  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 equation 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.

Computational Physics;Machine Learning;Fluid Dynamics