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Volume 35, Issue 4
A Data-Driven Scale-Invariant Weighted Compact Nonlinear Scheme for Hyperbolic Conservation Laws

Zixuan Zhang, Yidao Dong, Yuanyang Zou, Hao Zhang & Xiaogang Deng

Commun. Comput. Phys., 35 (2024), pp. 1120-1154.

Published online: 2024-05

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  • Abstract

With continuous developments in various techniques, machine learning is becoming increasingly viable and promising in the field of fluid mechanics. In this article, we present a machine learning approach for enhancing the resolution and robustness of the weighted compact nonlinear scheme (WCNS). We employ a neural network as a weighting function in the WCNS scheme and follow a data-driven approach to train this neural network. Neural networks can learn a new smoothness measure and calculate a weight function inherently. To facilitate the machine learning task and train with fewer data, we integrate the prior knowledge into the learning process, such as a Galilean invariant input layer and CNS polynomials. The normalization in the Delta layer (the so-called Delta layer is used to calculate input features) ensures that the WCNS3-NN schemes achieve a scale-invariant property (Si-property) with an arbitrary scale of a function, and an essentially non-oscillatory approximation of a discontinuous function (ENO-property). The Si-property and ENO-property of the data-driven WCNS schemes are validated numerically. Several one- and two-dimensional benchmark examples, including strong shocks and shock-density wave interactions, are presented to demonstrate the advantages of the proposed method.

  • AMS Subject Headings

68T07, 35L65, 65M06

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COPYRIGHT: © Global Science Press

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@Article{CiCP-35-1120, author = {Zhang , ZixuanDong , YidaoZou , YuanyangZhang , Hao and Deng , Xiaogang}, title = {A Data-Driven Scale-Invariant Weighted Compact Nonlinear Scheme for Hyperbolic Conservation Laws}, journal = {Communications in Computational Physics}, year = {2024}, volume = {35}, number = {4}, pages = {1120--1154}, abstract = {

With continuous developments in various techniques, machine learning is becoming increasingly viable and promising in the field of fluid mechanics. In this article, we present a machine learning approach for enhancing the resolution and robustness of the weighted compact nonlinear scheme (WCNS). We employ a neural network as a weighting function in the WCNS scheme and follow a data-driven approach to train this neural network. Neural networks can learn a new smoothness measure and calculate a weight function inherently. To facilitate the machine learning task and train with fewer data, we integrate the prior knowledge into the learning process, such as a Galilean invariant input layer and CNS polynomials. The normalization in the Delta layer (the so-called Delta layer is used to calculate input features) ensures that the WCNS3-NN schemes achieve a scale-invariant property (Si-property) with an arbitrary scale of a function, and an essentially non-oscillatory approximation of a discontinuous function (ENO-property). The Si-property and ENO-property of the data-driven WCNS schemes are validated numerically. Several one- and two-dimensional benchmark examples, including strong shocks and shock-density wave interactions, are presented to demonstrate the advantages of the proposed method.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0162}, url = {http://global-sci.org/intro/article_detail/cicp/23096.html} }
TY - JOUR T1 - A Data-Driven Scale-Invariant Weighted Compact Nonlinear Scheme for Hyperbolic Conservation Laws AU - Zhang , Zixuan AU - Dong , Yidao AU - Zou , Yuanyang AU - Zhang , Hao AU - Deng , Xiaogang JO - Communications in Computational Physics VL - 4 SP - 1120 EP - 1154 PY - 2024 DA - 2024/05 SN - 35 DO - http://doi.org/10.4208/cicp.OA-2023-0162 UR - https://global-sci.org/intro/article_detail/cicp/23096.html KW - WCNS, neural network, machine learning, scale-invariant, shock-capturing. AB -

With continuous developments in various techniques, machine learning is becoming increasingly viable and promising in the field of fluid mechanics. In this article, we present a machine learning approach for enhancing the resolution and robustness of the weighted compact nonlinear scheme (WCNS). We employ a neural network as a weighting function in the WCNS scheme and follow a data-driven approach to train this neural network. Neural networks can learn a new smoothness measure and calculate a weight function inherently. To facilitate the machine learning task and train with fewer data, we integrate the prior knowledge into the learning process, such as a Galilean invariant input layer and CNS polynomials. The normalization in the Delta layer (the so-called Delta layer is used to calculate input features) ensures that the WCNS3-NN schemes achieve a scale-invariant property (Si-property) with an arbitrary scale of a function, and an essentially non-oscillatory approximation of a discontinuous function (ENO-property). The Si-property and ENO-property of the data-driven WCNS schemes are validated numerically. Several one- and two-dimensional benchmark examples, including strong shocks and shock-density wave interactions, are presented to demonstrate the advantages of the proposed method.

Zhang , ZixuanDong , YidaoZou , YuanyangZhang , Hao and Deng , Xiaogang. (2024). A Data-Driven Scale-Invariant Weighted Compact Nonlinear Scheme for Hyperbolic Conservation Laws. Communications in Computational Physics. 35 (4). 1120-1154. doi:10.4208/cicp.OA-2023-0162
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