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A third-order finite difference weighted essentially non-oscillatory scheme with shallow neural network

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Title: A third-order finite difference weighted essentially non-oscillatory scheme with shallow neural network
Authors: Kwanghyuk Park, Xinjuan Chen, Dongjin Lee, Jiaxi Gu, Jae-Hun Jung
Source: Applied Numerical Mathematics. 218:1-21
Publication Status: Preprint
Publisher Information: Elsevier BV, 2025.
Publication Year: 2025
Subject Terms: FOS: Computer and information sciences, Computer Science - Machine Learning, FOS: Mathematics, Computer Science - Neural and Evolutionary Computing, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), Neural and Evolutionary Computing (cs.NE), Machine Learning (cs.LG)
Description: In this paper, we introduce the finite difference weighted essentially non-oscillatory (WENO) scheme based on the neural network for hyperbolic conservation laws. We employ the supervised learning and design two loss functions, one with the mean squared error and the other with the mean squared logarithmic error, where the WENO3-JS weights are computed as the labels. Each loss function consists of two components where the first component compares the difference between the weights from the neural network and WENO3-JS weights, while the second component matches the output weights of the neural network and the linear weights. The former of the loss function enforces the neural network to follow the WENO properties, implying that there is no need for the post-processing layer. Additionally the latter leads to better performance around discontinuities. As a neural network structure, we choose the shallow neural network (SNN) for computational efficiency with the Delta layer consisting of the normalized undivided differences. These constructed WENO3-SNN schemes show the outperformed results in one-dimensional examples and improved behavior in two-dimensional examples, compared with the simulations from WENO3-JS and WENO3-Z.
Document Type: Article
Language: English
ISSN: 0168-9274
DOI: 10.1016/j.apnum.2025.07.005
DOI: 10.48550/arxiv.2407.06333
Access URL: http://arxiv.org/abs/2407.06333
Rights: Elsevier TDM
CC BY NC ND
Accession Number: edsair.doi.dedup.....790b2eec87cf90affe56527c7e52f31c
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  Data: A third-order finite difference weighted essentially non-oscillatory scheme with shallow neural network
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  Data: <searchLink fieldCode="AR" term="%22Kwanghyuk+Park%22">Kwanghyuk Park</searchLink><br /><searchLink fieldCode="AR" term="%22Xinjuan+Chen%22">Xinjuan Chen</searchLink><br /><searchLink fieldCode="AR" term="%22Dongjin+Lee%22">Dongjin Lee</searchLink><br /><searchLink fieldCode="AR" term="%22Jiaxi+Gu%22">Jiaxi Gu</searchLink><br /><searchLink fieldCode="AR" term="%22Jae-Hun+Jung%22">Jae-Hun Jung</searchLink>
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  Data: <i>Applied Numerical Mathematics</i>. 218:1-21
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– Name: Abstract
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  Data: In this paper, we introduce the finite difference weighted essentially non-oscillatory (WENO) scheme based on the neural network for hyperbolic conservation laws. We employ the supervised learning and design two loss functions, one with the mean squared error and the other with the mean squared logarithmic error, where the WENO3-JS weights are computed as the labels. Each loss function consists of two components where the first component compares the difference between the weights from the neural network and WENO3-JS weights, while the second component matches the output weights of the neural network and the linear weights. The former of the loss function enforces the neural network to follow the WENO properties, implying that there is no need for the post-processing layer. Additionally the latter leads to better performance around discontinuities. As a neural network structure, we choose the shallow neural network (SNN) for computational efficiency with the Delta layer consisting of the normalized undivided differences. These constructed WENO3-SNN schemes show the outperformed results in one-dimensional examples and improved behavior in two-dimensional examples, compared with the simulations from WENO3-JS and WENO3-Z.
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