On the approximation of the solution of partial differential equations by artificial neural networks trained by a multilevel Levenberg-Marquardt method
| Autor: | Henri Calandra, Serge Gratton, Elisa Riccietti, Xavier Vasseur |
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| Přispěvatelé: | Total E&P, Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Institut National Polytechnique (Toulouse) (Toulouse INP), École normale supérieure - Lyon (ENS Lyon), Institut Supérieur de l'Aéronautique et de l'Espace (ISAE-SUPAERO), ANR-19-P3IA-0004,ANITI,Artificial and Natural Intelligence Toulouse Institute(2019), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), École normale supérieure de Lyon (ENS de Lyon) |
| Rok vydání: | 2019 |
| Předmět: |
Algebraic multigrid method
Artificial neural network FOS: Computer and information sciences Computer Science - Machine Learning Computer Science::Neural and Evolutionary Computation Partial differential equation Multilevel optimization method Numerical Analysis (math.NA) Machine Learning (cs.LG) Levenberg-Marquardt method Optimization and Control (math.OC) FOS: Mathematics [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] Mathematics - Numerical Analysis [MATH]Mathematics [math] Mathematics - Optimization and Control [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] |
| Zdroj: | Optimization Methods and Software Optimization Methods and Software, Taylor & Francis, 2020 HAL Optimization Methods and Software, 2020 |
| ISSN: | 1055-6788 1029-4937 |
| DOI: | 10.48550/arxiv.1904.04685 |
| Popis: | International audience; This paper is concerned with the approximation of the solution of partial differential equations by means of artificial neural networks. Here a feedforward neural network is used to approximate the solution of the partial differential equation. The learning problem is formulated as a least squares problem, choosing the residual of the partial differential equation as a loss function, whereas a multilevel Levenberg-Marquardt method is employed as a training method. This setting allows us to get further insight into the potential of multilevel methods. Indeed, when the least squares problem arises from the training of artificial neural networks, the variables subject to optimization are not related by any geometrical constraints and the standard interpolation and restriction operators cannot be employed any longer. A heuristic, inspired by algebraic multigrid methods, is then proposed to construct the multilevel transfer operators. Numerical experiments show encouraging results related to the efficiency of the new multilevel optimization method for the training of artificial neural networks, compared to the standard corresponding one-level procedure. |
| Databáze: | OpenAIRE |
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