物理信息编码解码卷积长短期记忆网络求解时变偏微分方程

Physics-Informed Encoder-Decoder Network Based on ConvLSTM for Solving Time-Dependent Partial Differential Equations

摘要: 本文基于ConvLSTM，构建了一种编码–解码的物理信息深度学习网络框架(PIED-Net)，用于求解含时域的偏微分方程。首先，PIED-Net基于卷积网络提取图像信息中不同层次的空间特征，并使用转置卷积对提取的特征进行表达，实现从图像中学习物理约束的空间信息。为了保证时域信息的连续性，PIED-Net采用ConvLSTM模块，有效地捕捉输入序列的时序关系，提高在预测时间步骤中网络的精度。其次，控制方程、边界和初始条件被用来构建损失函数，并利用有限差分方法来计算各阶导数，进一步提高网络的准确性和稳定性。最后，将所提出的PIED-Net网络框架用来求解2D Burgers方程和2Dflowmixing方程。数值实验结果表明，该网络框架在求解含时域的偏微分方程方面具有出色的精度和运算效率。这为利用深度学习方法解决物理问题提供了一种新的途径。

Abstract: This article presents PIED-Net, a deep learning network framework based on ConvLSTM, for solving time-domain partial differential equations. First, convolutional networks are used to extract spatial features at different levels from image data and transpose convolutions are used to represent these features, enabling the learning of spatial information constrained by physics. ConvLSTM modules are utilized to effectively capture the temporal relationships of neighboring sequences to maintain temporal continuity, and enhance the accuracy of the network in predicting time steps. Besides, governing equations, boundary and initial conditions are employed to construct the loss function, and the finite difference method is employed to calculate derivatives of various orders, further en-hancing the accuracy and stability of the network. Finally, the proposed PIED-Net framework is ap-plied to solve 2D Burgers’ and 2D flow mixing equations, respectively. Numerical experimental re-sults demonstrate that this network framework exhibits outstanding accuracy and computational efficiency in solving time-dependent partial differential equations, presenting a new deep-learning approach for physical problems.