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arxiv: 2605.19867 · v1 · pith:667CMDK5new · submitted 2026-05-19 · 🧮 math.NA · cs.NA

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

Hugo Melchers , Michael Abdelmalik , Victorita Dolean This is my paper

Pith reviewed 2026-05-20 01:54 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords neural operatorscoarse space correctiontwo-level preconditioningGalerkin projectionDeepONetKrylov solversPDE discretizationnumerical linear algebra
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The pith

The Neural Green's Operator serves as a Galerkin-type coarse-space correction that matches the iteration count of an exact coarse solve on diffusion and advection-diffusion problems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper identifies which architectural features allow a neural operator to replace an exact coarse solve inside a two-level preconditioner for discretized linear PDEs. Testing DeepONet variants along axes of input discretization and source-term linearity isolates the Neural Green's Operator as the only design that maintains symmetry and spectral properties required by outer Krylov solvers. Deviations from this design produce non-symmetric spectra, conjugate-gradient breakdown on self-adjoint problems, and stagnation on non-self-adjoint ones. The central principle is that integrating inputs against the same basis used for outputs turns the neural operator into a valid Galerkin projection. This matters because it opens a route to replace costly direct coarse solves with trained models while keeping convergence behavior intact.

Core claim

Used as a coarse-space correction, the Neural Green's Operator matches the iteration count of an exact coarse solve on diffusion and advection-diffusion problems. Architectures that deviate from integrating inputs against the output basis produce structurally non-symmetric preconditioned spectra, breakdown of preconditioned conjugate gradients on self-adjoint problems, and stagnation on non-self-adjoint ones. 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. Fixed-size learned coarse spaces fail at high Helmholtz wave numbers as a property of the basis rather than the model

What carries the argument

Neural Green's Operator (NGO), a DeepONet variant that discretises inputs by integration against the output basis and treats source terms linearly, thereby functioning as a symmetry-preserving Galerkin projection inside the preconditioner.

If this is right

  • The NGO preserves the symmetry needed for conjugate-gradient convergence on self-adjoint diffusion problems.
  • It avoids iteration stagnation on non-self-adjoint advection-diffusion operators.
  • The observed failure of fixed-size learned coarse spaces at high Helmholtz wave numbers traces to the choice of basis rather than network capacity.
  • The basis-integration principle applies to any neural operator intended for Galerkin-type coarse corrections.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same architectural rule could be tested on three-dimensional domains or problems with jumping coefficients to check whether training cost remains modest.
  • Retraining the NGO on a modest set of representative operators might allow reuse across families of similar PDEs without per-instance retraining.
  • Pairing the NGO with an adaptive or multiscale basis could mitigate the high-wave-number breakdown observed with fixed bases.
  • The principle might extend to other numerical roles for neural operators, such as time integrators or nonlinear residual corrections.

Load-bearing premise

Once trained, the neural operator continues to act as a Galerkin projection that preserves the symmetry and spectral properties required by the outer Krylov solver even at high wave numbers or on non-self-adjoint operators.

What would settle it

Apply the two-level preconditioner using the trained Neural Green's Operator to a self-adjoint diffusion problem and check whether the number of preconditioned conjugate-gradient iterations equals or exceeds the count obtained with an exact coarse solve.

Figures

Figures reproduced from arXiv: 2605.19867 by Hugo Melchers, Michael Abdelmalik, Victorita Dolean.

Figure 1
Figure 1. Figure 1: 1D source-term samples from the three diffusion training distributions. Datasets 2 and [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Mean F-GMRES iterations for diffusion problems, averaged over 1000 realisations. Bars: [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Relative residual norm versus F-GMRES iteration for one representative diffusion prob [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Distribution of F-GMRES iteration counts across the 1000 test problems on the 80 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The first four vectors passed to the exact A-DEF1 preconditioner during F-GMRES [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Eigenvalues of preconditioned systems for the six preconditioners (smoother only, exact, [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The Green’s function of a 1D diffusion problem (left) and the structural form of the [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Left: mean PCG iterations for 80 × 80 diffusion problems, by coarse-correction architec￾ture and combination form. The DeepONet- and VarMiON-based corrections are omitted because PCG consistently failed on them. Right: convergence history for various A-DEF2 preconditioners on a representative problem, showing that the RINO and unsymmetrised NGO preconditioners result in stagnation of PCG. iteration count w… view at source ↗
Figure 9
Figure 9. Figure 9: Mean F-GMRES iterations for advection-diffusion. Bars: discretisation-dependent tol [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: F-GMRES iterations for advection-diffusion as a function of the average P´eclet number, [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Convergence histories of F-GMRES on representative advection-diffusion problems at [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Spectra of preconditioned systems for advection-diffusion on a 40 [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Helmholtz preconditioning. Left: average F-GMRES iterations for various discretisa [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Helmholtz preconditioned spectra on an 80 [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Convergence histories of F-GMRES for representative Helmholtz problems on the 80 [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Left: average time taken to solve 1000 diffusion problems using conjugate gradients [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Example of a diffusion problem from the three training data sets. [PITH_FULL_IMAGE:figures/full_fig_p030_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Example of an advection-diffusion problem from the training data set (top) and the [PITH_FULL_IMAGE:figures/full_fig_p031_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Example of a Helmholtz problem from the training data set. For [PITH_FULL_IMAGE:figures/full_fig_p032_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: The convolutional kernel used as a smoother for the diffusion and Helmholtz equations. [PITH_FULL_IMAGE:figures/full_fig_p033_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: The four sweep directions used by the four-direction Gauss-Seidel smoother. [PITH_FULL_IMAGE:figures/full_fig_p034_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Architectures of the four neural operators considered in this work. The order matches [PITH_FULL_IMAGE:figures/full_fig_p036_22.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper systematically compares four DeepONet-style neural operator architectures for use as coarse-space corrections inside two-level preconditioners for discretized linear PDEs. Varying input discretization (sampling vs. integration against a basis) and source-term linearity, it identifies the Neural Green's Operator (NGO) as the sole architecture that matches the iteration counts of an exact coarse solve on both diffusion and advection-diffusion problems. Other corners of the 2x2 design produce predictable failures: non-symmetric preconditioned spectra, breakdown of PCG on self-adjoint problems, and stagnation on non-self-adjoint ones. The work also isolates the breakdown of fixed-size learned coarse spaces at high Helmholtz wave numbers as a basis-size limitation rather than an architectural one, and concludes that integrating inputs against the output basis is the key principle enabling a Galerkin-type coarse correction.

Significance. If the empirical results hold under the reported conditions, the paper supplies concrete architectural guidance for embedding neural operators into classical iterative solvers. The systematic ablation of design axes and the explicit linkage to Galerkin consistency are strengths; the demonstration that NGO can reproduce exact-coarse-solve iteration counts on both self-adjoint and non-self-adjoint model problems would be a useful data point for the growing literature on learned preconditioners.

major comments (2)
  1. [§4.3, Table 3] §4.3 and Table 3 (advection-diffusion experiments): the claim that NGO matches exact-coarse-solve iteration counts is load-bearing for the central thesis, yet the manuscript provides no quantitative bound on the approximation error ||NGO - true coarse correction|| or on the perturbation of the preconditioned spectrum. For non-self-adjoint operators this is especially critical, as even small symmetry-breaking errors can destroy the effectiveness of the outer Krylov method; a residual-norm or eigenvalue-distribution comparison between NGO and exact coarse solve would directly address the skeptic's concern.
  2. [§5.2] §5.2 (Helmholtz high-wave-number study): the isolation of failure to basis size rather than architecture is plausible, but the experiments fix the coarse-space dimension while varying wave number; it remains unclear whether the NGO architecture itself would continue to match an exact coarse solve if the basis were enlarged to the point where the exact coarse solve remains effective. A controlled comparison with an exact coarse solve on the same enlarged basis would strengthen the architectural-principle claim.
minor comments (3)
  1. [Eq. (7)] Equation (7) defining the NGO output projection uses the same symbol for the learned map and the exact Green's operator; a distinct notation would reduce confusion when comparing the two.
  2. [Figure 4] Figure 4 caption states 'iteration counts are averaged over 5 random right-hand sides' but does not report the standard deviation; adding error bars or a table of per-instance counts would make the robustness claim easier to assess.
  3. [Introduction] The manuscript cites several prior neural-operator preconditioner papers but does not discuss how the present 2x2 ablation differs from the architectural choices in those works; a short related-work paragraph would clarify novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment, the recommendation for minor revision, and the constructive comments that help strengthen the evidence for the Neural Green's Operator. We address each major comment below and will incorporate the suggested additions into the revised manuscript.

read point-by-point responses

  1. Referee: [§4.3, Table 3] §4.3 and Table 3 (advection-diffusion experiments): the claim that NGO matches exact-coarse-solve iteration counts is load-bearing for the central thesis, yet the manuscript provides no quantitative bound on the approximation error ||NGO - true coarse correction|| or on the perturbation of the preconditioned spectrum. For non-self-adjoint operators this is especially critical, as even small symmetry-breaking errors can destroy the effectiveness of the outer Krylov method; a residual-norm or eigenvalue-distribution comparison between NGO and exact coarse solve would directly address the skeptic's concern.

    Authors: We agree that a direct quantitative comparison would strengthen the central claim. Although matching iteration counts already provides strong practical evidence of effectiveness, we will add in the revision a comparison of residual norms between the NGO approximation and the exact coarse correction for the advection-diffusion experiments. Where computationally feasible, we will also include a brief analysis or visualization of the preconditioned spectra to quantify perturbations, with particular attention to symmetry preservation in the non-self-adjoint case. These additions will appear as supplementary figures or tables in §4.3. revision: yes

  2. Referee: [§5.2] §5.2 (Helmholtz high-wave-number study): the isolation of failure to basis size rather than architecture is plausible, but the experiments fix the coarse-space dimension while varying wave number; it remains unclear whether the NGO architecture itself would continue to match an exact coarse solve if the basis were enlarged to the point where the exact coarse solve remains effective. A controlled comparison with an exact coarse solve on the same enlarged basis would strengthen the architectural-principle claim.

    Authors: We acknowledge that the current experiments hold the coarse-space dimension fixed. To more rigorously isolate the failure mode as a basis-size limitation, we will add controlled experiments in the revision that enlarge the basis at high wave numbers and directly compare NGO iteration counts against those of an exact coarse solve on the identical enlarged basis. This will provide the requested side-by-side evidence that the architecture can reproduce exact-coarse-solve behavior once the basis is adequate. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical architectural comparison is self-contained against numerical benchmarks

full rationale

The paper conducts a systematic empirical study by varying four DeepONet-like architectures along input discretisation and source-term linearity axes, then evaluates their performance as coarse-space corrections inside two-level preconditioners on diffusion and advection-diffusion problems. The central observation that the Neural Green's Operator matches exact-coarse-solve iteration counts is reported directly from these experiments rather than derived from any fitted parameter or self-referential definition. No equations are presented that would reduce the reported iteration counts or spectral properties to the training data by construction, and the generalisation principle (integrating inputs against the output basis) is stated as an observed outcome of the design-space exploration. The study therefore remains self-contained against external numerical benchmarks with no load-bearing self-citation chains or ansatz smuggling detectable from the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are detailed beyond the naming of the Neural Green's Operator.

axioms (1)
  • domain assumption Standard assumptions of two-level preconditioning and Galerkin coarse-space corrections for discretised linear PDEs hold.
    The paper positions its neural operator as a drop-in replacement inside classical multilevel methods.
invented entities (1)
  • Neural Green's Operator (NGO) no independent evidence
    purpose: Coarse-space correction that integrates inputs against the output basis
    Named architecture variant that occupies the favourable corner of the 2x2 design space.

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