Bridging Equivariant GNNs and Spherical CNNs for Structured Physical Domains
Authors: Colin Kohler
Purvik Patel
Nathan Vaska
Justin Goodwin
Matthew C. Jones
Robert Platt
Rajmonda S. Caceres
Robin Walters
Venue: NeurIPS 2025 poster ()
Keywords: Equivariance
Geometric
Fourier
Spherical Signals
SO(3)
Radar
Policy Learning
Primary area: applications
TL;DR: This paper introduces and evaluates G2Sphere, a general method for mapping object geometries to spherical signals using equivariant neural networks and the Fourier Transform.

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.

Bridging Equivariant GNNs and Spherical CNNs for Structured Physical Domains"
Colin Kohler
Purvik Patel Northeastern University Boston, MA 02115 pu. pu. Chothhastetn edu
Northeastern University
Boston, MA 02115 kohler www.rrheasteernnncoomm edu
Nathan Vaska
Justin Goodwin
Matthew C. Jones
MIT Lincoln Laboratory
MIT Lincoln Laboratory Lexington, MA 02421 jgoodwine11. mit.edu
MIT Lincoln Laboratory
Lexington, MA 02421 nathan. vaskaøl1.me edu
Lexington, MA 02421 matthew. jonesooone0111 mit. edu
Robert Platt Northeastern University Boston, MA 02115 rplatt@ccs neu. edu
Rajmonda S. Caceres
MIT Lincoln Laboratory
Lexington, MA 02421
rajmonda, caceres@11 mit edu
Robin Walters Northeastern University
Boston, MA 02115
waltersschottheastenn edu
Abstract
Many modeling tasks from disparate domains can be framed in the same way, com- puting spherical signals from geometric inputs, for example, computing the radar response of different objects or navigating through 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-spperica 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 various challenging domains, including radar response modeling, aerodynamic drag prediction, and policy learning for manipu- lation and navigation. We find that G2Sphere outperforms competitive baselines in terms 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: https://github com Colinoooleeeeereeeeeeee / http://olo.noooeeeeeeeeettrrpeep http///emmmtttttrrreeeeeeee
"DISTRIBUTION STATEMENT A. Approved for public release. Distribution is unlimited. This material based upon work supported by the Under Secretary Defense for Research Engineering under Air Force Contract No. FA8702-15-D-008
or FA8702-25-D-B002. Any opinions, findings, conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the Under Secretary of Defense for Research and Engineering.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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[a] Radar Response
(b) Acrodynamics
[c] Control
Figure I: Spherical Domains. Across many different domains learning high-frequency spherical signals a challenging task. (a) the radar response domain, we learn function Fu,u.. which maps the input geometry X and the radar viewing angle u, to the radar response. (b) In the aerodynamics domain, we learn a function Fx (u:). which maps the input geometry X and the airflow direction to the corresponding drag coefficient. (c) In the control domain, we learn function Fx (u:), which maps the environmental geometry X and the control action u, to their value.
Introduction
Many 3D modeling problems from different domains require solving a common problem, mapping detailed local geometric information to global spherical functions. In this paper, we consider several examples, such as simulating the radar response of an object, predicting the drag characteristics of a vehicle, modeling an optimal control policy for manipulation (Fig T.. Modeling these types of signals is challenging because it requires very fine discretization of the domain to accurately capture high-frequency information, such as sharp specular reflections in radar. This can make traditional numerical solvers and simulators expensive
2 In practice, these approachs limits the potential applications they are too slow for use in many domains such as inverse radar modeling or high-frequency robot controllers.
Deep learning is a potentially more efficient approach, but current methods also struggle with inefficiency and overfitting. For example, one method of using neural networks model spherical output signals is to output a tensor of regularly sampled values across the sphere using a spatial grid such as Healpix 25.. However, a large number of samples is necessary to attain a high spatial resolution over the sphere and even then, the model can easily overfit to the specific sample locations. Furthermore, these large input and output spaces can lead to other compounding issues, such as training instability and computational costs [56]. An alternate strategy is to use implicit models, which represent functions in a continuous manner and can be queried at any coordinate rather than outputting values predefined coordinates. Implicit models have the advantages of being resolution independent, but are relatively inefficient, requiring both large amount of data and training to achieve good performance [2). Furthermore, implicit models have been shown to be prone to overlitting to the training coordinates 18] and lose the geometric relationship between the input and output spaces.
In this work, we seek to address these challenges by introducing G2Sphere, a more accurate, general, and efficient approach to mapping object geometries to spherical functions. We model the output signal in Fourier space linear combination of spherical harmonic (SH) basis functions, enabling simultaneous prediction of the continuous output function at any resolution. Although previous work has modeled mappings from geometric inputs to spherical functions using spherical harmonics, they rely on predefined mappings to convert geometry to spherical input 220 26). In contrast, G2Sphere uses learned spherical embeddings for geometric inputs that are both more general and more flexible. Furthermore, G2Sphere maintains the geometric structure between the input geometry and the output signal by operating entirely in Fourier space and enforcing the end-to-end S0(3)- equivariance. G2Sphere leverages equivariant graph convolutions 24 40 to encode the geometry, spherical convolutions [15] to render the spherical signal, and pointwise MLP non-linearities model information in the output space. This combination alleviates issues with existing equivariant GNN approaches which have been limited to fneer-freuuency modeling, primarily due to computational constraints.
Our contributions are as follows:
A novel -euuierqni architecture, G2Sphere, for mapping complex geometric inputs to high frreueecy, continuous spherical functions, utilizing Fourier decomposition.

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Empirical demonstration of the improved accuracy. flexibility, and efficiency of G2Sphere across wide range of challenging domains, including radar response and aerodynamic drag of 3D meshes, and policy learning for 21) and 3D control.
Demonstration of zero-shot super- repputresoltiion ability and generalization to novel meshes.
In policy learning, an inference time of only 9ms compared the 156ms of Diffusion Policy, a leading state-of-the the-art approach, enabling our policy to achieve -ree-freueney controller.
2 Related works
Fourier Transform in Machine Learning The Fourier transform (FT) is an important tool in a wide range of mathematical and engineering applications ranging from solving differential equations [16] to quantum mechanics and signal processing [3. Fourier transforms have also been extensively utilized in machine learning. In image processing and pattern recognition the FT has been used to
speed up convolutional neural networks [42), to improve feature representation [47,, and to increase the resolution of generative adversarial networks [1. Lee-Thorp et al. [37 uses the FT in the more modern transformer architecture to improve tokenization and improve performance. Li et al. [8 combine neural operators with the Fourier transform to create the Fourier Neural Operator (FNO) to solve a number of challenging PDE systems and Bonev et al. 7 extend this work to the Spherical FNO, which they use to model weather systems. G2Sphere and Spherical FNO have a number of notable differences. First, the trainable layers of G2Sphere operate entirely in Fourier space, whereas SFNO also has parallel trainable layers operating in the spatial domain. Additionally, SFNOS are typically used to model dynamical systems which have similar continuous input and output spaces, while G2Sphere maps discrete geometric inputs to continuous outputs.
Equivariant Neural Networks Equivariant neural networks constrain their layers to respect transformations under a symmetry group [4 23 61]. Neural networks equivariant to the 3D rotation
group S3(3) are used for classifying shapes 20 15
classifying protein structures [62]. determining
3D orientation [36], and predicting features of atomic systems [54 g 39]. Similar to our method, all
these approaches use steerable kernel bases and perform equivariant operations, such as convolutions or tensor products, but almost all are limited to harmonics of low degree (5 10). As such, G2Sphere is capable of capturing significantly more detail and fineggrained structure due to higher maximum frequency (+e 40). Esteves et al. (20] and Cohen et al. [15] model data over the sphere, but a predefined mapping to convert the input into a spherical signal. They also focus on learning functions with low-dimensional discrete outputs, e.g. classification. Ha and Lyu [26] also uses predefined mapping to convert brain geometry spherical signals and a spherical UNet with frequency up-sampling to do segmentation. Lee and Cho [36) extracts features from 2D images using resnet and projects them onto sphere before applying spherical convolutions. They also operate a fixed low maximum frequency of 5. Unlike these methods, G2Sphere uses a learned mapping to convert object geometries to spherical signals, enabling the broad applicability of G2Sphere, enabling the use of the same model for a wide range of applications.
of much
Spherical Domains Spherical functions naturally arise in number of disparate domains including physical modeling and decision-making tasks. In the radar response domain (Fig. [b], the reflected response from a 3D object is function defined over the sphere, where each direction corresponds to
an incident wave direction. In the aerodynamic domain (Fig. [T], the drag function can be represented as spherical signal, capturing how air resistance varies across orientations of a body in flow. Current machine learning methods for reconstructing radar and aerodynamic signatures do not explicitly model the inherent geometric and physical symmetries of 3D objects, e.g., rotational and permutation symmetries [63] 46 64, [9]. Geometric deep learning has emerged as a class of algorithms that can address some these challenges by directly encoding symmetry and other physical priors into neural networks [60 9 6 23 but applications to settings that map fine 3D geometries to corresponding high- high-frequency signals like radar and aerodynamics has been lacking. Similarly, control policies can be framed as functions over spherical action domains (Fig. [Tb). particularly when considering continuous action spaces and geometric state spaces. While several recent works have explored
incorporating geometric structure into policy learning (58 28 29 29 they primarily focus on extending prior methods using equivariant neural networks, whereas G2Sphere redefines the value function composition of harmonic basis functions.
3

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3 Background and Problem Statement
Problem Setting We consider the problem of mapping 3D geometric data, such as object meshes or point clouds, to spherical signals representing physical systems, such as radar response or drag. We consider spherical signals as continuous functions f: S + R where S is the 1-sphere or 2-sphere and the number of channels in the signal. Given some input geometry X C RR, the objective is to learn functional mapping F X + fx. Since full observations of fx are rare in practice, we assume a data set consisting only of partially observed output. That is, a dataset D = {(x,, {ug) ). (. 1)
where samples consist of meshes or point clouds X,, set of viewing directions (uiz), where each thj 5 and the value of the spherical signal Myy fx, (u,) E I= at each U4j. Since the output signals are only partially observed, we consider generalization to new viewing directions Wij and new geometries X,
Equivariance The value fx (u) may be considered as a property of the geometry X defined with respect to viewing direction u E 5². We assume that both the geometry X and the direction u are defined with respect to the same coordinate frame. Thus if the coordinate frame is rotated, both X and u are rotated, and fx (u) = fax (Ru) should be invariant. This property is equivalent to the equivariance of the functional mapping F; if X is rotated to RX, the spherical signal fx rotated to JRX = Rfxx preserving the relationship between the input geometry and the output signal. That is, ((RX)= = (.)... This property can be enforced in the model architecture to ensure that the learned model respects this property. Since the composition of equivariant functions is equivariant, it is sufficient to ensure that each component of the architecture is equivariant.
(3( Equivariant Neural Networks. For physical 3D systems, since the orientation of the coordinate frame is arbitrary, many task functions should be (33u)uequivariant. Broadly speaking, there are two types of -eeui-qquiaariant models; those which focus on modeling high frequency spherical signals such as Spherical CNN or Spherical Fourier Neural Operator
75] which have been applied to weather prediction and GNN-based approaches such as Equiformer (39 which specialize in lower frequency signals over geometric graphs, e.g. in materials property prediction. In spherical CNNs, features are defined signals f:: X - R, where X' = Sa or X' = SO(3). The model architecture consists of alternating layers of SO(3) group convolutions and pointwise activations applied on SO(3) or S². In GNN-based approaches, features are defined as node embeddings that transform according to (3))-irreducibbl representations. Linear layers are implemented via messaging passing, where node features are tensored with spherical harmonic embeddings of their relative positions and then projected back to irreps using Clebsh-Gorde Gordan coefficients (23]. Non- linearities are typically implemented radial gated equivariant activations.
Spherical Fourier Features Cohen et al. 15] provide an efficient solution for handling SO(3)- equivariant signals in the Fourier domain using the truncated basis of spherical harmonics Y for signals defined over S and Wigner D-matrix coefficients D for signals over SO(3). Writing f: SO(3) + R in terms of the D I' and then truncating to given frequency < L gives the approximation f(g) E Σd+1 Σn=o C (g). The SO(3) group convolution can be efficiently computed in the Fourier domain in terms of the coefficients (cmm by convolution theorem. See Cohen et al. [15] for additional details on the SO(3) group convolution.
4 Method
A G2Sphere model consists of two key components: (1) an encoder, which compresses the input data into a compact latent representation, and (2) decoder, which decomposes this latent representation into the spherical output using spherical harmonics (Fig. ( Encoder-decode architectures such as U-Net [65] and Transformers [55] are attractive methods for prediction tasks with high -igeeiinnsiional inputs, such as 3D geometric data, due to latent space compression. The encoder-decoder design a good fit for our problem since we map between two different types of geometric data. also allows us to match the geometry of the input and output spaces to the architecture of the encoder and decoder. Using these learned and generalizable mappings from geometry to spherical signals, G2Sphere can be applied to tasks that were previously unattainable or computationally prohibitive

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Ocorratii
Spttrical
Spterical
Spherical
Inverso Foutter
Sphercaa
Spicce
Spretical
Hammo
Paaig
Figure 2: G2Sphere Architecture. Illustration of the G2Sphere (G2S) model. First, the geometric input is converted from real space (red dotted-line) to Fourier space (blue dotted line) using an -equivaaaaa encoder. We decode this spherical latent representation into set of Fourier coefficients é, representing the continuous spherical signal. Finally, these coefficients are combined with grid pre-computed harmonic basis functions to convert back to real space.
under previous methods [20
15 40
We detail the exact implementations utilized in our experiments
in Appendix G
4.1 Encoder
In G2Sphere model, the first step to encode the local and global information of the 3D input into Fourier space, thereby ensuring that Euclidean and permutation symmetries are correctly respected within our model. This is accomplished by utilizing an equivariant neural network encoder, with the exact architecture varying depending the type of input data. Broadly speaking, we define 3D geometric data as information that describes the shape, position, and spatial propertics of objects in virceddimensional space. This can include various types of data including point clouds, voxel grids, object meshes, and RGB-D images. When dealing with dense inputs, such as point clouds or object meshes, equivariant transformers such as Equiformer V2 440] used. However, when dealing with voxel grids or RGB-D images, E(n) ((qi)--quiaaiian Steerable CNNs [1] are a more natural choice. These types of models the computational costs of basis. In each case, the input signal is initially spatially dispersed in R and the end of the encoder is aggregated into a single feature vector representing the multi-channel spherical signal in Fourier space.
constrained
low-frequency spherical features, i.e. lmar 10, due features scaling significantly when applied on a per node
In Figure we illustrate a G2Sphere model with mesh inputs which uses Equiformer V2 [40], SO(3)- eequivrriant transformer commonly used 3D geometric data, the encoder. The mesh is embedded as a geometric graph with node and edge features. The node feature at node is the position x; R and the edge features are the spherical harmonic embedding of the relative positions Eij = Y (xj x:), where y": R3 + 222+1 are the spherical harmonics and i, are vertices connected by an edge. Then set of node-wise spherical features are learned over the graph which are mean pooled to form a multi-channel spherical latent space.
4.2 Decoder
Once the input has been projected into Fourier space using the encoder, we use operations that preserve the SO(3) symmetry of the latent representation. Specifically, we use 50(3)-equival) group convolutions to form a Spherical CNN [15) decoder. The output of the final layer of the decoder
represents N-channel signal over S in the Fourier domain. That is, fr (c ) m=0.1= are the coefficients of the spherical harmonies up frequency L for 1 < k < K. To convert the signal back to real space, we apply the inverse Fourier transform and evaluate these K signals on the 2-sphere, at position specified by spherical coordinates (0, Φ). That is,
L 21+1
K
((0 =
Σ Σ NOC
( d)
k=1
where Vi are the spherical harmonics of degree / and order m. In practice, we pre-compute the spherical harmonies basis functions at the desired grid resolution to allow for fast evaluation. See Appendix D for additional details on harmonics implementations.
5

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Table I: Mesh to Spherical Functions. Mean square error of G2Sphere (G2S) and baselines on the radar and drag domains. Performance is averaged across 3 random seeds with = standard error.
Domain
Mesh Type
Transformer
Equiformer
Spherical CNN
G2S
G2S+TSNL.
Frusta
0.201 = 20-4
0.271 + 3o-4
0.438 lc-6
0.221 lo-5
0.195 9c-4
Radar
Asym
0.179 = 5e-4
0.257 + 6e-4
0.496 + 1e-6
0.123 + 2c-4
0.128 + 4p-4
Drag
Pods
0.064 + 1e-2 0.0672 4e-3
0.075 + 3e-4
0.062 + 2c-3
0.064 + 6e-3
One drawback to using harmonics to approximate functions in this manner is that if the maximum frequency I. is low, then their resolution will also be low. The SH excel capturing smooth, low- frequency components, but struggle with sharp features or discontinuous behavior unless a large number of harmonics are used, i.e. high L. As result, traditional equivariant decoder architectures, which operate with a relatively small number of harmonics (L S 10), struggle in domains where higher output fidelity is required. G2Sphere uses two techniques improve output f frequency up-sampling where we gradually increase the frequency from the encoder's 's frequency Lene to the output frequency Ldec and trainable spherical non-linearities (TSNL) 77. where an MLP is applied pointwise across the sphere resulting in arbitrarily high frequency. We implement more general version of TSNL. than Bonev al. 7 by applying inverse Fourier transform for each frequency | separately, giving the MLP more input channels.
Trainable spherical non-linearity (TSNL) To further increase L in the output space, we apply a learned MLP non-linearity pointwise to the spherical output signal. That is, given decoder outputs
d we apply the FT each frequency separately yielding f = [22++] y km)) x.t" a K L- channel spherical signal f 22 - RK-L Let M be trainable MLP M: R- -> R.. The final output is then evaluated at E S as (f()).. This allows the maximum frequency of the output signal M o f to be unbounded when using typical non-linearities inside M. While this could also be achieved by setting M to be a single fixed non-linearity, a trainable M is more controllable and can introduce higher frequencies without adding artifacts.
The TSNL is equivariant up to the sampling error introduced by the inverse Fourier Transform used to convert from the frequency domain features to real space signals a fixed spherical grid. Since M is applied identically at each point in the sphere, it is SO(3) equivariant by construction. While the use of finite spherical sampling can introduce minor numerical artifacts, these lead to negligible equivariance error in practice. This approximation is common in many equivariant architectures such SFNO 771. Spherical CNN [15), and Gauge Equivariant Mesh CNNs [17). Notably, Cohen et al. explicitly measure this equivariance error and find it to be minimal (< 2e 6).
15
Frequency Up-sampling Inspired by Ha and Lyu 226). while increasing L in the decoder, we gradually reduce the number of channels in our latent representation (See Appendix G. To maintain equivariance during this operation, we need to account for the relationship between the Fourier and real space. This is accomplished through the use of regular non-linearity (as in [15) [77), where we map the signal to real space using the inverse Fourier transform (IFT), apply a point-wise non-linearity to the values of the spherical signal, and then perform the Fourier transform (FT), with higher frequency resolution, to convert back to the frequency domain. As with the TSNI., this operation is -quuieqqui up to the same minor sampling error introduced by the IFT, which is negligible in practice. By combining a number of these non-linearities in our decoder, we are able to achieve much higher maximum frequency (up to L 40) than previous works using equivariant architectures with dense geometric inputs, e.g. object meshes S 33 This frequency upsampling mechanism is a key architectural component that enables our model to produce high-fidelity, fine- grained reconstructions, directly contributing to the improved high-resolution radar outputs shown in Figure and discussed in Section
5 Experiments
To demonstrate the generality of G2Sphere, we systematically evaluate across three different domains: two supervised learning domains, where we learn mapping from object meshes to spherical radar and

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(c) Ground Truth
(f) G2S
(a) Ground Truth (b) Transformer
(c) Equiformer
(d) G2S
Figure 3: Left: Radar Response. Outputs generated by the ground truth first- prittprrneiples simulator, two baseline methods and G2Sphere. G2Sphere does a good job in reconstructing the overall projected shape characteristics and correctly capturing both types of radar scattering responses whereas the Transformer model underfits and the Equiformer model overfits. Right: Zero-Shot Super-Resolution. G2Sphere trained on 60 20 and tested on 180 20.
drag signals, and policy learning domain, where we learn mapping between states and state-action values. These evaluations benchmark the performance of G2Sphere on dense (e.g., radar response) and sparse (e.g., drag coefficient) spherical signals and with complex (object mesh) and simple (object keypoint) input geometries. G2Sphere consistently outperforms the prior state-of-the of-tth--rttttttt in all domains. In the following, we provide overview of each domain, our evaluation methodology. and our key findings.
5.1 Radar and Drag Modeling from Meshes
Many modeling tasks take object mesh input and predict physical property of that object. If the property depends on a direction vector, that is, a spherical coordinate (0, Φ), then the output may be represented as continuous function defined the 2-sphere. We evaluate the performance of G2Sphere against baselines on two such tasks, predicting radar response and aerodynamic drag.
Radar Prediction Due the expense of collecting real-world radar data, we simulate radar responses for a variety of different meshes using a physical optics method to approximate the electromagnetic waves [4] Specifically, we generate two mesh datasets, Asym and Frusta. See Fig. 10 for examples. The Asym dataset is collection of randomly scaled basic shapes (cubes, cylinders, and spheres), with randomly generated protrusions to make the radar response asymmetric. The Frusta dataset represents collection of complex 3D shapes, where each mesh is a combination of several basic components stacked together form roll-symmetric objects. Due to this symmetry, we evaluate each frusta mesh over 360 orientations
the non-symmetric axis, 0 E [0, 2n] to generate the associated radar responses. The Asym meshes are evaluated over both axes, E [0, 2n). Φ E [0, T].
where 0 is discretized over 60 values and á over 20.
Drag Prediction We use high- petfrrrrrormccc Computational Fluid Dynamics (CFD) simulator to generate Pods dataset of mesh and drag coefficient samples of pod-like geometrical shapes. There are some differences relative to the radar dataset. First, as drag-coefficiemt are of most interest for flying objects (e.g. aireraft) the drag coefficients
only simulated for cone in the front of the objects with 0, à E [-20, 20). Secondly, a large contributing factor in the drag coefficients are the of flight conditions (altitude, speed, etc.) which we latent representation after encoding the mesh. The Pods dataset contains 10, 000 samples (mesh, flight conditions, drag) each with a single drag coefficient for specific angle (0, Ø), which defines the orientation of the incoming flow relative to the centerline of the Pod shape. Additional details on the radar and drag datasets can be found in Appendix
global parameters and append the
G2Sphere Architecture We use Equiformer v2 440 to encode the input mesh into set of latent representations for each vertex in the mesh. These latent vectors are then combined into a single
feature vector using global average pooling layer. The decoder is spherical CNN with several
layers of spherical convolutions followed by the spherical non- con-lineerrty introduced in See. 4.2 to upsample the frequency from Lene to Ldee- We use Lene 5 for both tasks but use Laee 40 for

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Ground Truth
G25
Equiformer
Transformer
63
-22
-3
-01
-03
-03
-03
-03
Figure 4: Generalizing to Unseen Objects. Full drag prediction on test sample not included in the Pods dataset. Unlike in training dataset, we compute the entire drag prediction from -20 to 20 degrees (shown in radians here). Despite being trained on single coordinate points, G2Sphere is able to reconstruct the general shape of the drag function, whereas the baseline implicit models overfit to specific points.
the radar domain and Teee = 5 for the drag domain. We examine two variants of G2Sphere, one with the trainable spherical non- non-linearity (G2S+TSNL), and one without (G2S).
Baselines We compare the performance of G2Sphere against three competitive baselines for processing meshes, Transformer, Equiformer, Spherical CNN. The Transformer model is inspired by other prominent mesh-based transformer architectures 52 21 It tokenizes the mesh into spatial and structural descriptors in 21], and uses transformer encoder with MIP decoder generate the predicted response. The Equiformer [40] model resembles the G2Sphere. but directly maps the latent representation the discrete set of spherical values. The Spherical CNN (20) model uses ray-based method to map the input geometries onto the sphere, followed by a series of spherical convolutions. This can be considered G2Sphere model where the learned encoder mapping is replaced with a predefined map. We use explicit models for the radar prediction task, where the baseline models output the full radar response for given mesh. For the drag prediction task, we use implicit models where the coordinates are passed as input to the baseline models and appended onto the latent representation. This is necessary as, unlike in the radar domain, our drag dataset only has partially observed outputs and therefore lacks the dense targets required by explicit models. See Appendix G for additional information on model architectures and training.
G2Sphere performance We report the performance of G2Sphere and the baselines on both radar and drag tasks in T G2Sphere obtains the lowest mean squared error compared any of the baseline methods. Further, as the complexity of the task increases from the roll-symmetrie Frusta shapes to Asym shapes, the difference in performance between G2Sphere and the baselines increases. This implies that G2Sphere is more adept at predicting dense output spaces than methods that output discrete values. While exact error rates for radar prediction vary depending on the task specifications, the Transformer baseline and G2Sphere fall within the most commonly cited 10 to 20% error range Fig. (Left) shows examples of the spherical radar signals generated by each model using meshes from the Asym dataset. When comparing the predictions generated by the equivariant models (Equiformer and G2Sphere), we can clearly see that G2Sphere captures significantly more detail and fine-grained structure which we attribute to the higher maximum frequency. Additionally, we can see that the Spherical CNN does very poorly, suggesting that the ray-based mapping from geometry to sphere does not capture the geometric information required for the radar prediction task. Similar to the radar task, we find that G2Sphere is able to achieve good performance on the drag task despite being trained around 6%, which falls within the single-digit percentage errors benchmark for aerodynamics [43] In the following, we explore two interesting zero-shot capabilities of G2Sphere.
30
sparse set of coordinate values. We note that all of models achieve error of
Zero-Shot Super-Resolution G2Sphere outputs coefficients of Fourier basis functions and thus gives a continuous spherical signal which can be evaluated at arbitrary resolution. Therefore, G2Sphere can be trained on only low resolution radar data and evaluated at higher resolution, i.e. a zero-shot super-resolution task. Fig. 3 (Right) shows an example where we train G2Sphere 61 x 21 (0 x Φ) resolution radar data and transfer to 180 21 resolution. G2Sphere is the only model among the benchmarks (Transformer, Equiformer) that can do zero-shot super-resolution.
8

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Table 2: Policy Performance. Comparison of G2Sphere (G2S) against baselines the PushT and PyBullet Drone domains. We compare max performance and, in parentheses, the average of last 10 checkpoints, each averaged across 50 different initializations. G2Sphere significantly outperforms both an equivariant IBC model and the Diffusion Policy.
Domain
Task
IBC
E-IBC
Diffusion
NE-G2S
G2S
Fixed
0.85 (0.81)
0.98 (0.95)
0.95 (0.91)
0.97 (0.95)
1.0 (0.98)
PushT
Random
0.62 (0.54)
0.81 (0.76)
0.71 (0.69)
0.74 (0.70)
0.92 (0.89)
FlyToTarget
0.94 (0.87)
0.98 (0.92)
0.96 (0.94)
1.00 (0.95)
1.00 (0.97)
PyBullet Drones
Fly FlyThroughGate
0.90 (0.83)
0.94 (0.87)
0.94 (0.92)
0.92 (0.90)
0.98 (0.95)
Generalization to unseen Object Geometries. Our drag dataset contains only single drag coefficient corresponding to a single (0, Φ) for each object. In this experiment, we evaluate G2Sphere's s ability to generalize
objects held out during training and predict the full drag response cone, (0, Φ) E 20 20). Fig. 4 shows the generalization capability of G2Sphere when compared to the baselines. When compared to these implicit baseline models, G2Sphere demonstrates a stronger ability to generalize, effectively capturing the overall shape of the drag cone and producing more accurate predictions. Although all the baseline models perform well on the training data, here we see that they have overfit to specific (0, Φ) coordinates, resulting in poor generalization.
5.2 Policy Learning
In order to demonstrate the flexibility G2Sphere for high- frequency continuous spherical signal learning, we apply G2Sphere on four policy learning tasks from two benchmarks [22 48 Although there are many potential policy learning algorithms to which we could apply G2Sphere, this work explore the application of G2Sphere behavioral cloning (BC) [49]. BC casts the policy learning task as a supervised learning problem where the agent learns to imitate a set of expert demonstrations. Following Florence et al. (22). we formulate BC conditional energy- problem. EMBs define a policy through an energy function which assigns an energy value to each state-action pair. We compare G2Sphere to prominent EBM policy learning methods, Implicit Behavior Cloning (IBC) 22 and Diffusion Policy [33, and demonstrate that G2Sphere outperforms these baselines both in terms of final performance and sample efficiency.
modeling (EBM) (35)
Training spherical value functions We use techniques from the EBM literature to train our G2Sphere EBM. Given a dataset of state-action samples as, where s, E I and a, R we aim to learn an energy function E : R N + R. We train using contrastive learning [34] by generating set of non-expert actions a' and utilizing an InfoNCE-style loss function 45 where the model must predict the true expert action a; from the non non-expert actions a'z.
The actions a are 2D or 3D velocity vectors normalized to be in the unit ball. In order to model the energy function in Fourier space, we decompose into action magnitude ||a|| and an action direction à a/ |a]l e Sd. For 2D actions, the energy function is evaluated
E)(s, = Σ (Fo(s))) (,,
L,m
where Fg is a G2Sphere model outputting Fourier coefficients ci and Bi, are polar harmonics (Appendix E),, which gives basis of functions over the unit ball. For 3D actions, the magnitude is treated as an implicit variable and Eg(s, a) = (F,,(Fsss [|al))) y. (a).
we use keypoint observations which reduce the underlying geometries of objects in the scene to a set of keypoints that capture the spatial and structural information in the environment. Although we could still use an equivariant GNN encoder, given the small size and lack of variability of the input, an equivariant MLP is simpler. the PushT domain, we have 2D keypoints and actions and therefore use -quivvaraa linear layers. The PyBullet Drones domain a 3D navigation task, and we use (eu))--quv linear layers. The decoder remains spherical CNN but with small modification for the 2D domains to perform
G2Sphere architecture Following Chi et al. 13

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Table Inference Speed (ms). Comparison of inference speeds (ms) on an Nvidia Titan.
Domain
IBC
E-IBC
Diff
NE-G2S
G2S
Drones
32
83
156
convolutions on the I-sphere. To separate the impact of equivariance and representing the output in Fourier space, we consider non-equivariant version of G2Sphere (NE-G2S) where we replace the equivariant MLPs with normal MLPs but still output the Fourier coefficients. Additional architectural and training details for both G2Sphere and the basclines can be found in Appendix
PushT Adapted from 22 3 the PushT task (Fig. 13a requires pushing a T-shaped block (gray)
to fixed target (green) with a circular end-effector (blue). Object geometries represented by series of keypoints denoting the pose of the agent, block, and goal. We examine two variants: a fixed- goal variant where the target pose is always set to the pose in Fig. 13a and a randomized -agal variant where the target pose is randomly sampled within the workspace. The metric for performance is target coverage area between the block and the goal pose. We compare the performance of G2Sphere against two versions of IBC, standard (IBC) and equivariant (F-IBC), and Diffusion Policy. The results are shown in Table [] We find that G2Sphere outperforms all other methods using the best checkpoints almost always achieves perfect score in the fixed-variant. Additionally, we can see that the addition of equivariance stabilizes training instability leading to more consistent performance.
PyBulle Drones We use the PyBullet Drones benchmark Panerati et al. 48 to study the perfor- mance of G2Sphere in 3D positional control domain. Specifically, we examine two tasks of varying difficulty, FlyToTarget (Fig. [3b] and FlyThroughGate (Fig. [3c). In the FlyToTarget task, the drone must take off from a landed configuration and fly the larget position. In the FlyThroughGate task, the drone is initialized in flying configuration and must fly through a gate which is randomly posed around the drone. The observation space is composed of 4 keypoints for the drone, 4 keypoints for the gate, and single keypoint for the target position. The action space is delta motion from the current position which is passed to PID controller. The agent operates at 30hz and the PID controller at 240hz. The results are shown in Table 2 We see that similar to the PushT results, G2Sphere outperforms all other methods and provides increased training stability.
Inference Speed It is critical to have fast inference speed for many closed-loop real-time control tasks, such as robotic control. One of the known drawbacks to diffusion that the denoising process requires number of steps to optimize the trajectory. This is somewhat mitigated in the Denoising Diffusson Implicit Models (DDIM) approach[13] which has enabled relatively fast inference times of 100ms. However, this does limit diffusion policies from many high-freeuency control tasks, such as policies enabled with feeddbcke G2Sphere, on the other hand, has extremely fast inference time due the energy distribution in a single forward pass. demonsssesssssaaees the differences in inference speed for G2Sphere, IBC, and diffusion, with equivariant G2Sphere taking only 9ms compared to 156ms for Diffusion Policy.
use of grids of pre-computed harmonic function values, which can accurately predict
6 Conclusion
In this work, we present G2Sphere, a novel approach to learning mappings from complex geometric inputs to continuous spherical signals. We evaluated G2Sphere across a wide range of challenging domains including radar response prediction, drag modeling, and policy learning for navigation and demonstrated improved performance on all tasks. By employing a fully -0(3)-equivariant architecture that learns the mapping from object geometry directly the spherical Fourier domain, G2Sphere helps close the gap in learning high-fidelity spherical signal representations from geometric data. Additionally, we demonstrated G2Sphere is capable of zero-shot super-resolution, full-resolution generalization to unseen geometries, and vastly improved inference times.
10

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Acknowledgments
This work supported by the Under Secretary of Defense for Research and Engineering under Air Force Contract No. FA8702-15-D-000 or PA8702-25-D-BOO0 We acknowledge the MIT Super- Cloud, the Lincoln Laboratory Supercomputer Center, and the Northeastern University Discovery HPC cluster for providing resources that contributed to the research results in this work. R. Walters was supported by NSF awards #2442658, #2134178, #2314182, #2107256.
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A Reproducibility statement
To ensure reproducibility, we have documented and made accessible all resources utilized by this work. Our source code, datasets, and visualization methods are provided for transparency. We have provided additional training and model architectural details for both G2Sphere and the various baselines Appendix G The techniques utilized to generate our high-quality synthetic data used in the experiments in Sec. 5.1 is detailed in Appendix [ Similarly, the details for the expert data used for policy learning experiments can be found in Appendix Klalonsiide additional environmental details such as the sources for the environment implementations. Finally, the design decisions and implementation details of our harmonics implementation tailored to equivariant neural networks are discussed in Appendix
B Limitations and discussion
There are number of limitations which future work could improve upon. First, in regards to the radar and drag domains, the output fidelity of G2Sphere is heavily dependent on the maximum harmonic frequeney of the model. Unfortunately, current inplementations of the spherical harmonics in equivariant network architectures result in am cubic increase in computational requirements as the harmonic frequency increases, thereby limiting the expressivity of our model. However, there do exist more efficient methods for calculating the spherical harmonics which would allow for increased maximum frequencies in future work [57] Similarly, in this work model the radial component of the spherical coordinates by either outputting a discrete set of spheres for different radii or by including it as implicit variable. An alternaee method would be to incorporate the radial component into the Fourier basis functions allowing for efficient computation of the all spheres continuously within some defined radial bounds (59]. Finally, in this work we only explore 2D and 3D positional control policies and have omitted full 6DoF pose control. The model can be extended to 6DoF by applying inverse FT over SO(3) instead of 5².
C Group theory
Equivariance A function is equivariant if it respects the symmetries of its input and output spaces. Specifically, a functtin F: X + Y is equivariant with respect to a symmetry group G, if commutes with all transformations g E G, (p)((g)z) of the group G that define how the group element E G acts on x E X and y E Y respectively. An equivariant function is mathematical way of expressing that F is symmetric with respect to G; if we evaluate F for various transformed versions of the same input, we should obtain transformed versions of the same output.
Ff()) where Pr and Py the representations
Equivaaauva Neural Networks For 3D physical systems, since the orientation of the coordinate frame arbitrary, many task functions f should be (03u)-equivariant. For neural networks to incorporate this sort of geometric reasoning, it is necessary to parameterize signals f:X- + R. where X' 5" or X SO(3) in a way that is both computationally efficient and easy to apply rotations to. Cohen et al. [15 provide effective solution using the truncated basis of spherical harmonics Y for signals defined over 5 and Wigner D-matrix coefficients D'n,. for signals over 8O(3). Writing f: SO(3) -> in terms of the D! man. and then truncating to given frequency < gives the approximation f(g) R can be efficiently computed in the Fourier domain in terms of the coefficients (cm..) by convolution theorem. See Cohen al. [15 for additional details on the SO(3) group convolution.
221+1 224+1 2.nn DI
(ttnnn))) (g). The SO(3) group convolution
Circular and spherical harmonics The Circular Harmonics (CH) describe functions the 1-sphere (i.e. the circle s*)). They are solutions to Laplace's equation over S* defined (0) cito where the angular coordinate [59]. In terms of group theory, the circular harmonics define irreducible representations for the 2D rotation group SO(2). Therefore, they are useful for constructing -eeuiaaanaa neural networks by representing circular signals as combination of fixed CH basis functions and learnable CH coefficients [61].
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The Spherical Harmonics (SH) extend the CH to describe functions on the surface of the 2-sphere S Similar to the CH, they are orthonormal solutions of Laplace's 's equation on S defined:
21 + 1 (1 mm)
Y (0, -
pl. (cos 0) Amp.
4x (1 + m)!
where 0 is the polar angle, & is the azimuth angle. is the degree, m is the order and Pl is the associated Legendre function (T. The SH describe irreducible representations of the group SO(3) of 3D rotations. They are useful for creating learning [23].
features for ( (33))-qquiaarian deep
D Harmonies implementations
In this section, we describe the details of our PyTorch harmonics packages which implements the Polar and Spherical Harmonics used in this work. While there number of existing spherical harmonics implementations [7]. they are tailored to the tasks that they solve, e.g. PDEs, and therefore do not integrate well with the equivariant neural networks used in this work (e.g. e3nn or escnn). Additionally, there are, to the best of our knowledge, no Py Torch implementations of the Polar Harmonies (detailed in Appendix E). To address these shortcomings, we developed our differentiable implementations of the Polar and Spherical harmonics in PyTorch which we utilize throughout this work. The design goals of this package were fold: (1) allow for the efficient integration of the harmonics with the output of eănn or escnn and (2) enable the calculation of the harmonics on both a pre-defined grid of coordinates and specific coordinates.
There are two considerations to make when tailoring our package to fit together with e3nn and escnn. First, these equivariant neural networks model the Fourier coefficients as the irreducible representations of the symmetry group and, therefor, our implementation needs to use the same representations. We note that this restriction makes number of previous implementations such as Bonev et al. 7 poor fit as they use different derivations of the harmonics with different Fourier coefficient specifications. Secondly, because these equivariant models typically operate in real space, we need the harmonics to be in real space as opposed to complex. For example, we use the sin and cos form of the circular harmonics (Appendix B to match the dimensionality of the SO(2) irreducible representations.
The second goal of our package is enable efficient batch computation of the harmonics many values. We require the ability to both compute the harmonics at specific coordinates (commonly used during training to match the partial views in our dataset) and to pre-compute grid of harmonics values at pre-defined coordinates (typically used at inference time to decrease computation time). In order to achieve this, when instantiating the harmonics, grid of harmonics values are pre-computed at the desired resolution and a number of intermediate variables are saved during this process. These intermediate variables are specific to the type of harmonies but, in general, they are components of the harmonics independent to the coordinates such as the normalization constants and Fourier mode components. This pre-computation allows for the efficient evaluation of the harmonics and reduces training times.
E
Polar Fourier transform
In this section we derive the Polar Harmonics Basis functions used in 2D policy learning experiments by performing Fourier analysis on the Polar coordinates. We would ideally like for these functions to be decomposed into radial and angular components such that we view this decomposition as an extension of the normal Fourier transform. We can fulfill this requirement by requiring the basis functions to take the ceparation-of-varialee form:
[ Φ) = (())
(1)
E.1 Basis functions
The angular component of the basis function is simply:
( =
imp
(2)
2m
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where / is integer. However, this definition resides in the complex domain, and when using equivariant neural networks, it is advantageous be the domain of real numbers. Therefore, will instead use the sine-cosine form:
(Φ)
(cos(T))) + sin(l))),
v 2m
where 7 2 0. The associated transform in angular coordinates is simply the normal 1D Fourier Transform.
We will rely Bessel functions to match the functions of arbitrary spatial frequency required by the radial component the Polar basis function. Using the Bessel function as a basis gives rise to the 1-th order Fourier-Besse series:
1
Pin (p)
J((kimp),
VNim
where No. is normalization constant and kim is the 1hth zero of J1.
E.2 Polar Fourier transform
By combining these radial and angular components we can form the complete basis function for the polar Fourier transform:
im (p. Φ) Pim (p)
where Pim. defined by Eq. 4 and ,, by Eq. 2 As v im forms an orthonormal basis on the region p a, we refer it the Polar Harmonics. A function defined over polar coordinates, {(, o). can be expanded using the polar Fourier transform with the Polar basis function, V mmm. and Polar coefficients, Pim:
inf
f(p
Pim Im (p, 3 mmdm,
(6)
where
2x Pin {e f(p, Wi pdpdddo.
While this infinite transform is of theoretical interest, in practice we instead use the transform defined on a finite region, c.g. the unit-circle. That is, function f(p, o), where P S. a can defined as:
inf
() = Σ
Pim V Vim(p, 15),
(8)
where
Pim 1 S () (p, d)pdpdo.
F
Spherical Fourier Transform
Functions defined on the 2-sphere 52 are commonly expanded in a basis of spherical harmonics Y/, yielding coefficients that compactly capture angular variation up degree 4maz- They are orthonormal solutions of Laplace's equation on S defined:
21 1 (1 mm)!
(0, 0) =
pl (cos ime
4
(1 m)!
where 0 is the polar angle, o is the azimuth angle. is the degree, m is the order and pl is the associated Legendre function [1] The SH describe irreducible representations the group SO(3) of 3D rotations. They are useful for creating (eeui-eqni features for (0eu)-equivariant deep learning 23 and underlay many spherical convolution methods, where learnable filters act on these coefficients to extract multi-scale features from spherical signals.
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(f((u)) =-o
(u,u)
Explex Fo(
impiec Fo(x, ui)
f
F2(22)
Y
2
(a) Explicit
(b) Implicit Model
(c) G2Sphere Model
Figure 5: Modeling Spherical Functions. (a) The explicit model outputs a fixed grid of predictions
approximate the continuous function fa- (b) The implicit model learns fz by conditioning both input z and coordinate point u. (c) G2Sphere decomposes fa into a set of learned Fourier coefficients of" and spherical harmonics Y This enables prediction of fe any coordinate point u.
G Training details
In this section we provide additional details about the model architectures and training process for the experiments in Sec. [5 Broadly speaking our models fall into three categories: explicit models (Fig. 5k), implicit models (Fig 5p), and our G2Sphere model (Fig 5p. ). In the radar prediction domain, the baselines are explicit models using grid of regularly spaced spherical viewing angles. As mentioned in Section 5. the drag prediction domain is limited by the data and, therefor, implicit models are used for the baseline methods. The same base architecture is used when modifying the models to be explicit or implicit with a few small changes to modify the input/ output spaces (see Fig. f additional details). Finally, in the policy learning experiments, the IBC baseline uses implicit models whereas the diffusion policy uses explicit model. All models were trained until convergence measured by loss in the supervised learning domains and policy performance in the policy learning domains. All of our experiments are run on high-performance cluster using either one or two Nvidia Xeon-g6-volta GPUs with 32GB of memory. Fach node has Intel Xeon Gold 6248 2.4 GHz CPU with 128GB of RAM and ITB of local storage. Each experimental run required approximately 8 GPU-hours, and the total compute for all reported experiments was around 300 GPU-hours. Including preliminary experiments and ablations not included in the final paper, the full research project used approximately 500 GPU-hours.
G.1 Mesh-to-Sphere
G.1.1 Transformer
The Transformer architecture composed of a mesh face embedding layer, transformer encoder, and an MLP decoder. The mesh face embedding layer takes individual mesh faces as input and uses geometric features like vertex position and normal direction to calculate an embedded representation of the face. The application of this layer to the mesh generates sequence of embedded faces; learned classification embedding is prepended to this sequence. standard transformer encoder as introduced in Vaswani et al. [55] is then applied to the embedded sequence to transfer global information about the mesh faces to the classification embedding; after this operation, the MLP decoder is used to convert the classification embedding into the radar response. We train using the Adam optimizer [32] with the best learning rate and its decay were chosen to be 5e -4 and le
respectively. We use batch size of 4.
G.1.2 G2Sphere
For the mesh-to-sphere domain, the G2Sphere model uses Equiformer v2 to encode the object mesh into Fourier space. We use the equiformer architecture from Liao al. [40] with layers where each layer feature dimension of 128, Imar of 5, and 4 attention heads. We then use global mean pooling head to compress the node features returned by Equiformer into a single latent representation of size 128. The decoder is 5 layer Spherical CNN [I5] with the feature dimensions [128, 64, 32, 16, K]. where K is the number of output channels the spherical signal, and Imaz dimensions [5, 10, 15, 20, 40). We use the ReLU [] activation in our (0(3)-active) activation layers in the
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Spherical CNN. We train using the Adam optimizer [32 with the best learning rate and its decay were chosen be 1 1-4 and 0.95 respectively. We use a batch size of
G.1.3 Equiformer
The Equiformer baseline has the same encoder architecture as G2Sphere but slightly different decoder architecture. The Spherical CNN decoder in Equiformer, has 6 layers where the feature dimensions [128, 64, 32, 16, K), where K is the number of output channels in the spherical signal, and maz dimensions [5, 5, 5, 5, 5/- We use the ReLU [] activation in our in the Spherical CNN. At the end of the Spherical CNN, final spherical convolution converts the latent representation into the explicit grid of output values. We use the ReLU [] activation in our 50(3)-activatio) activatiin layers in the Spherical CNN. We train using the Adam optimizer [32 with the best learning rate and its decay were chosen to be 1 and 0.95 respectively. We use batch size of 4.
layers
G.2 Policy learning
G.2.1 G2Sphere
In the 2D and 3D policy learning domains, our observations are a set of 2D/3D keypoints which describe the objects in the scene. Therefore, we use much simpler models than in the mesh-to- sphere domain. Specifically, we use SO(2) and SO(3) -equivarrv MLPs as both our encdoers and decoders. In the PushT domain, we use 4-layer (eeeiurquiaariin MLP for our encoder with a 5 $12-dimensional latent representation and a 4-layer, 512-fcature dimension, S0(2) S0(2)-equivariant spherical CNN with as the decoder. This S0(2) spherical CNN is essentially a series of spherical convolutions on the 1-sphere as opposed to the 2-sphere. Both encoder and decoder use Immaz 3. Similarly in the PyBullet Drones domain, we use 4-layer -0(3)--quivarainn MLP for our encoder with 256-dimensional latent representation and a 4-layer, 256-feature dimension, spherical CNN decoder. Both encoder and decoder use Lmaz of 5. Both models use a dropout (53) 0. while training. We train using the Adam optimizer [32] with the best learning rate and its decay were chosen to be 1 and 0.95 respectively. We use a batch size of 256 and train our contrastive loss using 256 negative samples.
G.2.2 IBC
We use two IBC models in our policy learning experiments: non-equivariant version which uses standard MLPs and equivariant version which uses SO(2) or -equvvvraa MLPs. The non- equivaraint IBC is comprised of a 4 layer MLP encoder and a 4 layer MLP decoder both with 1024 feature dimensions. The implicit actions are appended onto the latent representation after it has been encoded. The equivariant version of IBC, has the same structure as the non-equivariant model but uses feature dimension of 512 and I. Iamaz of and (for the 2D and 3D domains respectively). We train using the Adam optimizer [32] with the best learning rate and its decay were chosen to be 1 and 0.95 respectively. We use a batch size of 256. The contrastive loss is generated using 256 negative samples following Florence et al. 222 We use dropout [53] of 0.1 while training. At inference time, we perform 3 iterations of Derivative-Fres Optimization 22 using 4096 samples to select the best action.
G.2.3 Difftusion
We use the implementations from [3] [MIT License] for our Diffusion models, specifically, we use the transformer--en based architectures. The transformer has layers with 4 heads where each token is embedded as a 256 feature vector. We use the Square Cosine Schedule proposed in iDDPM [4] which cited as the best performing noise scheduler Chi et al. 13]. We train using the Adam optimizer 32 with the best learning rate and its decay were chosen to be 1 and 0.95 respectively. We use a batch size of 256. We use 100 denoising diffusion iterations for both training and inference. At inference time the model predicts the next 16 actions of which 8 are executed.
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Table 4: dec Ablation. Mean square error of G2Sphere using increasing maximum output harmonic frequencies. As the output frequency of the decoder increases the performance of the G2Sphere model improves. Performance averaged across 3 training seeds.
Asym
Lace = 5
dec = 10
Ldec 25
Ldee = 40
G2S
0.383
0.218
0.148
0.123
(a) Ground Truth
(b) Ldec
5
(c) Liten 10
(d) Liter 25 (e) Lite 40
Figure 6: Effect of Ldec on Fidelity. As the maximum frequency of the decoder increases so does the accuracy of the G2Sphere prediction. At Laee 5 we see that most of the features of the signal are lost and we are only capable of modelling the general high/low signals. However, as we increase to Ldee > 5 we can the features getting sharper the output frequency increases.
H Maximum frequency ablations
A unique capability of G2Sphere is its ability leverage the maximum harmonic frequency L to control bias of the model. A higher L captures more fine details and the underlying function which can lead to more complex and accurate approximation. Therefore when trying to predict the eigh-frequency radar response function, we find that the higher the L the more accurate our predictions become (Table 4 Fig- (). However, we note that this is only possible due to the large scale radar datasets which have responses over the entire sphere for each mesh. In contrast, in domains where we lack these dense outputs, such the drag and policy learning domains, we find lower L can combat overfitting by providing a smoother, lower-detailed approximation of the underlying function (Fig.
components of
I Multimodality
Learning from human demonstrations presents a significant challenge to BC due to the challenge of
modeling the multi-modal distributions common in human actions [2 51 27 13, One advantage of G2Sphere is the ability to control the maximum multimodality of the policy via the maximum frequency. At a high level, the maximum frequency describes the amount of energy in the basis function associated with that maximum frequency. From practical point of view, this means that the higher the frequency the more complicated function you are able to model. G2Sphere takes advantage of this by setting the maximum angular frequency to at least the amount of multimodality
Ground Tuth
L03
L425
-24
63
2200
02
-224
-040
122
0.2
-03
Figure 7: Effect of Laec on Generalization. As the maximum frequency of the decoder increases, G2Sphere starts to overfit to the sparse data samples in the Drag domain. As result, the G2Sphere model with lower output frequency generalizes better to new data.
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(a) N-Paths
(b) IBC
(c) G2S
(d) Diff
Figure 8: Multimodal Behavior. From the start state (blue) there are two paths (top row) and four
paths (bottom row) to the goal state (green). When using high enough angular frequency (L = 2. = 4), G2Sphere learns paths and commits to path at the start of each rollout. Diffusion Policy exhibits similar behavior but shows some bias towards certain paths whereas IBC fails learn due to an equal distribution of expert path data.
(a) N=2, T.=2
(b) N=4,
Figure 9: Energy Landscape. We visualize the energy landscapes generated with the G2Sphere on the N-Paths domain with N = and N = 4. respectively.
present in the task. We illustrate this behavior using the simple N-Paths task where the agent must
navigate to the goal (green) using one of the N available paths. Fig. 9 shows that by setting the maximum frequency the multimodality present in 2-Paths and 4-Paths, the energy landscape generated by G2Sphere is able to model each the multimodal trajectories. Additionally, Fig. B shows example trajectories generated by G2Sphere, E-IBC, and Diffusion. We note that G2Sphere is the best at matching the multimodality present in the task and is equally likely to take any of the N-paths, while diffusion shows bias towards certain paths and IBC fails commit any of the paths.
(a) Asym Mesh
(b) Frusta Mesh
(c) Pods Mesh
Figure 10: Object Meshes. Example object meshes used in the Asym and Frusta meshes are used in the radar prediction domain and the pods meshes are used the aerodynamics prediction domain.
experiments. The
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175
Fristum
Froce StAgE
84
Rundom Surtace Scartt
Viewing angle (deg)
-20
9.5
PUX
Ampstodte
-00
10
23
-2.0 -0.5
-3.0
-1.5
Assil Location ledl
Rangee lre)
Range (m)
(e)
(b)
Figure 11: Radar Simulator. (a) Cross section of random 3D object mesh. (b) The range profile for 20 viewing angle. (c) The corresponding static radar pattern.
Dataset generation
J.1 First-Principle RF model
Due to the scarcity of available real-world and simulated radar data for training radar models, we simulate own benchmark dataset. To generate ground truth data, approximation method [4, which provides a linear approximation the more general and highly non- linear scattering formulation for electromagnetic waves. A simple operator that describes physical optics response across the illuminated section of an object for perfectly reflecting material as,
the physical optics
Fxx, k)
ik
(2k( dy,
2T
128
where the incident wave number is k = 2 À is the wavelength, and the observation unit vector is,
x = (sin cos Φ sin sin Φ, cos f),
for 0 [0, T] and [0, 2n.. We note that the approximation is valid only in high frequency regions such that k << 2T/ / D. where D is the length of the longest side of the object.
As particularly interested in far-field sensing, the physical optics approximation is useful as a fairly accurate and flexible simulation tool for training data generation. The simulation input is a parameterized mesh object (an example cross section shown in Fig. (Ta)). The simulation calculates the the physical optics response for given observation one-of-sight and frequency and the total radar response of the object is equal to the sum of the individual triangle responses that are visible the radar. Simulations for this work required generating the response across a linear set of frequencies to emulate Linear Frequency Modulated (LFM) waveform, where the center frequency is 3e9Hz and bandwidth is 4e8Hz using circular polarized waves with orientation RL.
The Radar Cross Sec. (RCS) for each triangle is calculated using legacy software 100. The simulation produces a radar observation, r E IR No (Fig. b)) for a given viewing angle. The observation is the normalized magnitude of the range-profile as described in Sec. III of Chance et al. [12]. All the normalized range profiles are then stacked across varying radar viewing angles to generate what is referenced radar static pattern (Fig. c)). Note that fixing the radar line-of-siglt and rotating the object would generate the same radar static response.
We utilize two different mesh datasets for our experiments. in this dataset is defined by series of stacked frusta objects, in which adjacent frusta share the same radius to ensure a continuous object. To generate diverse set of objects, frusta components and the radial parameters for each frusta. To introduce additional object and radar response diversity, we also vary the ends of the frusta object to either be flat planes, half spheres, or one of each.
The first is the Frusta dataset. Each object
vary both the number of
While the Frusta dataset provides objects with complex forms, it is also rotational symmetric along roll axis. To address this issue, we also generate the Asym. This dataset consists of three different base shapes; cubes, cylinders, and spheres. Each object is randomly scaled across it's independent dimensions; to ensure diversity, no random scaling repeated within object class. After the scaling, the mesh is duplicated and subdivided to decrease the relative size of mesh faces. group of adjacent
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Figure 12: Aerodynamics Simulator. Ground truth drag coefficients generated by the numerical CDF simulator.
faces is selected at random from the overall set of mesh faces on the subdivided copy; these faces are then extruded by a random amount in the average normal direction of the selection. This extruded protrusion is then added to the original mesh using the union operation, resulting in an object without any rotational symmetries.
Using these methods, we generate a dataset of objects for the Frusta dataset, and 30,000 objects for the Asym dataset, evenly distributed across the underlying classes. Each dataset is divided 90/10 between train/validation and test sets. Since the meshes resulting from the Frusta generation method large relative the capacity of neural network models (maximum size of 4500 faces), frusta meshes are decimated to 50% of their face count using quadratic decimation for training and testing.
J.2 Aerodynamic simulation and data
The aerodynamic dataset consists of "pod-like" shapes defined parametrically by varying shape parameters such length, diameter, bluntness, and cross-section asymmetry using Latin Hypercuhe Sampling. Flight conditions including altitude, Mach number and Reynolds number were also selected using Latin Hypercube Sampling, where altitude ranges from 0 50 kilofeet, Mach number ranges from 0.05 to 0.5, and Reynolds number ranges from 10th to 5 x 10 For each shape permutation high-quality computational grid is algorithmically generated. The minimum cell size is defined to provide over 100 cells across the length of each shape, and 50-100 100 volume cells to resolve the where y 1 is the wall coordinate commonly
boundary layer down to the viscous sublayer, i.e. y* used in defining the law of the wall. Generation of drag force data is based on Computational Fluid Dynamics (CFD) simulator for which we solve the Reynolds-Averagel Navier-Stokes (RANS) mass, momentum and energy equations through the NASA FUN3D simulation framework. Turbulence modeled using the Spalart-Allme Allmaras model with a freestream turbulence intensity of percent. The van Albada flux limiter with the low-diffusion flux splitting "LDFSS" flux construction method. Simulations are run until a relative residual tolerance of 10 met for the mass, momentum, energy, and turbulence governing equations.
The Pods dataset contains 10, 000 samples of (mesh, flight conditions, drag) tuples, each with single drag coefficient for specific angle f, & E [-20, 20), which defines the orientation of the incoming flow relative the centerline of the Pod shape.
K Policy learning environments
PushT We generate 100 expert demonstrations for both the fixed and randomized PushT tasks. All demonstrations were generated by a single human operator familiar with the task. Each episode terminates either after the task is completed or the maximum number of steps (500) is reached. We use the PushT implementation found in Chi al. [3] which can be found here:
MMT License].
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PushT.
Fly FlyTo Target..
(c) FlyThhoughGate.
Figure 13: Policy Learning Domains. The tasks from our policy learning benchmarks. The fixed- goal variant PushT is shown in (a). The target position to fly to is shown as red ball in (b) and the gate to fly through is shown in red in (c).
PyBullet Drones We generate 50 expert demonstrations for both the FlyToTarget and FlyThrough- Gate tasks. All demonstrations were generated by an expert policy which has access the underlying simulation state. We add small amount of uniformly sampled noise, e <~ UO, 1), to each action from the expert policy, Each episode terminates either after the task completed or the maximum number of steps (300) is reached. We use the PyBullet Drones implementation found in Panerati et al. github. ggmm- wttepppybuulltrrooeeeee
48 which can be found here: License].
[MIT
L Example radar response outputs
In Fig. I4 we highlight representative examples generated by the ground truth simulator, transformer, equiformer, and G2Sphere on the Frusta dataset. We note that although G2Sphere still produces the best predictions, the difference between G2Sphere and the transformer baseline is much less pronounced on this dataset. This aligns with the result in Table T which shows that the performance of the transformer baseline and G2Sphere to be similar.
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Figure 14: Frusta Radar Predictions. Radar response predictions on the Frusta dataset. Due to the roll symmetry in the Frusta dataset plot the spherical outputs as static patterns. G2Sphere is best able to capture both the overall structure of the response and also the speculars (bright horizontal
lines).
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Reviews:
[{"review_id": "Ejo0dXoc51", "reviewer": "Reviewer_gYBD", "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.", "questions": "There are several hyperparameters for implementing the proposed method, what are the best values for a task given the dataset?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "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.\n\nWeaknesses:  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.", "was_revised": true, "final_justification": "The authors addressed all my concerns and problems so I have increased my score accordingly.", "rebuttal": "Thank you for your review. We are working on a revision to incorporate your feedback, but we address your comments and questions here as well. \n\n---\n\n*“The proposed method consists of certain existing algorithms and is short of a systematic theory.”*\n\nOur primary goal in this work is not to create a systematic theory but to address a lack of methods for mapping 3D geometric inputs (e.g. point clouds and meshes) to spherical signals, despite the fact that this type of mapping is found across many different domains. Our proposed method is general enough to demonstrate strong performance across a range of diverse domains.\n\n---\n\n*“There are some English presentation and grammar errors in the paper.”*\n\nThank you for pointing this out. We will carefully revise the paper to improve clarity and grammatical correctness.\n\n---\n\n*“There are several hyperparameters for implementing the proposed method, what are the best values for a task given the dataset?”*\n\nG2Sphere was trained using the Adam optimizer with a learning rate of 1e-4 and a decay rate of 0.95. In general, we use the largest batch size available to us given the compute required for the domain. For example, when using large meshes, such as in the radar domain, we use a smaller batch size of 4. In the policy learning domains, where the geometric input is smaller, we can use a larger batch size of 64. Additional hyperparameters, including the number of layers in both the encoder and decoder, the layer-wise feature dimensions, and the choice of maximum SH frequency , $L_{max}$ are detailed in Appendix G. For completeness, the appendix also provides the hyperparameter settings used for all baseline methods."}, {"review_id": "2HcIlNjiS4", "reviewer": "Reviewer_anCc", "summary": "This 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.", "questions": "Question:\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?", "limitations": "N/A.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "This architecture offers a framework that provides new insights into designing autoencoders for mesh data. Experiments against point-cloud networks (e.g., PointNet++) and equivariant models (e.g., Equiformer) demonstrate its effectiveness. Overall, the paper presents a good step toward autoencoding 3D mesh data.", "rebuttal": "Thank you for your review. We are working on a revision to incorporate your feedback, but we address your comments and questions here as well. \n\n---\n\n*“It might be valuable to provide a discussion on the efficiency of the method. In particular, it would be helpful to clarify the maximum frequency (L_{max}) applied in this work. As 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\nWe agree that the computational costs of SO(3)-equivariant networks with high $L_{max}$ is an important consideration. This is especially true of equivariant GNNs where maintaining high $L_{max}$ across a large number of nodes quickly becomes computationally prohibitive. In contrast, this is less of an issue with spherical models as we only have a single node, e.g. the sphere itself. Therefore, in our GNN encoder we use smaller $L_{max}$, for example in our mesh-to-sphere experiments we use $L_{max}=5$ for all layers, before raising the $L_{max}$ in the spherical decoder layers. By combining this lower $L_{max}$ encoder architecture and the frequency upsampling technique (Sec 4.2) in our decoder we avoid incurring the large computational costs that make high $L$ GNNs intractable. \n\nAs a rough rule of thumb, we found frequency upsampling to be necessary when $L_{\\max} > 10$. This typically corresponds to domains requiring high-fidelity predictions, such as mesh-to-radar, where we push $L_{\\max}$ to the upper limits of what our compute allows. We include ablations showing the relationship between $L_{max}$ and output fidelity in Appendix H. In lower-complexity domains such as policy learning, we instead use small fixed values ($L_{max} \\leq 5$) to encourage generalization and avoid overfitting.\n\n---\n\n*“How does your approach compare to standard, strong point cloud or mesh baselines, e.g. PointNet++ and RepSurf, that are not SO(3)-equivariant?”*\n\nThanks for bringing these stronger mesh and point cloud baseline encoders to our attention. While we were unable to run these baselines within the current rebuttal timeline, our paper does include competitive non-equivariant baselines (e.g. transformers in the supervised learning settings and diffusion models in the policy learning domain) that serve as points of comparison.These models serve as reasonable baselines for our domains and help highlight the benefits of incorporating equivariance and frequency-based modeling. If additional results become available before the end of the discussion period, we will report and include them.\n\n---\n\n*“...the paper could emphasize the generated dataset which is also part of the contribution of this paper. And it may be helpful to emphasize the importance of these generated datasets and how they can contribute to real-world applications.”*\n\nWe agree that the generated datasets represent a valuable contribution of this work and are planning on releasing them alongside our code. The domains we consider in this work are generally lacking in publicly available datasets and we hope that our data can lay the foundations of a standardized and reusable testbed for future methods targeting spherical signal learning from 3D data. Additionally, the scale of these datasets enables the broader application of ML models in this space which has historically lagged behind other more mainstream applications such as text, image, and audio.\n\n---\n\n*“How did you handle the distance in the 3D meshes when performing the Fourier transformation? I understand that applying the Fourier transformation on the S2 space is natural. However, when working with mesh data, defining the origin of the system requires transforming the mesh points by considering their distances to the origin. In this case, I believe applying spherical harmonics alone is not sufficient. How did you address the handling of these distances?”*\n\nYou’re absolutely correct that the SH alone are not enough to encode this type of 3D geometric data as they only capture the angular ($S^2$) information. To incorporate the radial (distance) information, we use a radial network which transforms pairwise distances into feature embeddings. This design parallels approaches like Equiformer [1], which similarly uses Gaussian-based radial basis functions transformed via learnable MLPs to encode edge distance information in their equivariant attention layers.\n\n[1] Liao, Yi-Lun, and Tess Smidt. \"Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs.\" The Eleventh International Conference on Learning Representations.\n\n---\n\n*“In Figure 2, it is said it will use \"grid of pre-computed harmonic basis functions\". Does this mean that each channel will be corresponding to a certain fixed distance?”*\n\nThis question relates to the decoder stage of our pipeline, where we transform features from Fourier space back to real space. The phrase “grid of precomputed harmonic basis functions” refers to our use of a fixed sampling grid on the sphere, over which we evaluate the inverse spherical Fourier transform to recover real-space signals. These basis functions are precomputed purely for efficiency, e.g. to avoid recomputing SH values during every forward pass, and are not related to encoding different radial distances. To clarify: no, the channels do not correspond to fixed distances, they represent frequency components on the sphere, not spatial or radial positions.\n\nFor example, suppose $L_{\\max} = 10$. Then there are 45 total frequency components (since we have $\\sum_{l=0}^{10} (2l + 1) = 121$ modes, but due to conjugate symmetry for real-valued signals, we only need to store the 45 complex coefficients). If we are modeling a function $f: S^2 \\rightarrow \\mathbb{R}^{50}$, the frequency-domain representation would be a tensor of shape [B, 45, 50], where B is the batch size. After applying the inverse Fourier transform on a set of, say, 10,000 sampled points on the sphere, the real-space signal is recovered as a tensor of shape [B, 10,000, 50], representing the function values at each of those 10K spherical coordinates."}, {"review_id": "IfV3wgvx54", "reviewer": "Reviewer_pmav", "summary": "This 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.", "questions": "Please see the Weaknesses section above.", "limitations": "yes", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 4, "originality": 1, "strengths_and_weaknesses": "Strengths\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?", "was_revised": true, "final_justification": "NeurIPS’25 Post-Rebuttal Review\n\nI apologize for not being able to actively participate in the reviewer discussion period. However, to make up for this and ensure thorough feedback, I have put more effort into preparing this post-rebuttal review.\n\nPlease see my comments below:\n\nW1: Thank you for explaining the comparison with [A]. I now clearly understand the differences from G2S. Please include this comparison in the final manuscript.\n\nW2: I understand that SO(3)-equivariance is preserved; however, I am concerned about potential precision loss due to the truncation of the Fourier transform. It would be helpful if you could quantitatively show whether the equivariance error reported in [2] also applies to this work.\n\nW3: Demonstrating that spherical representations and frequency-domain approaches are effective in improving performance is reasonable. However, an ablation study on these two factors would further strengthen the paper’s evidence.\n\nW4: I have confirmed the effect of the maximum frequency level  L_{\\text{max}}  in Table 4. Thank you.\n\nW5: The results shown in Figures 2 and 4 provide some evidence of sample efficiency, but it would be better to show results across a wider range of sample sizes. As mentioned in the rebuttal, the authors should add experiments with various settings.\n\nW6: I expect the authors to follow through on their promise to carry out the revisions clearly.\n\nW7: I expect the authors to follow through on their promise to perform the comparison with HEALPix.\n\nIn conclusion, I am releasing my concerns for W1, W4, and W6, and partially releasing my concerns for W2, W3, W5, and W7.\n\nI appreciate the authors’ rebuttal and will raise my score by one level (4: Borderline accept).", "rebuttal": "Thank you for your review. We are working on a revision to incorporate your feedback, but we address your comments and questions here as well. \n\n---\n\n*“ A closely related architectural baseline is missing: [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.”*\n\nThank you for bringing this work to our attention, we will add it to our related work section and explain its relationship to our method. Indeed, [A] shares key architectural similarities with G2S, including the use of spherical representations and SO(3)-equivariant convolutions in the frequency domain, which support continuity, equivariance, and data efficiency. However, there are two important distinctions.  First, [A] begins from 2D image inputs, using a ResNet backbone followed by projection onto the sphere, whereas G2S is designed for 3D geometric inputs and operates directly in the spherical frequency domain from the outset. Second, [A] targets 3D pose estimation and thus operates effectively at a fixed low bandwidth ($L_{max} = 5$), while G2S must adapt $L_{max}$ to the demands of each domain, ranging from 5 in our policy learning tasks to 40 in radar prediction. Therefore, we employ various techniques such as a frequency upsampling strategy to scale output resolution efficiently.\n\n---\n\n*“The use of point-wise MLPs in TSNL raises questions about whether end-to-end SO(3)-equivariance is preserved.”*\n\nThe TSNL is equivariant up to the sampling error introduced by the inverse Fourier transform which we use to convert from frequency domain features to real-space signals on a fixed spherical grid. Because we apply this MLP pointwise to each of these points it is SO(3)-equivariant by construction. While it’s true that performing a Fourier transform with finite samples can introduce minor artifacts, these result in negligible equivariance error in practice. We note that this approach is common in many equivariant architectures such as the Spherical Fourier Neural Operator [1], Spherical CNN [2], and Gauge Equivariant Mesh CNNs [3]. Specifically in [2] they measure this equivariance error and find that it is $< 2e-6$ (see Sec 5.1 of [2]).\n\n[1] Bonev, Boris, et al. \"Spherical fourier neural operators: Learning stable dynamics on the sphere.\" International conference on machine learning. PMLR, 2023.\n\n[2] Cohen, Taco S., et al. \"Spherical CNNs.\" International Conference on Learning Representations. 2018.\n\n[3] De Haan, Pim, et al. \"Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs.\" International Conference on Learning Representations, 2021.\n\n---\n\n*“While the use of SO(3)-equivariant models is well-motivated, the rationale for spherical representation and frequency-domain approach remains vague.”*\n\nWe appreciate this comment and agree that the rationale for using spherical representations and the frequency domain deserves further clarification. Many of the signals we consider (e.g., radar and drag) are global in nature but direction-dependent. Representing them as functions on the sphere allows us to model this directional dependence explicitly, without tying the representation to the surface geometry of the 3D object. Figure 5c illustrates how G2Sphere models these directional signals by learning Fourier coefficients enabling continuous signal reconstruction via spherical harmonics. This approach is more efficient than spatial methods that either predict dense grids (5a) or require repeated coordinate queries (5b), as the cost of prediction is decoupled from the number of output points.\nModeling in the frequency domain also provides a strong structural prior that helps avoid overfitting to noisy or irregular signals. We observe this in the drag prediction experiments (Fig. 4), where spatial baselines tend to overfit, while G2S is biased toward smoother solutions due to its harmonic representation. Additionally, it provides a clean and scalable way to maintain SO(3) equivariance, resulting in an efficient end-to-end equivariant architecture. To demonstrate this, we do include competitive spatial-domain baselines such as transformers in the mesh-to-sphere experiments, and IBC and diffusion models in the policy learning tasks.\n\n---\n\n*“Although SH frequency is increased to 40, the paper does not isolate how this impacts precision.”*\n\nWe do include a study in Appendix H on the effect of frequency on precision. As shown in Table 4, increasing $L_{max}$ leads to a clear reduction in error (0.26 MSE reduction from $L=5$ to $L=40$), while Figure 6 provides qualitative evidence of improved predictions, e.g. critical specular radar effects are completely missing at lower $L$ but become observable as $L$ increases. However, as noted in Figure 7, higher frequencies can lead to overfitting in low-data regimes. Additionally, we observe that compute and memory usage increase substantially with larger $L_{max}$. We agree that analyzing the trade-off between SH frequency and performance is critical to understanding the role of $L_{max}$​ and we will move this study to the main body. \n\n---\n\n*“The paper mentions sample efficiency in the introduction, but does not provide supporting experiments.”*\n\nThe mention of sample efficiency in the introduction is supported by the policy learning experiments (Table 2) and the generalization tests on the drag domain (Fig 4). In all of our policy learning experiments we are in relatively low data regime settings with only 100 demos for each dataset. Therefore, when we see both the equivariant and non-equivariant versions of G2S outperforming the baselines, the primary reason for this is their increased sample efficiency. \nFor example, in the push-T task, the non-equivariant G2S achieves a success rate of 0.97, the equivariant version reaches 1.0, while the diffusion model scores 0.95. The differences are even more pronounced in the random push-T variant, where non-equivariant G2S achieves 0.74, equivariant G2S reaches 0.92, and diffusion drops to 0.71. These gains in sample efficiency are primarily due to the combination of equivariance, which allows the model to better generalize under rotations, and the frequency modeling, which provides regularization, stability, and avoids overfitting. This overfitting in particular can be seen in the generalization on the drag domain (Fig 4). That being said, we acknowledge that additional ablations on dataset sizes would further strengthen this claim, and we plan to include this to provide a more comprehensive analysis of sample efficiency.\n\n---\n\n*“ 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.“*\n\nWe thank the reviewer for the valuable feedback. While Section 5 contains some overlap with Appendix G, this redundancy was intentional to ensure the reproducibility of our experiments. Section 5 provides a high-level overview of key model parameters and important details, while Appendix G includes a more comprehensive set of specifications and additional information necessary for fully replicating the experiments. This design ensures that readers have both a concise summary in the main text and an exhaustive reference in the appendix for reproducibility. However, we acknowledge that there might have been too much overlap and will take care to streamline the content to reduce this.\nAdditionally we will do the following to address the other points brought up:\n\n- Separation of reused and new modules: We will revise Section 4 to more clearly differentiate between components adopted from prior work and those introduced as novel contributions in this paper.\n- Linking motivation and experiments: We will strengthen the connection between the motivation and the experiments by revising both the introduction and experimental sections to clearly articulate the purpose of each experiment and how it addresses our core research questions.\n- Cohesive flow: We will revise the transition from the methodology section to the experiments to ensure a more cohesive flow. In particular, we will add forward references from Section 4 to the relevant experiments in Section 5. For example, highlighting how the frequency upsampling technique introduced in Section 4.2 directly contributes to the improved high-resolution outputs shown in Figure 3. This will help clarify how specific design choices lead to observed performance gains.\n\n---\n\n*“ The paper critiques HEALPix spatial grids (L29–30) for overfitting to sample locations.“*\n\nWhile HEALPix is cited as an example of a possible grid for spherical output signals, we do not explicitly compare to it in this work. However, we evaluate several representative gridded methods that similarly predict or rely on regularly sampled points over the sphere. For example, our transformer baseline in the radar domain predicts a fixed grid of uniformly spaced outputs. In the drag prediction and policy learning tasks, implicit models such as IBC and the implicit transformer are conditioned on coordinate queries sampled from a fixed grid, which the model must learn to reconstruct. \n\nAs we stated in the introduction using fixed spatial grids, whether for direct prediction or coordinate conditioning, commonly leads to overfitting to the sample locations. This behavior is clearly visible in Figure 4 (drag prediction), where the implicit transformer baseline overfits and in Figure 8 (policy learning), where IBC shows severe overfitting. We will revise the paper to clarify that this overfitting tendency is characteristic of the broader class of gridded methods, of which HEALPix, our transformer baseline, and coordinate-conditioned implicit models like IBC are all representative."}, {"review_id": "7fhyTvimrG", "reviewer": "Reviewer_DYSg", "summary": "The 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.", "questions": "It'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?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The 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.", "was_revised": true, "final_justification": "The authors addressed all my concerns so I increased my score accordingly.", "rebuttal": "Thank you for your review. We are working on a revision to incorporate your feedback, but we address your comments and questions here as well. \n\n---\n\n*“It's not very clear to me how the 3D shapes are mapped to the radar signal in the Radar dataset. Similarly, why is the output signal defined over a sphere rather than over the mesh surface in the Drag Prediction dataset?”*\n\nFor the radar dataset, the 3D shape is processed as a point cloud and encoded using a GNN.  This is then decoded to a spherical radar signal. \n\nTo clarify, for all three applications, we are not projecting the mesh onto the sphere. Rather, we are modeling the directional dependency of each output (radar signal, drag prediction, control policy) as a spherical (angle-dependent) function. These are inherently global functions defined over the sphere of directions, not over the local mesh surface.\nIn the radar setting, the radar signature is dependent on the sensor viewing angle on the object, in the drag prediction task, the drag value depends on the direction of the airflow impacting the object. In the control policy learning task, the policy value depends on the agent’s motion angle.\nIf our explanation is still unclear, we would be happy to explain further.\n\n---\n\n*“Why using global averge pooling? … this choice still feels like it might discard a lot of important spatial information. Isn't there some better choice?”*\n\nWe use global pooling to aggregate spatial features across the mesh, enabling the model to produce outputs that depend on global geometry and direction. This is related to our answer for the previous question, the spherical output signal is actually a global signal, and thus pooling across the mesh is appropriate. While global average pooling is effective in this context, we also experimented with other pooling strategies, such as top-k pooling, and found similar results. That said, we agree that more expressive alternatives like attention-based pooling could better preserve spatial structure, and we are interested in exploring such methods in future work.\n\n---\n\n*“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 …”*\n\nWhile training on lower-resolution data does limit the signal to lower frequency components, our model's ability to perform zero-shot super-resolution stems from more than just interpolation. Because the spatial sample locations are randomized across the different meshes (due to the different mesh geometries and orientations), achieving good performance requires the model to learn the correlation between low and high frequency components across the dataset. This is only possible when there is consistent structure in the signals that the model can exploit, such as those arising from similar geometry or alignment across samples.\n\nWe additionally note that this has been seen in prior works like Fourier Neural Operators[1], where models trained on coarse data, PDEs in their case, can generalize to finer resolutions by learning global patterns. We also agree that this could have been made clearer and will revise the text to emphasize these details in the paper.\n\n[1] Z. Li et al., Fourier Neural Operator for Parametric PDEs, ICLR 2021.\n\n---\n\n*“Fig 4 is a bit hard to read.”*\n\nThank you for the suggestion. We will improve the readability of Fig. 4 by increasing the size in the final version. We will also revise the caption to provide more context and better highlight the key takeaways from the figure.\n\n---\n\n*“Tab 2: could you report the standard deviation over the 50 initializations?”*\n\nInstead of reporting the standard deviation across the 50 initializations, we follow the common practice in policy learning literature of reporting the mean performance of the 50 initializations over the final 10 evaluation rollouts. This reflects the stability of the final policy and aligns with established conventions in the field. We acknowledge that additional variance metrics (e.g.,standard deviation) could provide further insight and will consider including them in future work.\n\n---\n\n*“Tab. 3: why is the NE-G2S much faster than G2S?”*\n\nThe difference in runtime between G2S and NE-G2S in Table 3 is primarily due to the increased computational overhead of equivariant architectures. Specifically, SO(3)-equivariant models like G2S rely on spectral convolutions and spherical harmonics, which are inherently more expensive than the standard operations used in non-equivariant models such as NE-G2S. That being said we find that the increased performance of equivariant G2S to out-way the minute (6*ms*) increase in inference time.\n\n---\n\n*“TSNL: … 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.”*\n\nThank you for pointing this out. You are correct in that applying a Fourier Transform with finite samples can introduce minor artifacts, which in turn lead to a small equivariance error in the TSNL layer. While this is typically negligible in practice, we agree that the claim in lines 727 and 730 should be updated accordingly. We will revise the text to clarify this point and will additionally quantify the equivariance error introduced by TSNL in our revised version.\n\nAdditionally, we note that this approach is common in many equivariant architectures such as the Spherical Fourier Neural Operator [1], Spherical CNN [2], and Gauge Equivariant Mesh CNNs [3]. Specifically in [2] they measure this equivariance error and find that it is $< 2e-6$ (see Sec 5.1 of [2]).\n\n[1] Bonev, Boris, et al. \"Spherical fourier neural operators: Learning stable dynamics on the sphere.\" International conference on machine learning. PMLR, 2023.\n\n[2] Cohen, Taco S., et al. \"Spherical CNNs.\" International Conference on Learning Representations. 2018.\n\n[3] De Haan, Pim, et al. \"Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs.\" International Conference on Learning Representations, 2021."}]

Author rebuttal:

Meta-review: This paper introduces G2Sphere, a method that connects equivariant GNNs and spherical CNNs for physical domains. In the initial comments, reviewers raised concerns about missing baselines, clarity of the theory, and limited experiments.
In their rebuttal, the authors added stronger baselines (such as PointNet++), provided clearer explanations of the theory, and included more experiments and ablation studies. Reviewers agreed that these additions resolved their main concerns, and some raised their scores.
Overall, the idea is well-motivated, the paper is now clear and thorough, and the results are convincing. I recommend acceptance.
Decision: Accept (poster)