A physics-informed deep learning approach for solving strongly degenerate parabolic problems

IF 8.7 2区 工程技术 Q1 Mathematics
Pasquale Ambrosio, Salvatore Cuomo, Mariapia De Rosa
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引用次数: 0

Abstract

In recent years, Scientific Machine Learning (SciML) methods for solving Partial Differential Equations (PDEs) have gained increasing popularity. Within such a paradigm, Physics-Informed Neural Networks (PINNs) are novel deep learning frameworks for solving initial-boundary value problems involving nonlinear PDEs. Recently, PINNs have shown promising results in several application fields. Motivated by applications to gas filtration problems, here we present and evaluate a PINN-based approach to predict solutions to strongly degenerate parabolic problems with asymptotic structure of Laplacian type. To the best of our knowledge, this is one of the first papers demonstrating the efficacy of the PINN framework for solving such kind of problems. In particular, we estimate an appropriate approximation error for some test problems whose analytical solutions are fortunately known. The numerical experiments discussed include two and three-dimensional spatial domains, emphasizing the effectiveness of this approach in predicting accurate solutions.

Abstract Image

解决强退化抛物线问题的物理信息深度学习方法
近年来,用于求解偏微分方程(PDEs)的科学机器学习(SciML)方法越来越受欢迎。在这种模式中,物理信息神经网络(PINN)是一种新型深度学习框架,用于解决涉及非线性偏微分方程的初界值问题。最近,PINNs 在多个应用领域取得了可喜的成果。受气体过滤问题应用的启发,我们在此提出并评估了一种基于 PINN 的方法,用于预测具有拉普拉奇类型渐近结构的强退化抛物线问题的解。据我们所知,这是第一批证明 PINN 框架在解决此类问题方面功效的论文之一。特别是,我们估算了一些测试问题的适当近似误差,幸运的是,这些问题的解析解是已知的。讨论的数值实验包括二维和三维空间域,强调了这种方法在预测精确解方面的有效性。
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来源期刊
Engineering with Computers
Engineering with Computers 工程技术-工程:机械
CiteScore
16.50
自引率
2.30%
发文量
203
审稿时长
9 months
期刊介绍: Engineering with Computers is an international journal dedicated to simulation-based engineering. It features original papers and comprehensive reviews on technologies supporting simulation-based engineering, along with demonstrations of operational simulation-based engineering systems. The journal covers various technical areas such as adaptive simulation techniques, engineering databases, CAD geometry integration, mesh generation, parallel simulation methods, simulation frameworks, user interface technologies, and visualization techniques. It also encompasses a wide range of application areas where engineering technologies are applied, spanning from automotive industry applications to medical device design.
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