When can a neural operator replace a coarse solve? Architectural principles for two-level preconditioning

TL;DR

This paper investigates architectural principles for neural operators used as coarse-space corrections in two-level preconditioners for PDEs, identifying the Neural Green's Operator as optimal among tested designs.

Contribution

It systematically compares four DeepONet-like architectures, demonstrating that the Neural Green's Operator best preserves spectral properties and efficiency in preconditioning.

Findings

Neural Green's Operator matches iteration counts of exact coarse solves.

Moving away from NGO causes spectral and convergence failures.

Integrating inputs against the basis is key for neural operators to serve as Galerkin coarse corrections.

Abstract

Neural operators are increasingly used as drop-in accelerators inside classical numerical methods, but it is rarely clear which architectural ingredients matter for which role. We answer this question for one important role: the coarse-space correction inside a two-level preconditioner for discretised linear partial differential equations. By systematically varying four DeepONet-like architectures along two design axes - input discretisation (sampling versus integration against a basis) and source-term linearity - we show that the favourable corner of this 2$\times$2 design is occupied by a single architecture, the Neural Green's Operator (NGO), and that moving away from it produces predictable failure modes: structurally non-symmetric preconditioned spectra, breakdown of preconditioned conjugate gradients on self-adjoint problems, and stagnation on non-self-adjoint ones. Used as a coarse-space correction, the NGO matches the iteration count of an exact coarse solve on diffusion and advection-diffusion problems. We also characterise the failure of fixed-size learned coarse spaces at high Helmholtz wave numbers, isolating it as a property of the basis rather than of the architecture. The principle generalises: integrating inputs against the basis used for the output is what allows a neural operator to serve as a Galerkin-type coarse-space correction.