Evolutional deep neural networks (EDNN) solve partial differential equations (PDEs) by marching the network representation of the solution fields, using the governing equations. Use of a single network to solve coupled PDEs on large domains requires a large number of network parameters and incurs a significant computational cost. We introduce coupled EDNN (C-EDNN) to solve systems of PDEs by using independent networks for each state variable, which are only coupled through the governing equations. We also introduce distributed EDNN (D-EDNN) by spatially partitioning the global domain into several elements and assigning individual EDNNs to each element to solve the local evolution of the PDE. The networks then exchange the solution and fluxes at their interfaces, similar to flux-reconstruction methods, and ensure that the PDE dynamics are accurately preserved between neighboring elements. Together C-EDNN and D-EDNN form the general class of Multi-EDNN methods. We demonstrate these methods with aid of canonical problems including linear advection, the heat equation, and the compressible Navier-Stokes equations in Couette and Taylor-Green flows.

翻译：进化深度神经网络（EDNN）通过利用控制方程推进解场的网络表示来求解偏微分方程（PDE）。使用单一网络在大型域上求解耦合PDE需要大量网络参数并产生显著计算成本。我们引入耦合EDNN（C-EDNN）来求解PDE系统，其为每个状态变量使用独立的网络，这些网络仅通过控制方程进行耦合。我们还引入分布式EDNN（D-EDNN），通过将全局域空间划分为若干单元，并为每个单元分配独立的EDNN来求解PDE的局部演化。随后，网络在其界面处交换解和通量，类似于通量重构方法，并确保相邻单元之间的PDE动力学得到精确保持。C-EDNN与D-EDNN共同构成了通用的Multi-EDNN方法类别。我们借助典型问题（包括线性平流方程、热方程以及Couette流和Taylor-Green流中的可压缩Navier-Stokes方程）来演示这些方法。