Deep learning techniques, particularly neural networks, have revolutionized computational physics, offering powerful tools for solving complex partial differential equations (PDEs). However, ensuring stability and efficiency remains a challenge, especially in scenarios involving nonlinear and time-dependent equations. This paper introduces novel residual-based architectures, namely the Simple Highway Network and the Squared Residual Network, designed to enhance stability and accuracy in physics-informed neural networks (PINNs). These architectures augment traditional neural networks by incorporating residual connections, which facilitate smoother weight updates and improve backpropagation efficiency. Through extensive numerical experiments across various examples—including linear and nonlinear, time-dependent and independent PDEs—we demonstrate the efficacy of the proposed architectures. The Squared Residual Network, in particular, exhibits robust performance, achieving enhanced stability and accuracy compared to conventional neural networks. These findings underscore the potential of residual-based architectures in advancing deep learning for PDEs and computational physics applications.

中文翻译：

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稳定的权重更新：使用深度学习的可靠 PDE 解决方案的关键


深度学习技术，特别是神经网络，彻底改变了计算物理学，为求解复杂的偏微分方程（PDE）提供了强大的工具。然而，确保稳定性和效率仍然是一个挑战，特别是在涉及非线性和瞬态方程的场景中。本文介绍了新颖的基于残差的架构，即简单高速公路网络和平方残差网络，旨在增强物理信息神经网络（PINN）的稳定性和准确性。这些架构通过合并残差连接来增强传统神经网络，从而促进更平滑的权重更新并提高反向传播效率。通过对各种示例（包括线性和非线性、时间相关和独立偏微分方程）进行广泛的数值实验，我们证明了所提出的架构的有效性。尤其是平方残差网络，表现出强大的性能，与传统神经网络相比，稳定性和准确性更高。这些发现强调了基于残差的架构在推进偏微分方程和计算物理应用深度学习方面的潜力。