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arxiv: 2512.10723 · v3 · submitted 2025-12-11 · 💻 cs.LG

Generalized Spherical Neural Operators: Green's Function Formulation

Hao Tang , Hao Chen , Chao Li This is my paper

Pith reviewed 2026-05-16 23:02 UTC · model grok-4.3

classification 💻 cs.LG
keywords spherical neural operatorsGreen's functionspherical harmonicsrotational equivarianceparametric PDEsweather forecastingdiffusion MRIgrid invariance
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The pith

A designable Green's function on the sphere lets neural operators flexibly mix rotational symmetry with real-world asymmetry while keeping spectral speed and grid freedom.

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

The paper builds a generalized framework for spherical neural operators from a customizable Green's function and its spherical-harmonic expansion. This kernel supplies an operator-theoretic basis that supports both absolute and relative position dependence, letting the operator trade off exact rotational equivariance against the invariance needed for irregular data. The resulting GSNO uses a new spectral learning procedure to stay efficient and grid-invariant. A hierarchical architecture called SHNet combines multi-scale spectral blocks with spherical up-down sampling to improve global feature capture. Tests on diffusion MRI, shallow-water flow, and global weather prediction show consistent gains over prior spherical operators.

Core claim

A designable spherical Green's function together with its harmonic expansion supplies a solid operator-theoretic foundation for spherical learning; absolute and relative position-dependent versions of this kernel enable a tunable balance between equivariance and invariance; the resulting Green's-function Spherical Neural Operator (GSNO), equipped with a novel spectral learning method, adapts to non-equivariant systems while retaining spectral efficiency and grid invariance.

What carries the argument

The designable spherical Green's function and its harmonic expansion, which acts as the integral kernel of the neural operator and controls the equivariance-invariance trade-off through absolute and relative positional weighting.

If this is right

  • GSNO can model systems that violate rotational symmetry without losing computational efficiency.
  • The operator remains invariant to the choice of spherical grid discretization.
  • SHNet's multi-scale spectral modeling plus spherical sampling improves global feature representation.
  • Empirical results show outperformance on diffusion MRI, shallow-water dynamics, and global weather forecasting.

Where Pith is reading between the lines

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

  • Similar Green's-function constructions could extend the approach to other curved manifolds beyond the sphere.
  • The spectral learning procedure may reduce training cost for large-scale global simulation tasks.
  • The framework offers a route to hybrid operators that default to equivariance but relax it only where data demands it.

Load-bearing premise

A designable spherical Green's function and its harmonic expansion can be constructed to balance equivariance and invariance without introducing unaccounted distortions in complex real-world systems.

What would settle it

A controlled spherical PDE experiment where the learned GSNO produces larger errors or visible grid artifacts than a strictly equivariant baseline on data that breaks rotational symmetry.

Figures

Figures reproduced from arXiv: 2512.10723 by Chao Li, Hao Chen, Hao Tang.

Figure 1
Figure 1. Figure 1: Method comparison under the Green’s function framework (Details in Sec. 4.1). (Left) Ex [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The proposed GSNO block (left) and the architecture of SHNet (right). SHT and ISHT [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Predicted geopotential height (H) at 5h and 10h from different methods. Zoom in and [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Predicted V600 samples of different methods. Each image set shows the main prediction [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Predicted dMRI fiber orientation distribution samples of different methods in the same [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Predicted temperature field at 2m from the earth surface of GSNO w/o I [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
read the original abstract

Neural operators offer powerful approaches for solving parametric partial differential equations, but extending them to spherical domains remains challenging due to the need to preserve intrinsic geometry while avoiding distortions that break rotational consistency. Existing spherical operators rely on rotational equivariance but often lack the flexibility for real-world complexity. We propose a generalized operator-design framework based on the designable spherical Green's function and its harmonic expansion, establishing a solid operator-theoretic foundation for spherical learning. Based on this, we propose an absolute and relative position-dependent Green's function that enables flexible balance of equivariance and invariance for real-world modeling. The resulting operator, Green's-function Spherical Neural Operator (GSNO) with a novel spectral learning method, can adapt to non-equivariant systems while retaining spectral efficiency and grid invariance. To exploit GSNO, we develop SHNet, a hierarchical architecture that combines multi-scale spectral modeling with spherical up-down sampling, enhancing global feature representation. Evaluations on diffusion MRI, shallow water dynamics, and global weather forecasting, GSNO and SHNet consistently outperform state-of-the-art methods. The theoretical and experimental results position GSNO as a principled and generalized framework for spherical operator design and learning, bridging rigorous theory with real-world complexity. The code is available at: https://github.com/haot2025/GSNO.

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 / 2 minor

Summary. The paper proposes a generalized framework for spherical neural operators grounded in a designable spherical Green's function and its harmonic expansion. It introduces the Green's-function Spherical Neural Operator (GSNO) with a novel spectral learning method based on absolute and relative position-dependent Green's functions to flexibly balance equivariance and invariance. The work also presents SHNet, a hierarchical architecture combining multi-scale spectral modeling with spherical up-down sampling. Evaluations on diffusion MRI, shallow water dynamics, and global weather forecasting claim consistent outperformance over state-of-the-art methods, with code released.

Significance. If the central claims hold, particularly that the position-dependent Green's function construction yields an operator that is simultaneously non-equivariant and exactly grid-invariant with spectral efficiency, the work would supply a principled operator-theoretic foundation for spherical learning. This could meaningfully advance applications in climate modeling, fluid dynamics, and medical imaging on spherical domains, especially given the code release for reproducibility.

major comments (2)
  1. [Theoretical development] Theoretical development (Green's function formulation): The claim that the harmonic expansion of the absolute/relative position-dependent Green's function yields an operator that retains exact grid invariance and spectral efficiency for non-equivariant systems is not supported by an explicit truncation-error bound or a proof that learned coefficients remain consistent across arbitrary discretizations; any finite truncation produces a kernel that is no longer exactly diagonal in the spherical-harmonic basis, directly threatening the central claim once rotational equivariance is relaxed.
  2. [Abstract and evaluations] Abstract and experimental sections: The outperformance statements on the three tasks are presented without reported error bars, ablation studies isolating the spectral learning method, or explicit comparisons against strong equivariant baselines, leaving open whether post-hoc design choices or data-specific factors drive the reported gains.
minor comments (2)
  1. [Abstract] The abstract refers to 'designable spherical Green's function' without a concise definition or reference to the precise functional form used in the construction.
  2. [Theoretical development] Notation for the absolute versus relative position dependence should be introduced with a single equation early in the theoretical section to avoid later ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and outline the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Theoretical development] Theoretical development (Green's function formulation): The claim that the harmonic expansion of the absolute/relative position-dependent Green's function yields an operator that retains exact grid invariance and spectral efficiency for non-equivariant systems is not supported by an explicit truncation-error bound or a proof that learned coefficients remain consistent across arbitrary discretizations; any finite truncation produces a kernel that is no longer exactly diagonal in the spherical-harmonic basis, directly threatening the central claim once rotational equivariance is relaxed.

    Authors: We agree that an explicit truncation-error bound would strengthen the theoretical claims. The manuscript presents the harmonic expansion as exact in the infinite-dimensional spherical-harmonic basis, with the position-dependent Green's function (absolute and relative) designed so that the learned operator remains diagonal in that basis by construction. Finite truncation is an approximation shared by all spectral methods; grid invariance holds exactly for the continuous operator and is preserved up to the discretization error of the chosen quadrature. To address the referee's concern directly, we will add a new subsection deriving a truncation-error bound that quantifies the deviation from exact diagonality as a function of the number of retained harmonics, together with a consistency argument showing that the learned coefficients converge to the same continuous operator under refinement of the discretization. This revision will be included in the next version. revision: yes

  2. Referee: [Abstract and evaluations] Abstract and experimental sections: The outperformance statements on the three tasks are presented without reported error bars, ablation studies isolating the spectral learning method, or explicit comparisons against strong equivariant baselines, leaving open whether post-hoc design choices or data-specific factors drive the reported gains.

    Authors: We acknowledge that the current experimental section would benefit from additional statistical rigor and controls. In the revised manuscript we will: (i) report mean performance with standard error bars computed over at least five independent runs with different random seeds; (ii) add ablation studies that isolate the contribution of the absolute versus relative position-dependent Green's functions and the spectral learning procedure; and (iii) include direct comparisons against strong equivariant baselines (e.g., spherical CNNs and rotation-equivariant neural operators) on the same data splits. These additions will clarify the source of the observed improvements and will be placed in the experimental section with corresponding updates to the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity: Green's function framework presented as independent design choice

full rationale

The provided abstract and description frame the spherical Green's function, its harmonic expansion, and the absolute/relative position-dependent variants as a proposed operator-theoretic foundation chosen by design. No equations, derivations, or steps are exhibited that reduce any prediction or result to a fitted parameter, self-citation chain, or definitional tautology. The spectral learning method and SHNet architecture are introduced as novel extensions without visible reduction to the inputs by construction. This matches the default expectation that most papers are non-circular; the central claims rest on the design choice itself rather than any self-referential loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence of a designable spherical Green's function whose harmonic expansion yields a valid operator foundation; this is treated as a domain assumption rather than derived from first principles within the abstract.

axioms (1)
  • domain assumption Spherical Green's functions can be designed to preserve intrinsic geometry and support flexible equivariance-invariance balance
    Invoked as the basis for the entire GSNO framework in the abstract.
invented entities (1)
  • Absolute and relative position-dependent Green's function no independent evidence
    purpose: To enable tunable balance between rotational equivariance and invariance for real-world spherical data
    Newly proposed construct in the operator design framework

pith-pipeline@v0.9.0 · 5516 in / 1342 out tokens · 40124 ms · 2026-05-16T23:02:52.325235+00:00 · methodology

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    20 Published as a conference paper at ICLR 2026 Table 10: Parameters on dMRI. Models FOD-Net FOD-SFNO ESCNN FOD-GSNO Parameters 19.44 M 1.15 M 1.47 M 1.21 M Figure 6: Predicted temperature field at 2m from the earth surface of GSNO w/o I(T orig). Provide the residual (error) to the ground truth for clarity (darker is better). C.5 EXTRAEXPERIMENTS ANDINTER...

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