Discovering PDEs Corrections from Data Within a Hybrid Modeling Framework

Chady Ghnatios () and Francisco Chinesta
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Chady Ghnatios: Department of Mechanical Engineering, University of North Florida, Jacksonville, FL 32224, USA
Francisco Chinesta: Process and Engineering in Mechanics and Materials Laboratory, UMR CNRS—Arts et Metiers Institute of Technology, 75013 Paris, France

Mathematics, 2024, vol. 13, issue 1, 1-17

Abstract: In the context of hybrid twins, a data-driven enrichment is added to the physics-based solution to represent with higher accuracy the reference solution assumed to be known at different points in the physical domain. Such an approach enables better predictions. However, the data-driven enrichment is usually represented by a regression, whose main drawbacks are (i) the difficulty of understanding the subjacent physics and (ii) the risks induced by the data-driven model extrapolation. This paper proposes a procedure enabling the extraction of a differential operator associated with the enrichment provided by the data-driven regression. For that purpose, a sparse Singular Value Decomposition, SVD, is introduced. It is then employed, first, in a full operator representation regularized optimization problem, where sparsity is promoted, leading to a linear programming problem, and then in a tensor decomposition of the operator’s identification procedure. The results show the ability of the method to identify the exact missing operators from the model. The regularized optimization problem was also able to identify the weights of the missing terms with a relative error of about 10% on average, depending on the selected use case.

Keywords: scientific machine learning; model discovery; physics-informed neural networks; PINN; sparse SVD (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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