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

arXiv:2605.19867v2 Announce Type: replace Abstract: Neural operators are increasingly used as accelerators inside classical numerical methods, but it is rarely clear which architectural ingredients matter for which application. We answer this question for one important use case: the coarse-space correction inside a two-level preconditioner for discretised linear partial differential equations. We systematically vary four DeepONet-like architectures along two design axes: input discretisation (sampling versus integration against a basis) and source-term linearity. In doing this, 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. 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.