Automatic discovery of optimal meta-solvers via multi-objective optimization

TL;DR

This paper introduces ultra-fast meta-solvers for linear systems from PDE discretizations, combining neural operators with classical iterative or Krylov methods, optimized via multi-objective Pareto fronts for specific scenarios.

Contribution

It presents a novel approach to automatically discover optimal meta-solvers by integrating neural operators with traditional methods, leveraging spectral bias and multi-objective optimization.

Findings

Created Pareto fronts of optimal meta-solvers

Demonstrated effective selection of solvers for specific scenarios

Extended approach potential to nonlinear systems and time-dependent PDEs

Abstract

We design two classes of ultra-fast meta-solvers for linear systems arising after discretizing PDEs by combining neural operators with either simple iterative solvers, e.g., Jacobi and Gauss-Seidel, or with Krylov methods, e.g., GMRES and BiCGStab, using the trunk basis of DeepONet as a coarse preconditioner. The idea is to leverage the spectral bias of neural networks to account for the lower part of the spectrum in the error distribution while the upper part is handled easily and inexpensively using relaxation methods or fine-scale preconditioners. We create a pareto front of optimal meta-solvers using a plurarilty of metrics, and we introduce a preference function to select the best solver most suitable for a specific scenario. This automation for finding optimal solvers can be extended to nonlinear systems and other setups, e.g. finding the best meta-solver for space-time in time-dependent PDEs.