Approximating Partial Differential Equations with Physics-Informed Legendre Multiwavelets CNN
The purpose of this paper is to leverage the advantages of physics-informed neural network (PINN) and convolutional neural network (CNN) by using Legendre multiwavelets (LMWs) as basis functions to approximate partial differential equations (PDEs). We call this method Physics-Informed Legendre Multi...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-01-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/8/2/91 |
| _version_ | 1797298190741929984 |
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| author | Yahong Wang Wenmin Wang Cheng Yu Hongbo Sun Ruimin Zhang |
| author_facet | Yahong Wang Wenmin Wang Cheng Yu Hongbo Sun Ruimin Zhang |
| author_sort | Yahong Wang |
| collection | DOAJ |
| description | The purpose of this paper is to leverage the advantages of physics-informed neural network (PINN) and convolutional neural network (CNN) by using Legendre multiwavelets (LMWs) as basis functions to approximate partial differential equations (PDEs). We call this method Physics-Informed Legendre Multiwavelets CNN (PiLMWs-CNN), which can continuously approximate a grid-based state representation that can be handled by a CNN. PiLMWs-CNN enable us to train our models using only physics-informed loss functions without any precomputed training data, simultaneously providing fast and continuous solutions that generalize to previously unknown domains. In particular, the LMWs can simultaneously possess compact support, orthogonality, symmetry, high smoothness, and high approximation order. Compared to orthonormal polynomial (OP) bases, the approximation accuracy can be greatly increased and computation costs can be significantly reduced by using LMWs. We applied PiLMWs-CNN to approximate the damped wave equation, the incompressible Navier–Stokes (N-S) equation, and the two-dimensional heat conduction equation. The experimental results show that this method provides more accurate, efficient, and fast convergence with better stability when approximating the solution of PDEs. |
| first_indexed | 2024-03-07T22:31:22Z |
| format | Article |
| id | doaj.art-268006e8db134f10983db48af78a7e1a |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-07T22:31:22Z |
| publishDate | 2024-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-268006e8db134f10983db48af78a7e1a2024-02-23T15:17:11ZengMDPI AGFractal and Fractional2504-31102024-01-01829110.3390/fractalfract8020091Approximating Partial Differential Equations with Physics-Informed Legendre Multiwavelets CNNYahong Wang0Wenmin Wang1Cheng Yu2Hongbo Sun3Ruimin Zhang4School of Computer Science and Engineering, Macau University of Science and Technology, Macau 999078, ChinaSchool of Computer Science and Engineering, Macau University of Science and Technology, Macau 999078, ChinaSchool of Artificial Intelligence, Chongqing University of Technology, Chongqing 401135, ChinaZhuhai Campus, Beijing Institute of Technology, Zhuhai 519088, ChinaZhuhai Campus, Beijing Institute of Technology, Zhuhai 519088, ChinaThe purpose of this paper is to leverage the advantages of physics-informed neural network (PINN) and convolutional neural network (CNN) by using Legendre multiwavelets (LMWs) as basis functions to approximate partial differential equations (PDEs). We call this method Physics-Informed Legendre Multiwavelets CNN (PiLMWs-CNN), which can continuously approximate a grid-based state representation that can be handled by a CNN. PiLMWs-CNN enable us to train our models using only physics-informed loss functions without any precomputed training data, simultaneously providing fast and continuous solutions that generalize to previously unknown domains. In particular, the LMWs can simultaneously possess compact support, orthogonality, symmetry, high smoothness, and high approximation order. Compared to orthonormal polynomial (OP) bases, the approximation accuracy can be greatly increased and computation costs can be significantly reduced by using LMWs. We applied PiLMWs-CNN to approximate the damped wave equation, the incompressible Navier–Stokes (N-S) equation, and the two-dimensional heat conduction equation. The experimental results show that this method provides more accurate, efficient, and fast convergence with better stability when approximating the solution of PDEs.https://www.mdpi.com/2504-3110/8/2/91partial differential equationsLegendre multiwaveletsphysics-informed neural networkconvolutional neural network |
| spellingShingle | Yahong Wang Wenmin Wang Cheng Yu Hongbo Sun Ruimin Zhang Approximating Partial Differential Equations with Physics-Informed Legendre Multiwavelets CNN Fractal and Fractional partial differential equations Legendre multiwavelets physics-informed neural network convolutional neural network |
| title | Approximating Partial Differential Equations with Physics-Informed Legendre Multiwavelets CNN |
| title_full | Approximating Partial Differential Equations with Physics-Informed Legendre Multiwavelets CNN |
| title_fullStr | Approximating Partial Differential Equations with Physics-Informed Legendre Multiwavelets CNN |
| title_full_unstemmed | Approximating Partial Differential Equations with Physics-Informed Legendre Multiwavelets CNN |
| title_short | Approximating Partial Differential Equations with Physics-Informed Legendre Multiwavelets CNN |
| title_sort | approximating partial differential equations with physics informed legendre multiwavelets cnn |
| topic | partial differential equations Legendre multiwavelets physics-informed neural network convolutional neural network |
| url | https://www.mdpi.com/2504-3110/8/2/91 |
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