategy is employed to circumvent the integration issues of operating in multi-dimensional domains. Recently, physics informed neural networks (PINNs), a particular collocation schema where the unknown field is approximated by a neural network (NN), have emerged in the domain of scientific machine learning. PNNs combine the versatility of NN-based approximation with the ease of collocating pPDE. The present paper proposes a combination of both procedures to find an efficient solution for pPDE, that can either be viewed as an efficient collocation procedure for PINN, or as a monolithic PGD that bypasses the use of the fixed-point alternated directions.

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