This study presents a novel unsupervised convolutional Neural Network (NN) architecture with nonlocal interactions for solving Partial Differential Equations (PDEs). The nonlocal Peridynamic Differential Operator (PDDO) is employed as a convolutional filter for evaluating derivatives the field variable. The NN captures the time-dynamics in smaller latent space through encoder-decoder layers with a Convolutional Long-short Term Memory (ConvLSTM) layer between them. The ConvLSTM architecture is modified by employing a novel activation function to improve the predictive capability of the learning architecture for physics with periodic behavior. The physics is invoked in the form of governing equations at the output of the NN and in the latent (reduced) space. By considering a few benchmark PDEs, we demonstrate the training performance and extrapolation capability of this novel NN architecture by comparing against Physics Informed Neural Networks (PINN) type solvers. It is more capable of extrapolating the solution for future timesteps than the other existing architectures.
翻译:本研究展示了一个新的未经监督的进化神经网络(NN)结构,它与非本地互动解决部分差异(PDEs)的局部互动关系(PDEs)结构。非本地 Peririval差异操作器(PDDO)被用作评估场变量衍生物的进化过滤器。ND通过它们之间的进化-解码层和进化长期短期内存(ConvLSTM)层来捕捉较小潜在空间的时间动力。ConvLSTM结构通过使用新的启动功能来修改,以提高定期行为物理学学习结构的预测能力。物理学在NN和潜在(减少的)空间的输出中以治理方程式的形式被引用。通过考虑几个基准PDE,我们通过比较物理信息化神经网络(PINN)类型解答器,展示了这个新兴的NNC结构的培训性能和外推能力。它比其他现有结构更有能力外推出未来时间步骤的解决方案。