Approximating Partial Differential Equations with Physics-Informed Legendre Multiwavelets CNN Article Swipe

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Yahong Wang , Wenmin Wang , Cheng Yu , Hongbo Sun , Ruimin Zhang ·

YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.20944/preprints202311.0374.v1

Partial differential equations (PDEs) are used to describe a wide range of phenomena, such as heat transfer, ﬂuid dynamics, and quantum mechanics. By solving PDEs, we can ob- tain insights into the behavior of the system and make predictions about its future evolution. Conventional numerical methods for obtaining the approximate solutions of PDEs may re- quire extensive computational resources and time, especially for complex PDEs and large- scale problems. The recently developed physics-informed neural network (PINN) has shown promise in a variety of scientiﬁc and engineering ﬁelds by incorporating physical laws into the loss functions of the neural network (NN). In addition, the training of PINN does not require ground truth data, but it has poor generalization ability to unseen domains. On the other hand, a convolutional neural network (CNN) has fast inference and better generalization ability, but it requires a large amount of training data. Taking the advantages of PINN and CNN by using Legendre multiwavelets (LMWs) as basis functions, we introduce a new method to approach the PDEs in this paper, namely Physics-Informed Legendre Multiwavelets CNN (PiLMWs-CNN), in order to 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 pre- computed training data, simultaneously providing fast and continuous solutions that gener- alize 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, incompressible Navier-Stokes (N- S) equation, and 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.

Related Topics

Partial Differential Equation

Generalization

Convolutional Neural Network

Computer Science

Artificial Intelligence

Legendre Transformation

Algorithm

Theoretical Computer Science

Mathematics

Wavelet Transform

Mathematical Analysis

Discrete Wavelet Transform

Concepts

Partial differential equation Generalization Legendre polynomials Legendre wavelet Convolutional neural network Computer science Inference Artificial intelligence Applied mathematics Artificial neural network Legendre transformation Algorithm Theoretical computer science Mathematics Wavelet Wavelet transform Mathematical analysis Discrete wavelet transform

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Type: preprint
Language: en
Landing Page: https://doi.org/10.20944/preprints202311.0374.v1
PDF: https://www.preprints.org/manuscript/202311.0374/v1/download
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Cited By: 1
References: 56
Related Works: 10
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DOI: https://doi.org/10.20944/preprints202311.0374.v1

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Title: Approximating Partial Differential Equations with Physics-Informed Legendre Multiwavelets CNN

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Type: preprint

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Language: en

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Publication year: 2023

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Publication date: 2023-11-06

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Authors: Yahong Wang, Wenmin Wang, Cheng Yu, Hongbo Sun, Ruimin Zhang

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Landing page: https://doi.org/10.20944/preprints202311.0374.v1

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Concepts: Partial differential equation, Generalization, Legendre polynomials, Legendre wavelet, Convolutional neural network, Computer science, Inference, Artificial intelligence, Applied mathematics, Artificial neural network, Legendre transformation, Algorithm, Theoretical computer science, Mathematics, Wavelet, Wavelet transform, Mathematical analysis, Discrete wavelet transform

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Cited by: 1

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