Construction of Grid Operators for Multilevel Solvers: a Neural Network Approach
This work addresses a bottleneck in scientific computing by automating multigrid operator construction, though it appears incremental as it combines existing neural network techniques with multigrid methods.
The paper tackles the problem of constructing interpolation operators for multigrid methods in solving elliptic PDEs, which are typically unknown, by proposing deep neural network models to learn these operators, achieving results that enable an automatic and portable multilevel solver framework.
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