Discovering PDEs Corrections from Data Within a Hybrid Modeling Framework
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...
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MDPI AG
2024-12-01
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| author | Chady Ghnatios Francisco Chinesta |
| author_facet | Chady Ghnatios Francisco Chinesta |
| author_sort | Chady Ghnatios |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-f18231e092ce480eb55e843dbd62dc65 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-f18231e092ce480eb55e843dbd62dc652025-01-10T13:17:56ZengMDPI AGMathematics2227-73902024-12-01131510.3390/math13010005Discovering PDEs Corrections from Data Within a Hybrid Modeling FrameworkChady Ghnatios0Francisco Chinesta1Department of Mechanical Engineering, University of North Florida, Jacksonville, FL 32224, USAProcess and Engineering in Mechanics and Materials Laboratory, UMR CNRS—Arts et Metiers Institute of Technology, 75013 Paris, FranceIn 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.https://www.mdpi.com/2227-7390/13/1/5scientific machine learningmodel discoveryphysics-informed neural networksPINNsparse SVD |
| spellingShingle | Chady Ghnatios Francisco Chinesta Discovering PDEs Corrections from Data Within a Hybrid Modeling Framework Mathematics scientific machine learning model discovery physics-informed neural networks PINN sparse SVD |
| title | Discovering PDEs Corrections from Data Within a Hybrid Modeling Framework |
| title_full | Discovering PDEs Corrections from Data Within a Hybrid Modeling Framework |
| title_fullStr | Discovering PDEs Corrections from Data Within a Hybrid Modeling Framework |
| title_full_unstemmed | Discovering PDEs Corrections from Data Within a Hybrid Modeling Framework |
| title_short | Discovering PDEs Corrections from Data Within a Hybrid Modeling Framework |
| title_sort | discovering pdes corrections from data within a hybrid modeling framework |
| topic | scientific machine learning model discovery physics-informed neural networks PINN sparse SVD |
| url | https://www.mdpi.com/2227-7390/13/1/5 |
| work_keys_str_mv | AT chadyghnatios discoveringpdescorrectionsfromdatawithinahybridmodelingframework AT franciscochinesta discoveringpdescorrectionsfromdatawithinahybridmodelingframework |