Generalized Spherical Neural Operators: Green’s Function Formulation
Authors: Hao Tang
Hao Chen
Chao Li
Venue: ICLR 2026 Poster (2026)
Keywords: Fourier neural operator
Green function
Spherical harmonic
Primary area: learning on graphs and other geometries & topologies

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 designable 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 spherical geometry, 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 learning, bridging rigorous theory with real-world complexity. The code is available at: https://github.com/haot2025/GSNO.

Published as a conference paper at ICLR 2026
GENERALIZED SPHERICAL NEURAL GREEN'S FUNCTION FORMULATION
OPERATORS:
Hao Tang!
Hao Chen
Chao Li
University of Dundee, United Kingdom Universiit Cambridge, United Kingdom
ABSTRACT
Neural operators offer powerful approaches for solving parametric partial differ- ential equations, but extending them to spherical domains remains challenging due to the need to preserve intrinsic geometry while avoiding distortions that break ro- tational consistency. Existing spherical operators rely on rotational equivariance but often lack the flexibility for real-world complexity. We propose generalized operator-design framework based on designable Green's function and its har- monic expansion, establishing a solid operator-theorfs theoretic foundation for spherical learning. Based on this, we propose an absolute and relative position-dee dependent Green's function that enables flexible balance of equivariance and invariance for real-world modeling. The resulting operator, Green's 's- is-function Spherical Neu- ral Operator (GSNO) with a novel spectral learning method, can adapt to non- equivariant systems while retaining spherical geometry, 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 dynamies, and global weather forecasting, GSNO and SHNet consistently outperform state-of-the-art methods. The theoretical and experimental results po- sition 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:
INTRODUCTION
Solving parameterized partial differential equations (PDEs) is fundamental task across science and engineering. Applications such as weather forecasting, fluid dynamics, and neu- roimaging often involve high-order PDEs whose numerical solutions computationally expensive and intractable. Emerging neural operators provide a promising alternative by approximating solu- tion operators directly from data (Kovachki al. 2024). Earlier approaches (Lu et al. 2019 2021 Bhattacharya et al.) 2021) learned mapping between function spaces using neural networks but strug- gled scale nigh-dimensional PDEs. To address this, the Fourier Neural Operator (FNO) (Li et al. 2020a) leverages Fast Fourier transforms (FFTTs) to learn in the frequency domain, capturing global patterns and nigh-freeuency modes (Kovachki al. 2024). This inspired a new generation of operator learning methods and advanced large-scale applications such as high-resolution climate prediction (Pathak et al.) 2022 Liu et 2024).
Background:
However, FNOs rely on the standard Fourier transform and assume Euclidean geometry. On non- Fuelidean manifolds such as the sphere (Bonev al. 2023). FFT-based representations introduce distortions: small polar displacements can map to large Cartesian displacements, breaking spatial coherence and degrading performance. To address this, Spherical Fourier Neural Operator (SFNO) is proposed (Bonev et al.) 2023), replacing the FFT with the Spherical Harmonic Transform (SHT). By projecting functions onto spherical harmonic bases, SENO preserves rotational equivariance on the sphere, ensuring stability under arbitrary input rotations. SFNO-based methods have achieved strong performance on some spherical tasks, e.g., weather prediction (Lin et al. 2023) Mahesh et al.
2024a b: Hu et al. 2025).

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Published as a conference paper at ICLR 2026
For the solution to system: (( f(x), wes²
f(u)
Define Green's 's fumction the system: D(G(u, R)) 6(R (1)
g (u)
Input
Target
Solution system: g(u) Ig. ((u, R)f (Rn))
Existing methods for oquivariance: Assume G(u, R) as uu)
Ours for balancing equivariance and system modeling:
system field F F(u)
Design R) the system
((x) 1SHT (4)
solution system: g (u)
Designable Green's ''-Fasuttion Framework
Figure 1: Method comparison under the Green's 's function framework (Details in Sec. 4.1 (Left) Ex- isting approaches use SHT-based spectral learning (symmetric kernel). (Right) Our method extends sysyem characteristics by designable Green's 's function derive the system-constraint solution.
Despite these advances, fundamental challenges yet to be addressed for spherical operator learning:
(1) Theory: most spherical neural operators are rigorously constructed by Spherical Harmonic
Transform and spherical convolution theorem, thereby extending the FNO to the sphere, rather than derived from the integral solution of sphere-native PDEs (Bonev et al. 2023] Lin et al.] 2023 Ma-
hesh et al. 2024a b, Hu al. 2025). This lack of rigorous grounding obseures the physical meaning of the learned integral kernels and limits their generalization and extension in physical systems. (2)
Neural Operator Block: Physical systems involve non-equivariance and asymmetric constraints,
such as boundary effects, position-relevant patterns, or heterogeneous media (Ye et al.) 2022: 2023
Lucarini & Chekroun 2024 Behroozi al. 2025). which cannot be captured if operators depend solely on equivariance. However, by underlying design, most spherical operators enforce strict rota- tional equivariance to gain efficiency for spectral learning, overlooking such asymmetry and thereby limiting their ability to model complex, high-order nonlinear phenomena. (3) Neural Operator Net- work: Existing spherical operator networks use ResNet-style architecture [Bonev et al. (2023) op- erating single scale to capture global structure, but may limit the expressiveness in modeling climate dynamics (Zhao al., 2019
202T
al. 2025).
Our Approach: (1) We first derive the principled formulation of spherical operator learning that begins from a designable Green's function. This derivation extends the Green's function naturally from the sphere to the spherical harmonic domain, providing generalized operator-theoreti framework of spherical learning, which supports the simulation different complex systems by designing different Green's 's functions. (2) Based this foundation, enhance the flexibility of spherical operators on real-world systems by designing an absolute and relative position-dependen Green's function as explicit inductive bias, deriving the corresponding response explicitly bal- ance equivariance and invariance. The yielded Green's s-Function Spherical Neural Operator (GSNO) balances the spectral efficiency with the capacity to capture complex system constraints bound- aries and distortions while retaining spherical geometry and grid invariance, enabling the nonlinear modeling in real-world, -quivvrrvar physical systems. (3) Finally, we propose the Spherical Harmonic Neural Operator Network (SHNet), a hierarchical architecture that integrates GSNO with multi-scale spectral modeling through spherical up-down sampling, enhancing the expressive power while preserving geometric consistency across resolutions.
Contributions: This work mainly makes the following contributions:
generalized theoretical framework to derive system- based operator solutions based on the designable spherical Green's functions (Sec. 4.1
2. We design an absolute and relative position-dependent Green's function and derive the corre- sponding response to propose a novel operator (GSNO) that flexibly balances equivariance and invariance and models complex systems while retaining grid invariance(Sec. 4.2)
3. We design SHNet, multi-scale spherical network based GSNO (Sec. 4.3
2

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Published as a conference paper at ICLR 2026
4. GSNO and SHNet evaluated on the diffusion MRI modeling of brain microstructure, spher- ical shallow water equations and weather prediction, demonstrating consistent improvements in performance other state-of-the-ar models.
RELATED WORK
Geometric equivariance and complex constraints. Geometric equivariance has been explored through transformations such as translation, scaling, and dilation of filters, in both discrete and continuous domains (Xu et al. 2014] Sosnovik et al. 2021; Rahman & Ych. 2023 Chen et al., 2023). These approaches contribute to the framework of geometric deep learning by incorporating
rich symmetry groups (Cohen & Welling 2016 Bronstein et al., 2021 Several methods embed
inductive biases to enhance generalization (Wad et al. 2022 Han et al.
2022); for instance, CNNs inherently equivariant to translations (Li al. (021). Group-equivariant neural networks (Cohen
& Welling 2016) Cohen et al. 2018), along with spectral approaches such as DISCO (Ocampo et al. 2022). further exploit symmetries to improve learning efficacy, However, for real-world modeling, there are some components that relax rotational equivariance in most spherical CNN-based models,
e.g. residual pathways, position embeddings, activation functions and local operations (Cohen &
Welling, 2016; Cohen et al. 2018: Bonev et al.) 2023 Finzi et al. [021] Liu-Schiaffini et al. 2024).
Other methods are proposed further introduce complex constraints and relax strict equivariance
for better real-world modeling capabilities (Finzi et al. 2021 Huang et al. 2022 Wang et al. 2022
Duval et al. 2023 Pertigkiozoglou al. 2024 Zheng ct al. 2024 Li et al. 2024
Neural operators with multi-scale modeling. Recent extensions introduce multi-scale learning enhance feature representation. Some neural operators (Li al. [020c) Lütjens al., [022] Raonic et al. 2023; You et al., 22024] Liang et al. 2024 Liu et al. 2025) tackle this by combin- ing both upsampling and downsampling operations capturing multiscale information. However, the
operations may introduce aliasing artifacts (Ronneberger et al. 2015)] Karras et al. (021) and dis-
tortions, limiting modeling accuracy, especially spherical space (Zhao al. 2019 2021.
3 PRELIMINARY
3.1 SPHERICAL HARMONIC
The spherical harmonic function (Müller 2006) (0, d), with integer degrees 0 and orders
|m| S I, forms an orthonormal basis for square-integrable functions on the unit sphere 52, Spherical harmonic transform (SHT) decomposes a function f E (S(2)) into its harmonic coefficients:
(H.(f((.mm = J f(W)Y("(w) de, w E S
(1)
The orginal function f exactly reconstructed via inverse spherical harmonic transform (ISHT):
f(w) = ISHT(SHT((((,mm)
STT
1=U
Given two functions f and h defined on the sphere S², their spherical convolution is defined as (Driscoll & &eaaly 1994):
(f * h))
f(Rm)h(R-
(3)
50(3)
where S2 denotes the north pole, and R is an element of the three-dimensional rotation group (3(3). The SHT coefficients are given by:
SHT((S * h1](1, / (f ((*h)(W)Y" dw
(4)
3

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Published as a conference paper at ICLR 2026
3.2 SPHERICAL CONVOLUTION THEOREM
To derive the SHT coefficients of a spherical convolution, the result of the spherical convolution
theorem (Driscoll & Healy 1994) is follows:
4x 21 SHT SHT[fI(4, m).
SHT((f * h)]( m) = 2n
(5)
This result demonstrates that spherical convolution in the harmonic domain corresponds product of the SHT coefficients of f and h (with restricted to the m = 0 mode) in the frequency domain. The full derivation is in the Appendix A By replacing the classical convolution theorem with this spherical convolution theorem, SFNOs achieve Trequency-domall parameterization on the sphere (Bonev et al. [2233] Lin et al., 2023) Hu et (025). To enhance interpretability, we introduce two complementary SHT-based neural operator derivations in spherical space (Sec. 4.1)
4
METHODOLOGY
4.1 OPERATOR FRAMEWORK VIA SPHERICAL GREEN'S 's FUNCTION
The Green's 's function method offers classical strategy to solve PDEs, where the solution is pressed as a convolution integral with the Green's function as the kernel (Li et al. 2020b). Differ- ently, we define D, linear differential operator the sphere, and consider the following PDE:
D(g(u)) f(u), S².
where f(u) is the input function and g(u) is the target solution. The proposed spherical Green's function G, associated with D, is defined by the property:
{ 0.
00,
Rn = Rn +
D(G(u, R)) 5(R u)
where 8()) denoting the Dirac delta function defined on the sphere, and R SO(3) represents rotation from the north pole n. Using the Green function, the solution to Equation is:
g(u) = . G(u, (R) dR.
(8)
To verify this solution, we apply the operator D to both sides of Equation
D(g(u)) = (fg
g² G(u, R) dR
D(G(u, f Rn) dR
((R'u))(RRi))) dR
f(u).
In this framework, propose an operator design method based on the designable Green's function to simulate different systems. For example, classic spectral operators assume the strict rotational equivariant mapping/system: Design G(u, R) as G(R-Lu). -u). Under this relative-position dependent Green's 's function G ( 'u), the prediction target g(u) is given by:
g(u) = [ G(R 1 (Rn) dR.
(10)
Applying the spherical convolution theorem (Equation 5) the spherical harmonic transform of g()) is simplified as:
SHT[p(w)](, m) = SHT * G))
4x
= =2T
(HT (H))
(1I)
21
4

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Published as a conference paper at ICLR 2026
GSNO Block
GØ, ())
Wy
Ma.. MD
Desoder
SHT
Cb. (I)
Excode:
01.99
ISHT
ISHI
CRENO
GSNO
Poeiiin Enbelding
Add
Con 1x
Conoakeazz
GENO
Clasnds
Leindaal mazpk
W: Leuptatraiuusie
(Pasts Moni)
Sphericul Hurmonic Neural Operator Network (SIINet)
Figure 2: The proposed GSNO block (left) and the architecture of SHNet (right). SHT and ISHT represent spherical harmonic transformation and inverse transformation. Multi-layer perceptrons (MLPs) and two convolutional layers are used for channel interaction.
Thus, the target function g(u) is reconstructed via the ISHT (Equation22
g(u) (SHT( SHT]S[((,m))
(12)
where G1( denote the learnable spectral weights parameterized by the neural operator. This sistency with SFNOs verifies the feasibility this designable Green s-ffncctinn framework.
Spherical Harmonic Extension. The above derivation relies spherical convolution theorem. To extend, we propose to parse neural operators entirely from spherical harmonics and present a derivation based on the harmonic extension of Green's 's function (Groemer 1996). Green's function is extended as:
G(u)
(13)
L=0 m=-
Therefore, the designable form of the Green's 's function G(u, R) is naturally extended to harmonic domain without disrupting spherical geometry. Substituting into the definition of g we derive the SHT of target as follows:
SH Ssocs S(Rn) (1ss
G(R Y(T du dR
/ S0 f(Rm)
SHT[G] m)) m/)yy' (R (u)"") du dR
4T
2n
+1 SHT(G] SSTT[)] m).
(14)
21
Key derivation steps are detailed in Appendix Note that the derived result is also consistent
with the outcome obtained from Equation 11 (the target (g(u) follows Equation 12
Therefore, this section presents a comprehensive and solid theoretical framework simulate differ- ent systems based on designable Green's 's functions, deriving the corresponding operator solutions, shown the Figure Equation (11) and (14) are both the solution derived from the relative-position dependent Green's 's function, demonstrating the designable and scalable forms of Green's 's function: G(R u) and its spherical harmonic extension.
4.2 GREEN'S S-FUNCTION SPHERICAL NEURAL OPERATOR
Spherical spectral convolutions assume the Green's 's function depends solely the relative position R uu through rotational equivariance (G(R, u) = G(R w)). However, this strong assumption ig- nores the applicability to real-world non-equivariance, e-g., position-deeendent geophysical settings
5

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characterized by anisotropy. local heterogeneity, and non-periodic boundary conditions (Van Essen (et al. 2013; Lucarini Chekroun, 2024). To address this limitation, the Green's function is de- signed by combining the relative and absolute position-dependent terms. This enables modeling of invariant physical properties without sacrificing the efficiency of spectral-domain computation. We define the extended Green's function hypothesis as:
G u) = Σ SHT[G]] 17
[VT" (R- + bias(u))
I,mm
Σ
SHT[G [Y7 (R- + SHT(G2)1 W YT" (u)]
P.m'
S Y?" (R)u)) Σ P.mm
SHT[G
SHT G2lF Mo (u)
(15)
Original (Toig)
(Tour)
The total operator thus comprises two components: Tong is the original equivariant term, preserving the rotational equivariant characteristics, while Teurr novel, learnable, non-equivariant term that captures spatial constraints and heterogeneities, models complex constraint conditions. Based on this formulation, the SHT result of target g(u) is:
SHT (HT))( = SHT [J * +)+ SHT((( * ((,
m) ((Tang)
(16)
I(Torr)
Following the derivation in Sec. 4.1
the equivariant component
4m
I(Tori)= = V 21 +1
SHT(G) SHT(s((,,
= Gb (I) SHT
(17)
For the correction term:
I(Teom)
[(Rn)
Σ
SHT[G SHT( yo (u) Y (u) dR
S80(3)
52
P'mm
I Ssors) f(Rn)dR
Gb,y (l. Cj- G%2 (L.
(18)
where Cý is the spherical integral of the input function (f(u)). The original term I(Tani) and the correction term I(Teur) share Gb, (4). To reduce parameter redundancy between G (1. m) and G2 (1, m) and retain high computational efficiency, we simplify Equation 18 to:
I(Teun) G (1) Cj G (L.
(19)
Combining both components I(Torig) and I(Teurt) the final output is:
g(u) ((T I(Temr)) ISHTIG (1) (SHT[f](I,)) m)) Cy P.-Gh, m)))).
(20)
The parameter quantity of G),, (1) much larger than that of G3, (1, m) in applications because GJ,, (I), as the outer main weight, needs to represent cross-channel interaction, while G2, m). as the inner biased weight the same shape as (H, m), has no interaction requirement with Cy-
Based on Equation 20 GSNO block is designed (Figure 2 left). The input spherical feature f is first transformed into spherical harmonic coefficients through SHT. In parallel, the spherical integral Cj of input f is used to modulate the kernel G2, (1, m) obtain the complete correction term. Then, the sum of the spherical harmonic coefficient and the correction term undergo a generalized multiplication of the tensor contraction with G2, (!) obtain the transformed spherical harmonic

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Table MREJ (x10-3) on SSWE of test variables and their means at 5h and 10h across models. Bold indicates best performance. H, V and D are variables on SSWE tasks.
Method
MRE 5
MRE at 10
H
Average
V
D
Average
ClimaX Novyen 2023
30003.00
1000
FaurCaiter (PStli
3.40
7.60
302 2.00
310
737
8.95
SFNONe: Boxey 2 ai 2023
1,30
1 0.00
145 1 11.1
295 8.87
-
1.08
0.12
229
15.6
353
13.5
195
SHINet (Ours)
1.26 0.07
134
9.30
273 5.91
136 3.73
1.49 0.09
201
11.9
326
12.1
176 1 5.71
coefficient, which is finally converted to the spherical characteristics of the transformed output through ISHT. Thus, the complex transformations and parameter learning in the spherical space transformed into simple operations in the frequency domain. To further enhance nonlinear modeling, apply multi-layer perceptron (MLP) with two x convolutional layers and GELU activation function for channel interaction. Two additional light-weight convolutional layers are used for linear interaction, channel transformation, and skip connections (residuals).
GSNO combines equivariant response (G1) and invariant response (G2), enabling simultancous modeling of real non-equivariant systems. Specifically, G1 encodes more dynamic features, G2 provides mode-level spectral embedding, which explicitly encodes systematic and more stable con- straints, including local con-uniformity and boundary constraints (e.g., (e.g.weeather prediction affected by terrain) while retaining grid invariance and avoiding influence on the feature representation rely- ing on rotational equivariance by sufficient parameter learning. In addition, GSNO maintains high computational efficiency and low parameters SFNO. In summary, GSNO improves spherical op- erators by relaxing the strict (3(3) equivariance flexibly, allowing real-world modeling without disrupting spherical geometry.
4.3 SPHERICAL. HARMONIC NEURAL OPERATOR NETWORK
While spectral convolution excels in capturing global dependencies, it limited in multi-scale mod- cling. Building on GSNO, we propose a multi-scale spherical harmonic network, SHNet (Figure 2 right). SHNet adopts a U-Net structure (Zhao al. 2019 202T; Hu et al. 2023) for expansion and compression of spatial and channel, and incorporates position embedding to enhance global mod- eling Booev al. (2023). As core component, the GSNO block performs scale transformation of the spatial and spectral domains based on SHT and ISHT, and channel transformation via MLP. Specifically, we achieve scale transformation by modifying the number of sampling points along 0 (latitude) and @ (longitude) in Equation 2 and the degree / in Equation I which avoids distor- tions caused by traditional up- and down- wwwn-sampinng. The downsampling blocks reduce the sampling points and enhance feature expression through MLP, thereby realizing the abstraction of large-scale content, with skip connections providing direct
features. The upsampling restores nigher-frequent access to high-resolution information. This desgin allows SHNet to capture multi-scale interaction via geometric up- and down-sampling on the sphere (details in Appendix B.2
EXPERIMENTS
We compare GSNO and SHNet to other state-of-the-art methods under identical experimental setup and configuration (Liu et al. 2024), e.g., Transformer-based ClimaX (Nguyen et al. (023), FNO- based FourCastNet (Pathak et al. 2022). and SFNO-based SFNONet (Bonev et al., 2023) (detailed in Appendix C.I). The models are compared on diffusion magnetic resonance imaging (dMRI) modeling and two autoregressive spherical datasets. We also conduct ablation experiments further verify the effectiveness of the proposed GSNO block and SHNet structure. All experiments are implemented through PyTorch on 16GB A5000 GPUs.
5.1 SPHERICAL SHALLOW WATER EQUATIONS
Spherical Shallow Water Equations (SSWE) form a nonlinear hyperbolic PDEs system that model the motion of llynr fluids a rotating sphere. The underlying assumption is the shallow water approximation. where the vertical seale of the fluid layer is much smaller than the horizontal scale (Bonev et al. 2018). Following (Bonev al. 2018 2023). generate the SSWE simulation

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Published as a conference paper at ICLR 2026
Ground Truth (H)
NourCastNet
SFNONet
SHNet
Res
Res
10
Figure 3: Predicted geopotential height (H) at 5h and 10h from different methods. Zoom in and provide the absolute residual (Res) to the ground truth for clarity (darker is better).
Ground Truth (U600)
Climax
NoutCastNet
SENONet
SHNNet
Day:
Res
ACOMSE
94.07.331
93.277.00
2/.2/13.7
92.1/22
3 Day:
Res
ACOMSE
70,5334
75.7/27.8
10.N249
Day:
Res
ACCMS
48,5/5..3
47.2/50.7
85.2
55,2/45.6
Figure 4: Predicted V600 samples of different methods. Each image set shows the main prediction (left), zoomed-in region (top-right), and the absolute residual (Res) to the ground truth (bottom- right, darker is better). The performance is evaluated using ACC(x10-2) (x10-2).
using classical spectral solver (Giraldo 2001). This dataset is well-suited to verify our model due to its spherical geometric characteristics (Appendix C.2 spatial resolution of 256 x 512, time step of 60s, 3 channel dimensions: geopotential height (H), vorticity (V), and divergence (D)). All comparisons use the identical dataset and setting: 50 epochs containing 256 samples each, batch size of 16, Adam optimizer with learning rate of 1 x 10-3, and spherical weighted mean relative loss (details in Appendix C.I.J Models are tested on the dataset generated from 50 initial conditions (50 samples) and evaluated using MRE for each variable. The results (Table I show that our method achieves the best performance on all variables and time scales, where the average increase is 7.5% 5h and 9.7% at 10h. Besides, Besides, we conduct additional verifications under the exact same setting as SFNONet and at different resolution (128 x 256) for reference, the results are summarized 7 in Appendix. For the predicted geopotential height (Figure 3). SFNONet and
in the Table and FourCastNet show more errors in the zoomed-in region, compared to our model. ClmaX is not presented due to its poor results.
5.2 WEATHER FORECASTING
To further the performance of methods, we utilize WeatherBench (Rasp al.; 2020).
widely used autoregressive global weather prediction benchmark that offers data at multiple spatial resolutions. We select the dataset with a spatial resolution of 5.625 ( (32 x 64) and a temporal resolution of hour for evaluation. The training period spans 1979-2015, validation is conducted on data from 2016, and testing covers 2017-2018. The full dataset includes 24 meteorological variables, and we select six key variables as prediction targets (Liu et al.] 2024) (details in Appendix C.3) All models are under the same settings: batch size 128, 50 epochs, loss function of MRE, Adam optimizer with learning rate of 10-4 To evaluate multi-scale forecasting ability, we perform prediction at 24, 72, 120 autoregressive steps. Performance measured using anomaly correlation coefficient (ACC) and latitude-wett weighted mean square (MSE) (Nguyen al.) (023) (details in Appendix C.3). reported in Table and (details Appendix C.3). Besides, we conduct additional experiments different resolution (64 128) and the results summarized in the Table 6 in
8

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Table ACCT (in %) on WeatherBench for six variables and their average 1, 3, and 5 days. Rows grouped by forecast horizon. Bold indicates best performance in each row group. Gray blocks indicate the ablation experiments.
Prediction Variables
Method
Operater
Paramas
Average
2T
10E1
10V
16600
V400
T680
Day Forecast
Climax Nguves et al. 2023
5.4.
96.2 2.01
91.7 0.67
914 0.75
92.5 0.67
90.4 0.82
96.2 1.38
93.1 0.47
RouttastNerr
HNU
96.0
90.5
91.9
90.1
9
= 0.52
SPNONet Boney of al 1223
SENO
5.3
95.2 2.53
1.17+1811
82.1 1.78
56.4 1.50
8400
9.39
0.83
SHNet (Qurs)
GSNO
+
1.12
96.0
0.59
SHNONet Honey et al.
2025
GSNO
5.5 M
95.6 2.37
64.3 1.42
87.6 1.35
85.8 1.66
94.7
86.9
1.92
Duy Forecust
Climax Ngaven 2003
5.13
5
6.37
FourCa Putaa al. 2022
ENO
5.3
19.7 6.29
56.3
4.74
06.0 4,40
5:40
81.8 8.49
67.1
SPNONet Doney
6.48
SHNeet (Ours)
GSNO
4.0
92.5 4.63
61.7 4.68
60.8
70.2 4.21
636
85.8
7:.4
SFNONe 2025
GSNO
5.5
91.4
50.8 5.03
58.7
60.1 5:40
84.3
9:27
STINet
SENO
3.7 M
91.8 4.82
60.6 4.90
57.2 5.06
68.7 4.44
6.20
84.6 7.85
70.7 2.27
Day
Climax Npave et al
2013
5.4M
$4.0 9.63
9.35
2 8.23
35.7 10.0
8.57
6 18.4
41.6 4.59
ReurCaNNee
2023
20.2+88.65
10.5
64.5
4..5+4774
SFNONet
Bonev et all
2023
SENO
5.3
$6.9 7.76
29.5 8.74
45.3 9.33
26.5 9
71.2 16.0
4..2
SHNet
4.0
40.3
52.5
SFNONer Roney et al. 2023
GSNO
5.5 M
87.7+ 7.68
8.46
3
47.1 +.92
28.9
7 15.3
51.3+
SHNNe
Appendix. Our method achieves the best results on all variables and time scales, especially in the predictions for the third and fifth days. The wind velocity predictions (V600) and their residual plots (darker plots are better) also demonstrate the accuracy and stability of our method in Figure 4
To further evaluate superiority the spectral kernel G2 (1, m) as the spectral embedding GSNO, we construct extra operator-level comparison experiments, with all other architectures and settings identical for fair comparison. The results in the Table Clbarlyl demonstrate that our GSNO consis- tently and significantly outperforms the SFNOs with higher capacity and spatial position-embedding SNO (Go (z, v)) across all forecast lead times (details Appendix
5.3 DIFFUSION MRI MODELLING OF BRAIN MICROSTRUCTURE
Modeling brain microstructure with diffusion MRI is challenging due to sparse, anisotropic mea- surements across spherical shells (Van Essen al. 2013 Jeurissen al. 2014 Zeng et al. 2022); We choose dMRe-based Fiber Orientation Distribution (FOD) angular super-resolutin (Zeng al.
2022 Snoussi Karimi (0255) the specific task for evaluating model performance anisotropic system; see Appendix C.4 for challenge, motivation and preprocessing details. To further verify GSNO, we adopt the same architecture as FODNet (Zeng et 2022), only changing the convolu- tional layers, adapting it to SFNOs to obtain FOD-SFNO, and GSNOs to obtain FOD-GSNO. An- gular Correlation Coefficient (ACC) is used
evaluation metric for measuring the angular simi- larity between two spherical functions (Zeng et al. (022) (details in Equation 41 in Appendix CC44) The results in Table show that GSNO consistently outperforms other models, demonstrating strong generalization under irregular sampling and data sparsity. Figure the region of interest taken from the cingulate gyrus. Compared with the obviously false positive predictions by other methods, the proposed GSNO achieves better prediction of fiber orientation and density.
Table 3: Operator-level comparison WeatherBench. Bold is the best result.
Models
Channel
Parameters
Training Time
ACC 24h
ACC1 72h
ACC h20
SFNO
3.69
981
89.9%
70.7%
50.9%
SFNO with 80 channels
80
5.76 M
1138s
89.7%
70.6%
51.0%
SPNO with 96 channels
96
8.30 M
1267 s
89.6%
70.2%
50.7%
Spatial position-embeelding SNO
1037
90.1%
70.5%
50.3%
GSNO (ours)
64
4,00 M
1024s
91.5%
72.4%
52.5%

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Published as a conference paper at ICLR 2026
SSMT-CSD
FODNet
SENO
ESCNN
GSNO (ours)
MSMI-CSD
Figure 5: Predicted dMRI fiber orientation distribution samples of different methods in the same region of cingulate gyrus. Zoom in for clarity and the fibers at the same position have the same visualization perspective for clear fiber orientation and density comparison.
Table 4: Performance comparison on HCP dataset (ACC1). Bold represents the best result.
SSMT-CSD
FOD-Net
POD-SPNO
ESCNN
FOD-GSNO (Ours)
Methods
Khan et al 2020
Zong et 2022
Bonev et al. 2023)
Snoussi Karimi 2025
Parameters
19.44 M
1.15 M
1.47 M
1.21 M
White matter
0.7523 + 0.0256
0.8858 + 0.01
0.8995 1 0.0151
O 1.9006 + 0.0142
0.9083 + 0.0140
Whole brain
0.6640 + 0.0145
0.8250 = 0.0159
0.8334 + 0.0154
0.8362 + 0.0162
0.8517 + 0.0139
5.4 ABLATION EXPERIMENTS AND COMPUTATIONAL COSTS
To further evaluate the contributions of GSNO and SHNet, we conduct ablation studies to replace the GSNO block with classic SFNO block in the SHNet replace the classic SFNO block with GSNO block in the SFNONet, leading to performance degradation across all variables and time hori- zons (Table 2 and in Appendix C.3 This confirms the effectiveness of GSNO and the multi-scale architecture of SHNet. Further, we observe more pronounced performance gains from GSNO blocks longer-term predictions, suggesting that GSNO plays an increasingly important role in modelling long-range dynamics. To demonstrate the efficiency comparison, we provide the parameters of dif- ferent models in Table Detailed computational costs of different models are in Appendix C.6
We provide extra ablation experiments and interpretability analysis of the original term and correc- C.p))
tion term in our proposed GSNO on weather prediction (details Appendix C.5). As shown in the Figure 6 and Table 11 even if I(Torig) term is frozen, using only I(Tcor) term clearly outlines the temperature distribution framework linked to the Earth's topography, distinguishing the persistent low temperatures at the poles from the high-temperature pattern in regions like the equator. This explicitly demonstrates the effective representation of the I(Toor) for the topography constraints.
CONCLUSION
We have introduced a rigorous theoretical framework for spherical neural operators, grounded in the formulation of designable Green's function the sphere, and extended to the spherical har- monic domain. This framework is used to design different spherical operators and enables the principled incorporation of complex system constraint spherical operator learning. Building this framework, we design an absolute and relative position-ddeennent Green's function, yielding a novel spherical operator solution beyond existing equivariant operator solution to capture complex physical constraints for asymmetric conditions and position-dependent phenomena. The proposed Green's function leads GSNO, designed handle broader range of real-world scenarios by the flexible balance between equivariance and invariance while retaining spherical geometry and grid invariance. Further, we design a multi-scale architecture, SHNet, incorporating geometrically adaptive up-and-down sampling to enhance interactions across resolutions on the sphere. Both theo- retical analysis and extensive experiments on multiple spherical scenarios consistently demonstrate the superiority of our approaches over many state-of-the-art methods. This work establishes uni- fied, system-level perspective for designing spherical operator through designable Green's 's functions, suggesting further designs for the development of spherical modeling.
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A SPHERICAL CONVOLUTION THEOREM
A.1 PROPERTIES OF SPHERICAL HARMONICS
The explicit form of spherical harmonic function (Müller 2006) is:
(21 + 1)( m)! (cos f
Y!Th (8,
(21)
4 m))!
Spherical harmonic functions satisfy the orthonormality condition Müäller 2006):
/ Yi" (W)) (w) du di 8mm'
(22)
where 8 the Kronecker delta.
For subsequent analysis, recall how the spherical harmonics transform under rotation operation R from (3).). The rotation of the spherical harmonic function can be expressed as (Driscoll & Healy 1994):
(Im = A(R = Σ D (R (X)
(23)
|R|ST
where D denotes the Wigner D-matrix (Edmonds 1996) and A(R) represents the rotation oper- ator acting spherical functions. Taking the complex conjugate of Equation equation 23 yields:
(Rw)
Σ D k.m (R
(W)
(24)
A.2 INTERCHANGE OF INTEGRALS AND VARIABLE SUBSTITUTION
Interchange integral order after substituting the convolution definition into the spherical harmonic transform:
- 1s3 (JsO {(Rm))(R-"W) dR (w)
SHT * m)](m,m
-soro S(Rn) (1 WY dR.
(25)
Let w/ = R R then w Rw' with du = de' (measure preservation under rotation). The inner integral becomes:
(Ro'))
(26)
Substituting the expansion from Equation equation 24
f h(w') E D (R (R-1))
dw')
1 kmm (R-1)- (H,
(27)
A.3 ORTHOGONALITY CONDITION SCREENING NON-ZERO TERMS
Substituting Equation equation 27 into Equation equation 25 gives:
SHT[(f * h)] m) Σ
(HL
F(Rn)Di D (R- dR.
(28)
80(3)
Parameterize E SO(3) using Euler angles R (()))())u(() Driscoll Healy 1994). with inverse R = u(-v)aa( f))u( ()). The corresponding D-matrix is:
D. (R-1) cko (()) (cos (c0s
(29)
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which d is the Wigner d-matrix (Edmonds) 1996). Then Taking the complex conjugate:
( k,m
(R- = e the (()) k.m (cos(0)))
(30)
For any rotation u(v)) on the z-axis, substitute R - Ru(v). Due to right-invariance of the Haar measure dR (Driscoll & Healy 1994), the remaining integral is adjusted to:
7 =
[(Rn)) k.m (R-1)
(R-
50(3)
| f(Ru(()n) DE. ((Ru(v)))1) dR
(0(3)
J SSO13)
((Ru())n) D ( dR
f(Ru(D))) e D (R dR
S0(3))
k.m
-ike teb
f(Rn) k,m
(R-1) dd dR
S0(())
(31)
From the above equation:
I [11 - 6 (kp)) 0.
Therefore, if k + 0, choose v such that 1 - e ike + O, forcing I = 0. The only nontrivial solution is j = 0, where e ikk
Therefore, only the k 0 term survives. So the overall integral is simplified to:
SHT[(S m)) SHT [ SSOOO ((Rn) (R-11) dR.
(32)
A.4 RELATIONSHIP BETWEEN D-MATRICES AND SPHERICAL HARMONICS
Because the unitary characteristic of the Wigner D-matrix (Driscoll Healy 1994 we can get the relationship between D-matrix and spherical Harmonic function:
th D
4T
Y (0,
(33)
27
A.5 COMPLETE PROOF OF SPHERICAL CONVOLUTION THEOREM
Based on all the above theories, the complete derivation process as follows:
SHT[(f h)](1, = f(z (Isoos F(Rm))(R- dR (w) dw
S f(Rm) (1 (()))) (Rw) de d.R.
/ D( k.m. (R-1) ).R.
SHT[H](LKK)
S0(3)
SHT[M[(L0)
f(Rn)
u. (R- (-1)dR d.R.
5013)
4TT f(Rn)
SHT[M[(L,0)
Y dR
50(3)
21 +
4T
2T +1+1
SHT(M](L,0) -SH(()((,
(34)
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B DETAILS OF OUR MODEL
B.I COMPLETE OPERATOR DERIVATION VIA SPHERICAL HARMONIC EXTENSION
The spherical harmonic form of Green's function is:
G(u) Σ
(35)
1=0 m=-i
The detailed derivation process as follows:
SHT [S0(3)
[(Rn)
G(R (y)" dR
= Sso03) f(Rn)
Σ SHT m'mm (R"'u) du dR
Im'lsv
/ f(Rn)
(1 YF" (")),""))) du dR
SHT[G] m²)
80(3)
(ss Y (u))
= SHT(G](',m') / SSO(3)
f(Rm)
Σ
(1) (R- 1R7 (u) du dR
k.m
SHT[G](', m 10
[(Pn) (1)
(R-1)dl dR
50(3)
k.m
(L [SO(3)
{(Rm) D
( dR
0.m
4m
V 21 +1
(HT(G]((, (( m).
(36)
B.2 SPHERICAL HARMONIC NEURAL OPERATOR NETWORK
For WB and SSWE experiments, this study uses Spherical Harmonic Neural Operator Network (SHNet) with depth of 2, which performs two spherical downsampling operations. Specifically, the encoder first expands the input channel to C. In the first two GSNO blocks, the spatial res- olution, i.c., the number of latitudinal and longitudinal sampling points in the inverse spherical harmonic transform (ISHT), is reduced by half, while the channel dimension is doubled via MLPs and convolutional layers.
Subsequently, the next two GSNO blocks perform upsampling by doubling the number of samples in latitude and longitude while continuing transform the channels. The final GSNO block used as the output layer: it projects the features back to channel dimension C' without applying further spatial scaling. The implementations of MIPs,, encoders and decoders are consistent with those used in SFNONet (Bonev et al. 2023). The kernel of all convolution operations is 1 x 1 for channel adjustment. SHNet also includes three skip connections to enhance gradient flow and multi-scale feature fusion: The output of the final GSNO block is concatenated with the original input; The output of the fourth GSNO block is added to the input of the first GSNO block; The output of the third GSNO block is added the input of the second GSNO block.
C EXPERIMENTAL DETAILS
C.I MODEL IMPLEMENTATION DETAIL
The key model parameter settings used for the two experiments are presented Table 5 proposed SHNet with depth of 2 consists of five blocks, including two downsampling layers, two upsampling layers, and an output layer. Furthermore, the embedding dimension of SHNet set to smaller value (8 and 64), the depth increases, the embedding dimension gradually increases until
Our
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it is four times the original dimension. The depth of other models represents the number of blocks.
Table 5: Key model hyper-parameters Spherical Shallow Water Equations (SSWE) and Weather-
Bench (WB) experiments.
ClimaX
FourCastNet
SFNONet
SHNet
Hyperparameters
SSWE
WB
SSWE
WB
SSWE
WB
SSWE
WB
Depth
6
8
4
2
Embedding dimension
128
256
128
384
32
256
8
64
Activation function
GELU
GELU
GELU
GELU
GELU
GELU
GELU
GELU
MLP ratio
2
2
2
2
Patch size
4 x 4
4 x 4
4 X 4
4 x 4
1
/
/
/
Parameters
1.01
5.4 M
1.37
5.3 M
0.28
5.3 M
0.80
4.0 M
The loss function of the proposed method is the spherical grid weighted mean relative error be- tween the prediction F(X,.) and the target Yute, calculated following Bonus et al. (Bonev et al. 2023 as follows:
Eis an F()lF(X))EoAa)) (==r(=cs)))= Ui.5 Eis Disl DasiiV+tia (ate(Feas))
CIF(Xa). - Σ
(37)
where U:y is the products of the Jacobian sin(A;) (A; represents the latitude at grid point) and the quadrature weights (Bonev et al., (2023). C the number of predicted variables. is the index of the initial time step, / is the predicted autoregressive steps. This loss function is used in all the experiments.
C.2 SPHERICAL SHALLOW WATER EQUATIONS
Spherical Shallow Water Equations (SSWE) form a nonlinear hyperbolic PDE system that models the motion of thin-layer fluids on a rotating sphere. The core underlying assumption the shallow water approximation, where the vertical scale of the fluid layer is much smaller than the horizontal scale (Bonev et al. 2018).
The dataset is publicly available, which can be obtained and used for evaluation through Torch- Harmonics GitHub (Bonev et al. 2023). Consistent with Bonus el (Bonev al. 2023). we use mean relative error of each variable as the evaluation metric.
We also conduct the extra experiments on SSWE under the exact same setting as that of SFNO for reference (e.g., 150 time steps). The results are in Table
C.3 WEATHER FORECASTING
The dataset is publicly available at WeatherBench GitHub (Rasp et al. 2020). Following (Liu et al, 2024), the chosen 24 common variables detailed in Table including 10U, 10V, 2T, U50, U50, 0250, U500, U600, U700, U850, U925, V50, V250, V500, V600, V700, V850, V925, T50, T250, T500, T600, T700, T850, T925. And 10U, 10V, 2T, U600, V600 and T600 prediction target.
And the additional results (MSE) are in Table
The anomaly correlation coefficient (ACC) and the used to measure the performance of the model WeatherBench (WB). MSE between the prediction F(X,) and the target Y for evaluation is calculated following Rasp al. (Nguyen at. 2023)
mean square error (MSE)
MSE[P(X,), Yo4T] CxH W
ΣΣΣ wi (F((n)(Aedd)) - (Foid)))))
(38)
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Table Supplementary experiments on SSWE 128 256 resolution (MRE.) and WB 64 x 128 resolution (ACC1).
Methods
SSWE at 5h
SSWE 10h
WB at day
at 3day
WB at 5day
FNO
2.22
2.73
93.4
71.5
42.5
SFNO
0.74
0.87
91.6
73.2
49.2
GSNO
0.68
0.79
93.0
74.9
51.7
Table 7: Experimental results (I.3) SSWE under the exact setting for reference (e.g., 150 time steps).
Models
FNO
SFNO
GSNO
h
8.628 x
8.092 10
7.051 x 10 4
At 10
9.470 10-3
6.739 10-3
5.178 x 10-3
where the number of latitudes and W is the number longitudes. The latitude weight (Nguyen et al. 2023) are calculated as:
cOs (A:)
WT:
(39)
Li=1 cos(
ACC Nguyen et al.] 2023) is used to measure spatial correlation between predicted anomalies F(X,) and target anomalies X
Eass wi (F(Xn)ess' (Yntoeas)
ACC
(40)
Enig ws (F(Xn)eas)) Eeis we ((Yn+s)
Besides. to empirically compare the correction term of GSNO with positional embedding, we con- struct "Spatial position-emi -enbiddne SNO" model that strictly implements the SHT + Cj (G]e(z.))) path, maintaining all other architecture and settings identical for fair comparison.
The results the Table clearly demonstrate that our spectral implementation (GSNO) consistently and significantly outperforms the SFNO and spatial times. This indicates that spectral implementation derived from our Green's function framework is, in itself, more effective and different design. Moreover, the spatial position embedding added in each operator (SHT(f + Cy (G]o(r.)))) is not conducive to long-tent long-term stable prediction.
SNO across all forecast lead
C.4 DIFFUSION MRI MODELING OF BRAIN MICROSTRUCTURE
Detailed description. Diffusion MRI-based FOD angular super resolution in this study is distinct challenge in: (1) the input signals are sparse and anisotropic diffusion measurements acquired over spherical shells, exhibiting sharp angular variations corresponding to underlying fiber tract orienta- tions; (2) the spatial sampling is performed using HEALPix, resulting in nonuniform and incomplete coverage of the spherical domain.
The task motivation of dMRI-FOD rather than dMRI itself: The raw dMRI signals themselves, due noise, partial volume effects, and the aliasing of signals from multiple tissues, do not directly ex- hibit clear, anisotropic fiber structures. Learning directly from raw dMRI would require the model to simultaneously handle noise suppression, signal unmixing, and orientation estimation, introducing numerous confounding factors. This would make it difficult to cleanly evaluate the operator's 's core capability in modeling anisotropic geometric structures. Further, high angular resolution dMRI is costly and often infeasible in clinical settings, leading to poor-quality FODs. Therefore, enhancing dMRI quality from routine dMRI while while handling noise suppressiona and signal unmixing for estimating FOD is widely recognised and practical problem.
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Published as a conference paper at ICLR 2026
Table 8: MSEL (x10-3) 3) on WeatherBench for six variables and their average at 1, 3, and 5 days. Rows grouped by forecast horizon. Bold indicates best performance within each group. Gray blocks indicate the ablation experiments.
Prediction Variables
Method
Operator
Average
Day Forecast
Climax (guyyen et
2025
4.23 1 0.43
80.9 5.16
113 7.04
67.9 a 4.17
118
6.68 9.51
66.8 2,06
FourCasiNet Patbak ctal. X022
94.2 1.36
132 8.18
7
137 9.30
7.98 0.66 0.66
75.7 2:45
SENONet Boney ci
SENO
5.18 0.58
175 12.1
242 17.0
134 838
234 16.7
10.8 0.93
134 4.70
SHNet (Ours)
GSNO
4.22
124
173 10.4
7.18
11.3
8.75 0.81
97.5 3.18
SFNONet (oOIY al. SHNer
2023
GSNO
0.52
162 x: 10.5
221 = 14.9
127 s 8.32
215 : 1488
9.97 0.87
123 4.16
SPNO
4.76 0.48
156 + 9.90
208 = 14.1
122 + 8.34
204 + 14.1
9.89 + 0.86
117 3.97
Day Forecust
(Nuuyn 2025
11.6
384 35.0
56.1
28.8
61.4
31.4
306
FourCasiNet Pathak et al. 2022
12.7 + 1.66
402 + 33,1
594 = 54.4
322 28.0
616 62.3
4.59
330 15.6
SPNONet Boscy cl 2023
SENO
10.3 + 1.51
372 2 32.0
535 a 51.0
295 a 26.7
546 A 60.5
30.3 4.34
298 14,9
SHNet (Ours)
GSNO
24,0
492
26.7
SPNONet Bonev al. 2023
GSNO
9.59
357 + 29.3
501 & 48.1
281 & 25.5
524 à 56.2
28.5 4.10
284 14.0
SHNet
SENO
9.50 1.40
362 1 29.3
495 : 47.7
284 a 25.9
522 a 55.8
28.1
282 13.9
Day Forecust
(guuee
1.10
572
121
FouCCiNNNt Philbak M022
2019
510 41.77
725 70.7
471 s 39.3
811 : 99.5
58.9
433 22,5
SFNONet BoscY at 2026
SENO
14.57
502 1 1-6.5
726 a 81.6
4/0 42.9
791 1 105
18.1 7.76
421 21.6
SHNet (Curs)
GSNO
13.3
42.2
# 73.4
40.5
a 93.8
398
SPNONe: (onev
2023
GSNO
13.442.46
484 44.0
699 75.1
427 41.4
94.9
7.57
4088 22.6
SHNet
SENO
13.5
492 44,8
705 76.3
430 e 41,5
783 96.0
16.88t77.554
412 22,9
Table 9; Chosen Variables Abbreviation
Abbreviation
Description
10U
Zonal wind velocity at 10m from the surface
10V
Meridional wind velocity at 10m from the surface
2T
Temperature 2m from the surface
U--
Zonal wind velocity at pressure level Meridional wind velocity at pressure level
V--
T--
Temperature at pressure level
We randomly select dMRI images of 30 subjects the Human Connectome Project (HCP) (Van Es- sen et al. (013)), where 20 subjects were for training, 5 for validation, and 5 for test. Detailed pre- processing. The complete dMRI data of each subject contains 288 volumes (multi-shell HARDI,, including 270 volumes of b=1000, 2000, 3000 s/ s/mm² with 90 gradient directions for each shell and 18 bo volumes. Due to the wide application of low b-value with 32 gradient directions in clin- ical practice (Zeng al.] (2022), we subsample 32 volumes of b=1000 and volume of bo from the complete dMRI according to the HCP protocol, to obtain single-shell LARDI. Further, we use MSMT-CSD (Jeurissen et al. 2014) on the multi-shell HARDI obtain high angular resolution fiber orientation distribution (FOD) as the ground truth (HAR-FOD). Meanwhile, we use SSMT- CSD (Khan et al., 2020)) on the single-shell LARDI obtain single-shell low angular resolution FOD as the condition of GSNO (LAR-FOD). Imar is set to the default value of 8 to balance preci- sion and complexity (Zeng et 2022).
SSMT-CSD (Khan et al.) 2020), FOD-Net (Zeng et al. 2022). FOD-SFNO (Bonev et al. 202.3, ESCNN (Snoussi Karimi, 2025) and our proposed FOD-GSNOadopt the same network archi- tecture FOD-Net. The difference lies in that FOD-Net uses 3D convolutional layers to handle large number of voxels, while POD-SENO ESCNN and FOD-GSNO stack voxels on the channel dimension and then the corresponding operators for processing. To ensure fair comparison, FOD-SENO and FOD-GSNO have exactly the same hyperparameters, with the only difference being the types of operators. And their parameter comparison shown in Table TO Angular Correlation
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Published as a conference paper at ICLR 2026
Input
Output
Target
Input
Output
Errr
Target
Figure 6: Predicted temperature field at 2m from the earth surface of GSNO w/o I(Torig). Provide the residual (error) the ground truth for clarity (darker is better).
Coefficient (ACC) is used for evaluation:
Ei=0 E -immim
ACC(u,
(41)
Ei=o Σ Em Em Pm
where I, m represent degree and order of a spherical harmonic function. g', 9 are the ground-truth (MSMT-CSD) and the estimated FOD from different methods.
Table 10: Parameters on dMRI.
Models
FOD-Nee
FOD-SENO
ESCNN FOD-GSNO
Parameters
19.44 M
1.15 M
1.47 M
1.21 M
C.5
EXTRA EXPERIMENTS AND INTERPRETABILITY ANALYSIS
The additive form in Equation 20 allows us explicitly decouple the rotationally equivariant com- ponent. This separation reflects real-world scenarios where the global physical dynamics (e.g., geophysical flow) are predominantly symmetric, but local features (e.g., terrain, boundary anoma- lies) introduce perturbative effects. The additive form lets us modulate these independently and retain interpretability. Overall, this allows a clean decomposition and analysis of cach term, as validated in Tables II As shown in Figure 6 even if I(Toxig) term is frozen, using only I(Towr) term clearly outlines the fundamental temperature distribution framework linked to the Earth's 's to- pography, successfully distinguishing the persistent low temperatures at the poles from the overall high-temperattre pattern in regions like the equator. This explicitly demonstrates the effective rep- resentation of the correction term for the topography constraints. Furthermore, the finer undulations in high-temperature regions reflect stable, large-scale, non-equivariant patterns present in the un- derlying temperature field and its discretization, including: (I) Climatological zonal structure: temperature fields exhibit persistent, slowly varying zonal (cast-west) asymmetries arising from longitudinal land-ocean contrasts and stationary planetary waves. (2) Latitude-dependent variabil- ity: numerical discretization on an equiangular grid leads to resolution and dissipation patterns that vary with latitude. (3) Long-term statistical structure: even though instantancous dynamics vary, the climatological mean temperature field (which the model implicitly learns) contains smooth merid- ional and zonal gradients.
C.6 DETAILED COMPUTATIONAL TRADE-OFFS
We add the comparison of parameters, FLOPS, training time (seconds per epoch) and inference time for two models, SFNONet and SHNet. All results are obtained using the same hardware and dataset (Weatherbench).
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Published as a conference paper at ICLR 2026
Table 11: The ablation experiments of I(Torig) and I(Teor) items in GSNO are conducted verify
the effectiveness of ((Tong). The average ACC (%) of various variables is presented.
Models
GSNO
GSNO w/o ((Teu))
GSNO w/o ((Tong)
day
91.5
71.4
20.1
day 3
72.4
61.3
11.1
day 5
52.5
42.6
9.9
In the second and third columns, we compare the operators. GSNO introduces lightweight set of additional global spherical integration term compared to the standard SFNO. This results modest increase in model size and computational cost. Specifically, GSNO adds approximately 1.6% to the runtime, along with a slight increase in parameters and FLOPs, while delivering around 6% improvement in performance. In the comparison between the second and fourth columns, where the same operator is used but different networks are employed, SHNet demonstrates fewer parameters and lower FLOPs than SFNONet, owing to its multi-scal design. The slightly increased runtime in SHNet is primarily due to the added complexity of the GSNO operator, particularly the correction term. Since GSNO applied every layer (as shown in Figure 1), this results in approximate 10% increase in runtime. Nevertheless, this design achieves 2% performance gain while reducing overall memory and computational usage.
Table 12: Parameters and Inference Time on WeatherBench.
Models
SHNeet (SFNO) SHNet (GSNO) SFNONet (SFNO) SFNONet (GSNO)
Parameters
3.69 M
4.00 M
5.26 M
5.52 M
Training Time
981
1024 s
937 s
983
Inference Time
245.66 ms
249.82 .2.82 ms 52.5
217.32 ms
227. 18 ms
Performancef 50.9
49.2
51.3
Table 13: Extra performance and efficiency comparison, SFNO and GSNO use the same U-Net architecture for fair comparison.
Models
SFNO GSNO SFNO (80 C) SFNO (96 C) FourCastNet (FNO) Climax
Channel
64
64
80
96
384
256
Parameters
3.69 M
4.00 M
5.76 M
8.30 M
5.30 M
5.40 M
Training Time
981 s
1024 s
1138
1267 s
878 s
891 s
Performance?
50.9
52.5
51.0
50.7
41.5
41.6
D LIMITATIONS
Our experiments are conducted on the SSWE dataset (256 x 256 resolution), the WB dataset (5.625 resolution) and dMRI modeling of brain microstructure. While these results demonstrate the poten- tial of our method, they are limited in scope, such as other manifolds. The generalizability to other spatial resolutions and datasets remains be verified. In future work, plan evaluate our approach across a wider range of resolutions and on additional spherical scenarios.
E REPRODUCIBILIIT
We include the code for our model the supplementary material, containing README.md for guidance. We will release the complete code at
com wwt.coot//ttttacae GSNO
F THE USE OF LARGE LANGUAGE MODELS (LLMS)
We used Large Language Models (LLMs), such as GPT-4-based systems, as writing polishing as- sistants during the paper polishing stage to enhance clarity, readability, and precision of paper. And main contributions, such as scientific content, theoretic analysis, technical contributions, exper- imental design, and all results of this paper conceived, executed, and verified by the authors.
22


Reviews:
[{"review_id": "1wpW71yzkM", "reviewer": "Reviewer_pE1P", "summary": "Spherical neural operators (SNOs) map functions to functions on the sphere. To account for the spherical sampling, current SNOs use spherical rotation equivariant network layers to do this mapping. However, these equivariant networks are (a) not expressive enough for certain applications and (b) cannot model position-dependent features that are beneficial in structured datasets.\n\nSubmission 7686 develops an SNO framework that adds a non-equivariant learnable bias term that is position-dependent. This way, the learning framework can decide how much to weigh strict equivariance vs position dependency. Some preliminary experiments on weather forecasting, shallow water dynamics, and fiber orientation estimation in diffusion MRI are presented.", "questions": "In my opinion, the paper reads as incomplete in its current state (clarifications of technical details, depth of technical contributions, missing experimental information, etc.). However, IMO these are fixable issues and I am looking forward to the discussion period to clarify these points before finalizing a score. For specific questions, please see above.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The papers makes a good case for why such an approach that uses equivariant layers alongside components that are position-dependent are needed and sets up the needed motivation in the Introduction well.\n- Most of the methodological steps are very clearly laid out in the methods and illustrations.\n- Its experiments tackle a good, wide breadth of potential use cases.", "weaknesses": "(In no particular order)\n\n### Methodological concerns and questions:\n\n*Overstretched technical novelty (1)*: The core idea of the paper is a learnable tradeoff between two terms corresponding to strict equivariance and absolute position dependence in features, respectively. However, this soft equivariance idea has been explored quite extensively in the literature; see the [residual pathway priors](https://proceedings.neurips.cc/paper/2021/file/fc394e9935fbd62c8aedc372464e1965-Paper.pdf) paper as an example. A literature search for \"soft equivariance constraints\" will return many relevant pieces of related work, such as [1](https://proceedings.neurips.cc/paper_files/paper/2024/hash/98082e6b4b97ab7d3af1134a5013304e-Abstract-Conference.html), [2](https://arxiv.org/abs/2201.11969), and more.  \n\nOf course, this particular formulation, specifically designed for spherical neural operators, is new (as far as I am aware) and is appreciated, but previous work that presents similar additive approaches with learnable equivariance tradeoffs should definitely be acknowledged within the paper.\n\n*Overstretched technical novelty (2)*: L077--L081 is written in a way that makes it sound that studying Green's function on the sphere with spherical harmonics is a novel contribution. I'm assuming that this is not what is meant (spherical harmonics are widely used for this), and what is meant is that \"we use Green's function to develop a spherical neural operator network\". If not, please clarify.\n\n*Main technical descriptions in Sec 4.2:* Sec 4.2 is the primary contribution of the paper, yet a few things in it are not clear. For example, it is not described how it is possible to drop the index m in G^1 between equations 17 and 18 without any loss of generality.  Further, what are the convolutional layers doing in L307--310, and why are they used? Are they just regular convolutions? Do they change any equivariance properties? For example,  in the Diffusion MRI experiments, adjacent voxels are stacked as separate channels (unclear why), so it is important to justify their usage and define their properties in context.\n\n*Missing analysis of importance of position dependence:* The paper does not present an analysis of whether modeling position dependence actually improves performance due to position dependency or simply because of higher capacity. While a vague description of ablations is included, they simply report performance numbers, so it is unclear if absolute position is relevant here.\n\nFurther, position dependency leaks into all equivariant deep networks due to aliasing ([1](https://arxiv.org/abs/2210.02984)), and even the original [S2CNN](https://arxiv.org/pdf/1801.10130) paper indicated that it was not perfectly equivariant in Sec 5.1. Please reconcile the fact that equivariant spherical networks in practice still learn position-dependent features with the explicit positional modeling in this work and show that the proposed method actually exploits positions specifically (and doesn't just have better performance.)\n\n### Experimental concerns and questions:\n\n*Major missing details:* The paper provides no details about what is being reported, how baselines were chosen, and how/if they were tuned, nor any reasoning behind the experimental design. For example, what is ACC in Table 3? Accuracy is meaningless in Diffusion MRI analysis without context. No ablation details are reported in the main paper either. I read the appendix corresponding to the experiments and found that to also be light on details. This aspect of the paper definitely needs expansion.\n\n*Computational complexity:* The proposed method requires several forward and inverse spherical fourier transforms, which are highly computationally intensive and thus limit network capacity. I realize that this is how the original SFNO paper was also formulated, but other works, such as DeepSphere, have shown that these transforms are not strictly necessary for equivariance. Regardless, all experiments should also report training and testing runtimes for all benchmarked methods (and not just SFNO, as in the appendix).\n\n*Diffusion MRI experiments:* Looking at the appendix, the paper makes a few odd statements and choices in its experiments. \n- It states that diffusion MRI is captured on a HEALPIX grid -- this is not the case; the spherical grid points (b-vector patterns) are study-specific. \n- There are specific synthetic benchmark datasets with exact ground truth for diffusion MRI such as Fibercup, the ISMRM tractometer challenge, and other recent challenges. One of those datasets should be added to measure gold-standard performance vs. the current silver-standard experiments on HCP, as in recent work in equivariant FOD estimation ([1](https://proceedings.neurips.cc/paper_files/paper/2024/hash/5d4f5a2de6320641566be8722d5f78dc-Abstract-Conference.html)).\n- It significantly downsamples the angular resolution of the HCP dataset from hundreds of spherical sampling points to just 32 and also throws away two entire shells \"to emulate clinical conditions\". This is not well justified, as (a) it does not demonstrate that the proposed method has the memory efficiency to extend to the hundreds of spherical sampling points used in typical research data and (b) at this \"clinical\" resolution, the FODs have no asymmetry and are much simpler structures. \n\n### Presentation concerns and questions:\n\n*Characterization of previous work:* In the introductory paragraph corresponding to L053, the paper somewhat uncharitably characterizes previous work on spherical neural operators as \"not rigorous and constructed by analogy\". As far as I remember, the original SNO paper did construct the SNO from first principles. If specific non-rigorous arguments are being made in those papers, the submission should be specific about what it is referring to.\n\n*Significant portions can be abbreviated:* To add much more experimental detail to the main paper, several portions can be removed or abbreviated. For example, \n- The contributions list in the introduction is redundant and just restates the paragraph preceding it\n- Section 3 can be significantly shortened, as these are plain fundamentals\n- Section 4.3 is a long way of saying \"we constructed a UNet with our operator layers\"; it can also be shortened.\n\n\n### Minor/misc. comments:\nThese comments do not impact my score:\n- *Missing sanity check:* A common practical strategy in spherical equivariant learning on datasets with position-dependent structure (e.g., preferred orientation in natural 360-degree video) is to use the standard equivariant layers for most of the network and \"break\" equivariance near the output with 1--2 standard non-spherical layers (e.g., appendix E in [1](https://arxiv.org/pdf/2006.10731)). A similar strategy can be used with SNOs, and this paper would benefit from investigating whether the proposed method outperforms this simple baseline.\n- Following up on the above, spherical image segmentation might have been an interesting use case for the proposed method.\n- In the diffusion MRI experiments, FOD-Net is an odd choice to use the proposed GSNOs in, as the spherical harmonics are only used in a single portion of the network. In contrast, spherical UNets have been used for FOD estimation in [1](https://hal.science/hal-02946371/document), [2](https://arxiv.org/abs/2102.09462), among others; One of them would likely be a better fit for this analysis as the UNet architecture can simply be replaced by the proposed GSHN. \n- Minor typos. e.g. L177-178 \"denoting\" --> \"denotes\"; \"Equation equation\"", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "V1Q5zqjp1q", "reviewer": "Reviewer_Yp14", "summary": "This paper presents Generalized Spherical Neural Operators (GSNO), a method for learning neural operators on spherical domains. It introduces absolute-position-dependent spectral constraints for modeling real-world complexity on spheres and demonstrates improvements in diffusion MRI, fluid dynamics, and climate forecasting benchmarks over state-of-the-art models.​\n\nGSNO is strongly related to SFNO, which introduces the original approach based on the Driscoll-Healy ocnvolution theorem on the sphere, that the authors describe as the Greens-function based approach. The main novelty comes from the inclusion of a position-dependent bias term to handle anisotropy and boundary effects. GSHNet is a hierarchical, multi-scale network architecture that combines spectral domain learning with geometric up/down sampling for improved global feature representation.\n\nGiven the improvements that the paper makes, I do think that it can be considered for publication, but I strongly suggest the authors to adapt the language around the Green's formulation. While it is a helpful motivation, it ultimately motivates the existing SFNO architecture. The strong connection should be clearly pointed out and the authors should clearly communicate that the paper is about the inclusion of an extra bias term in spectral domain and how this can be motivated.\n\nAs such, my main concerns for this paper revolve around the messaging, which seems to inflate novelty and obscure the conceptual roots found in SFNO.", "questions": "- Is the hierarchical approach of GSHNet really necessary - there were some initial works using U-Nets based on the SFNO operator in the atmospheric science community and those did not seem to improve performance beyond that of the standard Resnet like SFNO backbone. This seems to be at odds to your ablations with SHNet with the SFNO operator. Can you comment more on this?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- improved architecture based on SFNO, which includes a spectral bias term which can address the limitations around the isotropic kernels in the original SFNO work\n\n- adds extra theoretical motivation with the Green's function viewpoint to the existing SFNO literature\n\n- the paper is well-written and has well-motivated experiments\n\n- shows improvement on various, simplified datasets (spherical shallow water equations, weather on low resolution, DMRI example)", "weaknesses": "- This work has a strong connection to SFNO and mainly introduces an extra bias term which allows it to alleviate the constraint to isotropic filters. Unfortunately, the introduction and the text in 4.1 are misleading and make it sound as if a) this paper introduces an entirely new approach based on the Green's formulation b) SFNO only replaces the FFT with an SHT, not leveraging the Driscoll-Healy convolution theorem. Given that \"Green's formulation\" advertised in the title and derived in 4.1 is the original SFNO, the messaging in both sections and perhaps the title should be adjusted.\n\n- A much more honest exposition would be that this work builds on top of SFNO, introducing an extra bias term in spectral domain which breaks the isotropy constraint and leads to better performance. While this may seem like a \"less novel\"  exposition it puts the research clearly into the context of the existing literature.\n\n- no code available for review, therefore limited reproducibility at time of submission.\n\n- Added complexity from dual spectral components increases computational cost (modest runtime and parameter overhead).​\n\n- Interpretability and ablation analyses mainly focus on the additive separation, which may not generalize.​\n\n- Reproducibility tested only on select public datasets at limited resolutions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HOvsrKM8bQ", "reviewer": "Reviewer_A988", "summary": "The paper proposes a new family of spherical neural operators derived from a Green’s-function formulation, termed Generalized Spherical Neural Operator (GSNO), and a hierarchical architecture GSHNet.\nThe authors aim to bridge theoretical rigor and practical modeling of spherical PDEs by introducing an absolute-position-dependent constraint into the Green’s-function kernel, thereby relaxing strict rotational equivariance while retaining spectral efficiency.", "questions": "1. What is the quantitative contribution of the bias/correction term alone? How is it better that just padding the grid?\n2. What is the computational complexity compared to baselines?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The authors present an interesting point that SFNO’s preservation of rotational equivariance on the sphere could actually restrict its usefulness in many real-world scenarios where such equivariance does not naturally occur.\n\n2. The authors include clear visualizations.", "weaknesses": "1. Theoretical contribution remains limited despite that the authors claim it. The Green’s-function formulation is largely symbolic and just an extension from GNO [1].\n\n2. Technical novelty is marginal. The main technique is to introduce a position-dependent correction term that is not equivariant to SFNO. The method could be viewed as an equivariant model plus an additive bias rather than a fundamentally new operator framework. The reframing via Green’s functions does not introduce new computational or representational concepts beyond existing spherical spectral models. Although, in FNO/SFNO, the grid padding can be viewed as a position-dependent correction term to some extent. \n\n3. The SSWE dataset still has rotational symmetries. Despite using the same SWE dataset, the settings are different from that of SFNO. \n\n4. The motivation behind the third claimed contribution, the multi-scale spherical network with a hierarchical architecture, is unclear to me. Typically, U-Net–style up- and down-sampling structures (see Appendix B.2, I believe that it's just an U-Net) are introduced to address the inability of local operators (e.g., standard convolutional layers in CNNs) to capture global spatial dependencies. However, models such as FNO, SFNO, and their variants inherently employ global operations. Therefore, it is not evident what benefits this U-Net–type hierarchical design provides in this context. Furthermore, the down-sampling path can introduce information loss and aliasing artifacts, which may further limit its effectiveness.\n\n\n\n[1] Neural Operator: Graph Kernel Network for Partial Differential Equations", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "269AoB5vIG", "reviewer": "Reviewer_vG13", "summary": "Spherical Fourier Neural Operators (SFNOs) are a form of neural network that take as input a scalar signal on the sphere and output a new scalar signal on the sphere. A major part of SFNOs are spherical convolutions that are equivariant under rotations. This convolution is implemented in the Fourier space of the sphere.\n\nThe main idea of the submitted paper is to add positional dependency to the spherical convolution, thus breaking the rotation symmetry.This is done in each convolution layer by adding a learned positional embedding to the signal before it is convolved by a filter. The addition of the positional embedding is also done in Fourier space.\n\nExperiments on three tasks (without full rotational symmetry) show that the added positional information can be beneficial.", "questions": "1. It seems from the hyperparameter table in the appendix that the original SFNO-nets and the new GSFNO-nets have different numbers of channels and layers. Why?\n2. It would be interesting to plot the learned auxiliary positional information $G^{(2)}$. Intuitively it should be symmetric around the axis of rotation in the shallow water experiments and perhaps have further structure in the weather prediction task (due to landmasses on Earth etc). Is this the case? Are there other structures apparent in the learned $G^{(2)}$? This would relate to the stated motivations for including $G^{(2)}$.\n3. The original SFNO-nets already have a positional embedding at the start of the network. Is the main reason for improvement in GSFNO-nets that positional information is more explicitly provided in each layer?\n4. Why is the performance worse with more trainable parameters in Table 2?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The proposed method is well-motivated, easy to implement and seems to work.\n2. The paper is clear and compares extensively to the most relevant baseline, i.e. networks built from ordinary SFNOs.", "weaknesses": "I have a few technical questions, see below. Other than that, the main weakness is that relevant prior literature is not sufficiently covered in my opinion.\n\n1. The idea to add auxiliary positional information is a spherical analogue of absolute positional embeddings as used in NLP/vision, and very similar to for instance “CoordConv” (Liu et al, An intriguing failing of convolutional neural networks and the coordconv solution, NeurIPS 2018).\n2. More generally, symmetry breaking by adding auxiliary inputs is a broad research direction that is not referenced in the submitted paper. See e.g. Smidt et al (Finding Symmetry Breaking Order Parameters with Euclidean Neural Networks, Phys. Rev. Research 3, 012002 (2021)) and later follow-ups.\n3. Spherical CNNs are not mentioned in the related work at all, except a reference to Cohen et al 2018, without explaining that it covers spherical networks. I believe spherical CNNs should be discussed in a bit more detail with references also to later works. When using spherical convolutions such as the ones in the submitted paper, the filters must be zonal (e.g. seen by formula 13, where $m\\neq 0$ parts of $G$ are not present). This is addressed in prior work on spherical CNNs for instance by lifting the signal from the sphere to SO(3) or by storing steerable features relative to a local gauge at each point on the sphere (Cohen et al, Gauge Equivariant Convolutional Networks and the Icosahedral CNN, ICML 2019). The potential for obtaining more expressive layers in this way should at least be mentioned in the submitted work. \n\nIt would make the contribution of the paper clearer if particularly points 1 and 2 above were discussed.\n\nMinor weaknesses/typos:\n\n1. Equation (7) is incorrect as the definition of $\\delta$ should be that integrating $f \\delta$ evaluates $f$ at a specific point as in equation (9).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)