Evolutional deep neural networks for accurate Navier-Stokes solutions and forecasts of turbulence

Evolutional deep neural networks (EDNN) are introduced for accurate solution of nonlinear partial differential equations (Du, Y., & Zaki, T. A. (2021). arXiv preprint arXiv:2103.09959). Training is only required for EDNN to represent the initial condition.  The network parameters are subsequently evolved, or marched, in time using the governing equations to provide accurate forecasts and without any further training.  Boundary conditions are treated as hard constraints that are enforced by the network design and therefore are exactly satisfied. For the solution of Navier-Stokes equations, the divergence-free constraint is embedded into the network structure. The solution of the momentum equation is thus guaranteed to be solenoidal, which eliminates the computationally costly pressure-projection step. Since EDNN represents the solution in space and evolves in time, the network architecture is compact and requires relatively low memory.  The evolutional nature of EDNN, where the parameters are updated using the equations rather than training within a specified time window, is suitable for forecasting the solutions of nonlinear chaotic systems, including turbulence, for indefinite time horizons.