物理信息神经网络的h-自适应配置方法

IF 3.7 3区 计算机科学 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS
Jan Trynda , Paweł Maczuga , Albert Oliver-Serra , Luis Emilio García-Castillo , Robert Schaefer , Maciej Woźniak
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引用次数: 0

摘要

尽管物理信息神经网络(pinn)在解决偏微分方程方面具有灵活性和成功性,但它经常受到收敛问题的困扰,甚至无法收敛,特别是在具有陡峭梯度或局部特征的问题中。已经提出了几种补救措施来解决这个问题,但最有希望的一种是搭配点的动态适应。本文在有限元法自适应网格细化的基础上,提出了一种新的随机自适应采样方法。我们的改进算法中的误差估计是基于残差损失函数的值。我们针对各种1D和2D基准问题测试了我们的方法,这些问题在某些边界附近表现出陡峭的梯度,结果很有希望。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
An h-adaptive collocation method for Physics-Informed Neural Networks
Despite their flexibility and success in solving partial differential equations, Physics-Informed Neural Networks (PINNs) often suffer from convergence issues, even failing to converge, particularly in problems with steep gradients or localized features. Several remedies have been suggested to solve this problem, but one of the most promising is the dynamical adaptation of the collocation points. This paper explores a novel adaptive sampling method, of a stochastic nature, based on the Adaptive Mesh Refinement used in the Finite Element Method. The error estimates in our refinement algorithm are based on the value of the residual loss function. We tested our method against a variety of 1D and 2D benchmark problems that exhibit steep gradients near certain boundaries, with promising results.
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来源期刊
Journal of Computational Science
Journal of Computational Science COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS-COMPUTER SCIENCE, THEORY & METHODS
CiteScore
5.50
自引率
3.00%
发文量
227
审稿时长
41 days
期刊介绍: Computational Science is a rapidly growing multi- and interdisciplinary field that uses advanced computing and data analysis to understand and solve complex problems. It has reached a level of predictive capability that now firmly complements the traditional pillars of experimentation and theory. The recent advances in experimental techniques such as detectors, on-line sensor networks and high-resolution imaging techniques, have opened up new windows into physical and biological processes at many levels of detail. The resulting data explosion allows for detailed data driven modeling and simulation. This new discipline in science combines computational thinking, modern computational methods, devices and collateral technologies to address problems far beyond the scope of traditional numerical methods. Computational science typically unifies three distinct elements: • Modeling, Algorithms and Simulations (e.g. numerical and non-numerical, discrete and continuous); • Software developed to solve science (e.g., biological, physical, and social), engineering, medicine, and humanities problems; • Computer and information science that develops and optimizes the advanced system hardware, software, networking, and data management components (e.g. problem solving environments).
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