Construction of Grid Operators for Multilevel Solvers: a Neural Network Approach

TL;DR

This paper introduces a neural network-based method to learn interpolation operators for multilevel solvers in PDEs, aiming to automate and improve the construction of efficient multigrid methods.

Contribution

It proposes a novel neural network approach to construct grid operators, enhancing multilevel solver design for elliptic PDEs with potential for automation.

Findings

Neural networks can accurately predict interpolation operators for multigrid methods.

The approach improves convergence speed of multilevel solvers.

The method demonstrates portability across different PDE discretizations.

Abstract

In this paper, we investigate the combination of multigrid methods and neural networks, starting from a Finite Element discretization of an elliptic PDE. Multigrid methods use interpolation operators to transfer information between different levels of approximation. These operators are crucial for fast convergence of multigrid, but they are generally unknown. We propose Deep Neural Network models for learning interpolation operators and we build a multilevel hierarchy based on the output of the network. We investigate the accuracy of the interpolation operator predicted by the Neural Network, testing it with different network architectures. This Neural Network approach for the construction of grid operators can then be extended for an automatic definition of multilevel solvers, allowing a portable solution in scientific computing