Theory-guided physics-informed neural networks for boundary layer problems with singular perturbation
Theory-guided physics-informed neural networks for boundary layer problems with singular perturbation
复制标题
用于解决奇异扰动边界层问题的理论指导物理信息神经网络
DOI：
10.1016/j.jcp.2022.111768
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发表时间：
2023
影响因子：
4.1
通讯作者：
D'Souza, Roshan M.
中科院分区：
文献类型：
--
作者：
Arzani, Amirhossein;Cassel, Kevin W.;D'Souza, Roshan M.
物理信息神经网络（PINN）是科学机器学习研究和微分方程建模的最新趋势。尽管PINN研究取得了进展，但大梯度和高度非线性模式仍然是建模的挑战。薄边界层问题是输运问题中常见的大梯度问题的突出例子。在这项研究中，边界层PINN（BL-PINN），提出了使薄边界层的解决方案，考虑他们作为一个奇异摄动问题。受经典微扰理论和渐近展开的启发，BL-PINN的设计复制了奇异微扰理论的过程。也就是说，不同的并行PINN网络被定义为代表不同的近似阶的边界层问题的内部和外部区域。在不同的基准问题（正问题和逆问题）中，BL-PINN方法表现出比传统PINN方法更上级的性能，能够得到精确的结果，而传统PINN方法不能提供有意义的解。BL-PINN还表现出比PINN的其他扩展（如扩展PINN（XPINN）方法）更好的结果。在BL-PINN的扰动参数的自然纳入提供了机会，以评估参数的解决方案，而不需要重新训练。BL-PINN演示了如何使用经典数学理论来指导深度神经网络的设计，以解决具有挑战性的问题。
Physics-informed neural networks (PINNs) are a recent trend in scientific machine learning research and modeling of differential equations. Despite progress in PINN research, large gradients and highly nonlinear patterns remain challenging to model. Thin boundary layer problems are prominent examples of large gradients that commonly arise in transport problems. In this study, boundary-layer PINN (BL-PINN) is proposed to enable a solution to thin boundary layers by considering them as a singular perturbation problem. Inspired by the classical perturbation theory and asymptotic expansions, BL-PINN is designed to replicate the procedure in singular perturbation theory. Namely, different parallel PINN networks are defined to represent different orders of approximation to the boundary layer problem in the inner and outer regions. In different benchmark problems (forward and inverse), BL-PINN shows superior performance compared to the traditional PINN approach and is able to produce accurate results, whereas the classical PINN approach could not provide meaningful solutions. BL-PINN also demonstrates significantly better results compared to other extensions of PINN such as the extended PINN (XPINN) approach. The natural incorporation of the perturbation parameter in BL-PINN provides the opportunity to evaluate parametric solutions without the need for retraining. BL-PINN demonstrates an example of how classical mathematical theory could be used to guide the design of deep neural networks for solving challenging problems.
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影响因子：
4.6
作者：
Brandt A. Belson;Onofrio Semeraro;C. Rowley;D. Henningson
通讯作者：
Brandt A. Belson;Onofrio Semeraro;C. Rowley;D. Henningson
DOI：
--
发表时间：
2000
期刊：
影响因子：
--
作者：
V. G. Vetekha
通讯作者：
V. G. Vetekha
DOI：
10.1016/j.cma.2021.113938
发表时间：
2021-06-02
影响因子：
7.2
作者：
Wang, Sifan;Wang, Hanwen;Perdikaris, Paris
通讯作者：
Perdikaris, Paris
影响因子：
4.6
作者：
Arzani, Amirhossein;Wang, Jian-Xun;D'Souza, Roshan M.
通讯作者：
D'Souza, Roshan M.
DOI：
--
发表时间：
2020
期刊：
影响因子：
--
作者：
J. Kutz
通讯作者：
J. Kutz