Accelerated Elliptical PDE Solver for Computational Fluid Dynamics Based on Configurable U-Net Architecture : Analogy to V-cycle Multigrid

A configurable U-Net architecture is trained to solve the multi-scale elliptical partial differential equations. The motivation is to improve the computational cost of the numerical solution of Navier-Stokes equations – the governing equations for fluid dynamics. Building on the underlying concept of V-cycle multigrid methods, a neural network framework using U-Net architecture is optimized to solve the Poisson equation and Helmholtz equations – the characteristic form of the discretized Navier-Stokes equations. The results demonstrate the optimized U-Net captures the high dimensional mathematical features of the elliptical operator and with a better convergence than the multigrid method. The optimal performance between the errors and the FLOPS is the (3, 2, 5) case with 3 stacks of U-Nets, with 2 initial features, 5 depth layers and with ELU activation. Further, by training the network with the multi-scale synthetic data the finer features of the physical system are captured.