Title: Bridging Equivariant GNNs and Spherical CNNs for Structured Physical Domains
Venue: NeurIPS 2025
Accepted: yes
Keywords: Equivariance

Geometric

Fourier

Spherical Signals

SO(3)

Radar

Policy Learning
Abstract: Many modeling tasks from disparate domains can be framed the same way, computing spherical signals from geometric inputs, for example, computing the radar response of different objects or navigating through an environment. This paper introduces G2Sphere, a general method for mapping object geometries to spherical signals. G2Sphere operates entirely in Fourier space, encoding geometric structure into latent Fourier features using equivariant neural networks and outputting the Fourier coefficients of the continuous target signal, which can be evaluated at any resolution. By utilizing a hybrid GNN-spherical CNN architecture, our method achieves much higher frequency output signal than comparable equivariant GNNs and avoids hand-engineered geometry features used previously by purely spherical methods. We perform experiments on various challenging domains including radar response modeling, aerodynamic drag prediction, and policy learning for manipulation and navigation. We find that G2Sphere outperforms competitive baselines in terms of accuracy and inference time, and we demonstrate that equivariance and Fourier features lead to improved sample efficiency and generalization.The source code is available at: https://github.com/ColinKohler/geometry2sphere.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes G2Sphere, a method to build efficient equivariant architectures able to output very high-resolution spherical signals.\nThis is achieved by 1) gradually increasing the spherical frequency of the features in the decoder and 2) densely sampling the resulting features on the sphere and applying an MLP pointwise on each location.\nThe paper demonstrates the benefit and the efficiency of the method on a variety of tasks.\n\nSTRENGTHS AND WEAKNESSES:\nThe manuscript is well written.\nWhile the main idea is in some sense incremental, it is well motivated and supported by many experiments.\nOverall, the proposed method seems solid.\n\nI only have two small doubt.\nFirst, it is not entirely clear to me why the output signal is defined over a spherical surface in the Drag Prediction dataset.\nI feel like the zero-shot super-resolution argument is not so different from interpolation.\nSee my questions below for more details.\nI would appreciate if the authors could comment on these points.\n\nQUESTIONS:\nIt's not very clear to me how the 3D shapes are mapped to the radar signal in the Radar dataset.\nSimilarly, why is the output signal defined over a sphere rather than over the mesh surface in the Drag Prediction dataset?\n\n\nLine 249: why using global averge pooling? while I understand why this works, this choice still feels like it might discard a lot of important spatial information. Isn't there some better choice? Maybe something attention based or maybe by preserving a low-resolution spatial dimension?\n\n\nLine 285-290: by training on lower resolution data, isn't the model simply going to learn only the low frequencies component of the signals? In this sense, the zero-shot super-resolution is not much more interesting than interpolation, in principle, right?\n\nFig 4 is a bit hard to read. You could consider inceasing the quality of the figure (e.g. the font sizes and the resolution) and include a more informative caption.\n\n\nTab 2: could you report the standard deviation over the 50 initializations?\n\nTab. 3: why is the NE-G2S much faster than G2S?\n\n\nTSNL: the output of the pointwise MLP applied on the spherical signal can have unbounded maximum frequency as stated in the paper. The authors, then, compute the Fourier Transform of the resulting signal, which was densely sampled on the sphere. However, because the resulting signal has unbounded frequency, a Fourier Transform with finite samples can not be exact and will always introduce some artifacts, thereby breaking the exact equivariance of this layer. This is in contrast with what the authors claim in lines 727 and 730. Could the author comment on this?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the problem of mapping local 3D geometric information to global spherical signals. The proposed method, G2Sphere, encodes the geometric structures into latent Fourier features using equivariant networks.\n\nThe encoder processes 3D input data via graph convolutions based on Equiformer v2 [39] (Sec. 4.1), while the decoder leverages spherical convolutions [15] to project features onto the sphere (Sec. 4.2).\n\nThe main contribution is the frequency upsampling module (L203–212), inspired by [7, 26], which integrates a Trainable Spherical Non-Linearity (TSNL) to increase the SH frequency from 5 to 40. This allows modeling of higher-frequency signals than previous equivariant GNN baselines [20, 26].\n\nG2Sphere demonstrates improved performance across several structured physical domains, including radar/drag modeling, and policy learning. The authors argue that equivariance and Fourier features reduce prediction errors in these three physics-based CV benchmarks.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n* S1. Technical Soundness: The combination of an equivariant GNN encoder [39] and spherical CNN decoder [15] is well-founded for SO(3)-equivariant modeling. The proposed frequency upsampling via a point-wise MLP allows increasing SH frequency levels within an iFT, but see the concerns of W2 in below.\n* S2. Interesting Applications: The chosen evaluation domains (radar/drag modeling and policy learning) effectively demonstrate the utility of SO(3)-equivariant framework in structured physics-based scenarios.\n* S3. Performance Gains: On synthetic datasets (Radar/Drag) and a policy learning dataset, G2Sphere outperforms baseline methods, including both equivariant and non-equivariant models.\n\n\nWeaknesses\n* W1. Missing Prior Work: A closely related architectural baseline is missing from the discussion: [A] uses a spherical encoder and SO(3)-equivariant CNNs [15], with a similar motivation of operating in the frequency domain rather than the spatial domain. Please cite [A] and compare both architectural and conceptual similarities and differences.\n\n [A] 3D Equivariant Pose Regression via Direct Wigner-D Harmonics Prediction (Lee & Cho, NeurIPS 2024) \n\n* W2. Maintaining Equivariance of TSNL: The use of point-wise MLPs in TSNL raises questions about whether end-to-end SO(3)-equivariance is preserved. Please clarify this point (L46–47), potentially by quantifying equivariance error module-wise or via ablation.\n\n* W3. Motivation for Spherical Representation & Frequency Domain: While the use of SO(3)-equivariant models is well-motivated, the rationale for spherical representation and frequency-domain approch remains vague. Please provide justification or empirical comparison with spatial-domain approaches.\n\n* W4. Frequency-level Choice: Although SH frequency is increased to 40, the paper does not isolate how this impacts precision. A direct analysis of the trade-off between SH frequency and performance would strengthen the claim. Is there an accuracy-efficiency trade-off analysis for different frequency levels?\n\n* W5. Sample Efficiency Claims: The paper mentions sample efficiency in the introduction, but does not provide supporting experiments. Clarify whether zero-shot (L285–290) or generalization tests (L291–306) demonstrate this, and if so, explain how.\n\n* W6. Redundant Writing: Sections 5 and Appendix G contain overlapping content and overly verbose explanations, which at times read like LLM-generated text. A more concise and well-structured narrative would greatly enhance clarity. \nIn addition, the manuscript would benefit from better organization around the following:\n1) Clear separation between reused modules and newly proposed components,\n2) Stronger logical linkage between the stated motivation and the experimental design,\n3) More cohesive flow from methodology to results and analysis.\nA refined and tightly written presentation would make the technical contributions more accessible and compelling to the reader.\n\n* W7.  Lack of HEALPix Comparison: The paper critiques HEALPix spatial grids (L29–30) for overfitting to sample locations. Is there an empirical comparison with HEALPix to substantiate this claim?\n\nQUESTIONS:\nPlease see the Weaknesses section above.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 2, "IMPACT": 4, "ORIGINALITY": 1, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper focus on using spherical fourier transformation and equivariant networks to handle the geometric data like Mesh data.\nSpecifically, the encoder takes SO(3)-equivariant networks fed into the equivariant networks to model the 3d mesh data points into the fourier frequencies denoted by the irreducible representations. Then, the decoder using this irreps as the coefficients for predefined spherical harmonic basis function to reconstruct the mesh input.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. This work extend the usage of SO(3)-equivariant networks into mesh data and consider the mesh problem which is applied in molecular or material modeling at the beginning.\n\n2. The Architecture provides a framework bringing new insights about how to build autoencoder for mesh data.\n\nDrawbacks:\n1. It might be valuable to provide a discussion on the efficiency of the method. \n In particular, it would be helpful to clarify the maximum frequency (L_{max}) applied in this work. \nAs is well-known in the literature, when L_{max} is high, the SO(3) equivariant networks can become computationally demanding, and it would be insightful to understand how this is handled in your setting.\n\n2. It could further strengthen the paper to include a discussion or comparison with widely recognized baselines for mesh or point cloud problems, such as PointNet++[1] and RepSurf[2].\nIncluding these would help better position the contributions of your work in the broader context of mesh-based (point cloud based) learning methods.\n\n\n3. As a suggestion, the paper could emphasize the generated dataset which is also part of the contribution nof this paper.\nAnd it may be helpful to emphasize the importance of these generated datasets and how they can contribute to real-world applications.\n\n\n\n[1] PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space.\n\n[2] Surface Representation for Point Clouds.\n\nQUESTIONS:\nQuestion:\n1. How did you handle the distance in the 3D meshes when performing the Fourier transformation?\nI understand that applying the Fourier transformation on the S2 space is natural. \nHowever, when working with mesh data, defining the origin of the system requires transforming the mesh points by considering their distances to the origin. \nIn this case, I believe applying spherical harmonics alone is not sufficient. \nHow did you address the handling of these distances?\nIn Figure 2, it is said it will use \"grid of pre-computed harmonic basis functions\". \nDoes this mean that each channel will be corresponding to a certain fixed distance?\n\nLIMITATIONS:\nN/A.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}

comments: SUMMARY:
This paper proposes  G2Sphere that maps object geometries to spherical signals by operateing entirely in Fourier space, encoding geometric structure into latent Fourier features using equivariant neural networks and outputting the Fourier coefficients of the continuous target signal at any resolution. The ideas are reasonable and natural. The algorithm is feasible and well demonsrated by the experiments.

STRENGTHS AND WEAKNESSES:
Strengths:   It is quite significant to bridge the equivariant GNNs and spherical CNNs for the modeling tasks in structured physic domians in this work. Moreover, the theoretical foundation is investigated and the complete framework G2Sphere is proposed. The experiments are carefully designed for the tasks of radar response modeling, aerodynamic drag prediction, and policy learning for manipulation and navigation and demonstrate the effectiveness of  G2Sphere.

Weaknesses:  The proposed method consists of certain existing algorithms and is short of a systematic theory. There are some English presentation and grammar errors in the paper.

QUESTIONS:
There are several hyperparameters for implementing the proposed method, what are the best values for a task given the dataset?

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-05