Lin Qiu, Fajie Wang, Wenzhen Qu, Yan Gu, Qing-Hua Qin
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引用次数: 0
摘要
本研究介绍了一种名为光谱集成神经网络(SINNs)的创新神经网络框架,用于解决三维空间中的正向和反向动态问题。在 SINNs 中,利用频谱积分技术进行时间离散化,然后应用全连接神经网络求解空间域的偏微分方程。此外,还采用多项式基函数来扩展未知函数,目的是提高 SINN 在处理逆问题时的性能。通过几个动态基准示例,包括线性和非线性热传导问题、线性和非线性波传播问题、热传导逆问题和长时间热传导问题,对所开发框架的性能进行了评估。数值结果表明,SINN 可以有效、准确地解决涉及热传导和波传播的正向和反向问题。此外,SINN 还能为时间持续较长的动态问题提供精确而稳定的解决方案。与常用的物理信息神经网络相比,SINN 在收敛速度、计算精度和效率方面表现出更优越的性能。
Spectral integrated neural networks (SINNs) for solving forward and inverse dynamic problems.
This study introduces an innovative neural network framework named spectral integrated neural networks (SINNs) to address both forward and inverse dynamic problems in three-dimensional space. In the SINNs, the spectral integration technique is utilized for temporal discretization, followed by the application of a fully connected neural network to solve the resulting partial differential equations in the spatial domain. Furthermore, the polynomial basis functions are employed to expand the unknown function, with the goal of improving the performance of SINNs in tackling inverse problems. The performance of the developed framework is evaluated through several dynamic benchmark examples encompassing linear and nonlinear heat conduction problems, linear and nonlinear wave propagation problems, inverse problem of heat conduction, and long-time heat conduction problem. The numerical results demonstrate that the SINNs can effectively and accurately solve forward and inverse problems involving heat conduction and wave propagation. Additionally, the SINNs provide precise and stable solutions for dynamic problems with extended time durations. Compared to commonly used physics-informed neural networks, the SINNs exhibit superior performance with enhanced convergence speed, computational accuracy, and efficiency.
期刊介绍:
Neural Networks is a platform that aims to foster an international community of scholars and practitioners interested in neural networks, deep learning, and other approaches to artificial intelligence and machine learning. Our journal invites submissions covering various aspects of neural networks research, from computational neuroscience and cognitive modeling to mathematical analyses and engineering applications. By providing a forum for interdisciplinary discussions between biology and technology, we aim to encourage the development of biologically-inspired artificial intelligence.