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This paper proposes the method to optimize restriction and prolongationoperators in the two-grid method. The proposed method is straightforwardlyextended to the geometric multigrid method (GMM). GMM is used in solvingdiscretized partial differential equation (PDE) and based on the restrictionand prolongation operators. The operators are crucial for fast convergence ofGMM, but they are unknown. To find them we propose a reformulation of thetwo-grid method in terms of a deep neural network with a specific architecture.This architecture is based on the idea that every operation in the two-gridmethod can be considered as a layer of a deep neural network. The parameters oflayers correspond to the restriction and prolongation operators. Therefore, westate an optimization problem with respect to these operators and get optimalones through backpropagation approach. To illustrate the performance of theproposed approach, we carry out experiments on the discretized Laplaceequation, Helmholtz equation and singularly perturbed convection-diffusionequation and demonstrate that proposed approach gives operators, which lead tofaster convergence.

Deep Multigrid: learning prolongation and restriction matrices