Evolutional deep neural network.

Abstract

The notion of an evolutional deep neural network (EDNN) is introduced for the solution of partial differential equations (PDE). The parameters of the network are trained to represent the initial state of the system only and are subsequently updated dynamically, without any further training, to provide an accurate prediction of the evolution of the PDE system. In this framework, the network parameters are treated as functions with respect to the appropriate coordinate and are numerically updated using the governing equations. By marching the neural network weights in the parameter space, EDNN can predict state-space trajectories that are indefinitely long, which is difficult for other neural network approaches. Boundary conditions of the PDEs are treated as hard constraints, are embedded into the neural network, and are therefore exactly satisfied throughout the entire solution trajectory. Several applications including the heat equation, the advection equation, the Burgers equation, the Kuramoto Sivashinsky equation, and the Navier-Stokes equations are solved to demonstrate the versatility and accuracy of EDNN. The application of EDNN to the incompressible Navier-Stokes equations embeds the divergence-free constraint into the network design so that the projection of the momentum equation to solenoidal space is implicitly achieved. The numerical results verify the accuracy of EDNN solutions relative to analytical and benchmark numerical solutions, both for the transient dynamics and statistics of the system.