PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs
PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs
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DOI：
10.1016/j.cma.2021.114399
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发表时间：
2021-06
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影响因子：
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通讯作者：
Pu Ren;Chengping Rao;Yang Liu;Jianxun Wang;Hao Sun-
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文献类型：
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作者：
Pu Ren;Chengping Rao;Yang Liu;Jianxun Wang;Hao Sun-
偏微分方程（PDE）在建模和模拟各种学科的问题中发挥着重要作用。深度学习的最新进展表明，物理信息神经网络（PINN）在解决偏微分方程（PDE）方面具有巨大潜力，可作为数据驱动建模和逆分析的基础。然而，大多数现有的PINN方法，基于全连接的神经网络，造成内在的限制，低维时空参数化。此外，由于初始/边界条件（I/BC）是通过罚函数软施加的，因此解的质量严重依赖于超参数的调整.为此，我们提出了新的物理信息卷积递归学习架构（PhyCRNet和PhyCRNet-s），用于在没有任何标记数据的情况下求解偏微分方程。具体而言，提出了一种用于低维空间特征提取和时间演化学习的编码器-解码器卷积长短期记忆网络。损失函数被定义为聚合的离散PDE残差，而I/BC在网络中被硬编码以确保强制满足（例如，周期性边界填充）。网络通过自回归和残差连接进一步增强，显式模拟时间推进。我们提出的方法的性能进行了评估，通过解决三个非线性偏微分方程（例如，2D Burgers方程，λ-ω和FitzHugh Nagumo反应扩散方程），并与最先进的基线算法进行比较。数值结果表明，我们所提出的方法的优越性，在解决方案的精度，外推性和推广。
Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep learning have shown the great potential of physics-informed neural networks (PINNs) to solve PDEs as a basis for data-driven modeling and inverse analysis. However, the majority of existing PINN methods, based on fully-connected NNs, pose intrinsic limitations to low-dimensional spatiotemporal parameterizations. Moreover, since the initial/boundary conditions (I/BCs) are softly imposed via penalty, the solution quality heavily relies on hyperparameter tuning. To this end, we propose the novel physics-informed convolutional-recurrent learning architectures (PhyCRNet and PhyCRNet-s) for solving PDEs without any labeled data. Specifically, an encoder–decoder convolutional long short-term memory network is proposed for low-dimensional spatial feature extraction and temporal evolution learning. The loss function is defined as the aggregated discretized PDE residuals, while the I/BCs are hard-encoded in the network to ensure forcible satisfaction (eg, periodic boundary padding). The networks are further enhanced by autoregressive and residual connections that explicitly simulate time marching. The performance of our proposed methods has been assessed by solving three nonlinear PDEs (eg, 2D Burgers’ equations, the λ-ω and FitzHugh Nagumo reaction–diffusion equations), and compared against the start-of-the-art baseline algorithms. The numerical results demonstrate the superiority of our proposed methodology in the context of solution accuracy, extrapolability and generalizability.