DualEqui: A Dual-Space Hierarchical Equivariant Network for Large Biomolecules
Authors: Junjie Xu
Jiahao Zhang
Mangal Prakash
Xiang Zhang
Suhang Wang
Venue: NeurIPS 2025 poster ()
Keywords: Geometric deep learning
Graph Neural Networks
GNNs
RNA property prediction
Primary area: machine_learning_for_sciences

Abstract: Geometric graph neural networks (GNNs) that respect E(3) symmetries have achieved strong performance on small molecule modeling, but they face scalability and expressiveness challenges when applied to large biomolecules such as RNA and proteins. These systems require models that can simultaneously capture fine-grained atomic interactions, long-range dependencies across spatially distant components, and biologically relevant hierarchical structure—such as atoms forming residues, which in turn form higher-order domains. Existing geometric GNNs, which typically operate exclusively in either Euclidean or Spherical Harmonics space, are limited in their ability to capture both the fine-scale atomic details and the long-range, symmetry-aware dependencies required for modeling the multi-scale structure of large biomolecules. We introduce DualEquiNet, a Dual-Space Hierarchical Equivariant Network that constructs complementary representations in both Euclidean and Spherical Harmonics spaces to capture local geometry and global symmetry-aware features. DualEquiNet employs bidirectional cross-space message passing and a novel Cross-Space Interaction Pooling mechanism to hierarchically aggregate atomic features into biologically meaningful units, such as residues, enabling efficient and expressive multi-scale modeling for large biomolecular systems. DualEquiNet achieves state-of-the-art performance on multiple existing benchmarks for RNA property prediction and protein modeling, and outperforms prior methods on two newly introduced 3D structural benchmarks demonstrating its broad effectiveness across a range of large biomolecule modeling tasks.

DualEquiNet: A Dual-Space Hierarchical Equivariant Network for Large Biomolecules
Junjie Xu', Jiahao Zhang', Mangal Prakash2, Xiang Zhang', Suhang Wang'
The Pennsylvania State University
2 Independent Researcher
(jujjiexu, jiahao zhang, xzz89, szw494) edu, mangalcrjjggailll
Abstract
Geometric graph neural networks (GNNs) that respect E(3) symmetries have achieved strong performance on small molecule modeling, but they face scala- bility and expressiveness challenges when applied large biomolecules such as RNA and proteins. These systems require models that can simultaneously capture fine-graines atomic interactions, long-range dependencies across spatially distant components, and biologically relevant hierarchical structure-sut ing residues, which in turn form higher-order domains. Existing geometric GNNs, which typically operate exclusively in either Euclidean or Spherical Harmonics space, are limited in their ability to capture both the fine-scale atomic details and the long-range, symmetry-ave -ware dependencies required for modeling the multi-scale structure of large biomolecules. We introduce DualEquiNet, a Dual-Space Hierar- chical Equivariant Network that constructs complementary representations in both Euclidean and Spherical Harmonics spaces to capture local geometry and global symmetry-ave aware features. DualEquiNet employs bidirectional cross-space message passing and a novel Cross-Space Interaction Pooling mechanism to hierarchically aggregate atomic features into biologically meaningful units, such residues, enabling efficient and expressive multi-scale modeling for large biomolecular sys- tems. DualEquiNet achieves state-of-the-art performance on multiple existing benchmarks for RNA property prediction and protein modeling, and outperforms prior methods its broad effectiveness across a range of large biomolecule modeling tasks. The source code is available lhttts:///// http://ggtttb.tt http://gggtuuuu.oo/un... wowmjjnjjee/eeee ceee-xudduee sul/dualqquinet
atoms form-
newly introduced 3D structural benchmarks demonstrating
Introduction
Large biomolecules such RNA and proteins play central roles in biological systems, enabling key functions ranging from enzymatic catalysis to molecular recognition and gene regulation [15] TT
61
38 77. Recent studies have explored diverse strategies improve protein and RNA modeling,
from geometric deep learning to physics and
neural architectures 59 (55) These molecules exist the atomistie scale, where complex
45 82 56
87
83
72
geometries, hierarchical organization (with atoms grouped into residues and domains), and long-range interactions jointly determine function. Crucially, molecular models must respect the underlying physical symmetries: predicted properties should transform consistently under global rotations and translations, remaining invariant for scalar quantities (such energy) equivariant for directional quantities (such as dipole moments). Designing computational models that can faithfully capture these geometric and symmetry-un aware aspects is essential for accurate biomolecular modeling [36] 71 (5.
Despite recent advances in geometric deep learning, current geometric graph neural networks (GNNs) face two critical limitations when applied large biomolecular systems. First, most existing
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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geometric GNNs model molecules purely at the atomic level, representing atoms as nodes and bonds
or spatial proximities as edges 80 62 23 35]. While effective for small molecules, this atomistic- only approach fails to capture the inherent hierarchical organization in large biomolecules, where atoms are naturally grouped into higher-order units such as nucleotides (in RNA) or amino acids (in proteins). Ignoring this hierarchical structure leads to the loss of essential contextual and relational information, ultimately constraining predictive performance. Second, large biomolecules exhibit rich long-range dependencies: distant residues nucleotides can interact and cooperatively determine global properties [22]. Most existing geometric GNNs focus on local neighborhoods in Euclidean space 66
16 limiting their ability to capture nonlocal interactions effectively. While some methods incorporate higher-order geometric features, such as using higher-degree spherical harmonics, they often suffer from significant computational overhead [69 [ 27], making them impractical for large-scale molecular systems.
63
12
To address the challenges of hierarchical structure and long-range dependencies in large biomolecules, we propose DualEquiNet, a novel Dual-Space Hierarchical Equivariant Network. Unlike existing methods which either use nonlocal attention using spherical harmonics [26] in Euclidean-space space neighborhoods or using high-degree steerable representations within a single geometric space [[] 6), DualEquiNet jointly constructs complementary representations in both Euclidean (EU) and spherical harmonics (SH) spaces: the EU space captures local geometric interactions crucial for atomic- level detail, while the SH space encodes rotation- equivarrinu features well-suited for long- long-range structural dependencies. Through bidircctional cross-space message passing, DualEquiNet enables structured information exchange between EU and SH spaces, integrating local and global geometrie contexts. Moreover, we introduce a novel Cross-Space Interaction Pooling (CSIP) mechanism that hierarchically aggregates atomic-level features into coarser scales (such as residue-level), expanding the receptive field without requiring deep stacking of atomic layers. By explicitly facilitating interaction between EU and SH representations during pooling, CSIP ensures efficient and expressive multi-scale modeling. Together, these innovations yield a scalable, symmetry-awen framework tailored the unique architectural and functional demands of large biomolecular systems.
We evaluate DualEquiNet across established RNA and protein property prediction datasets, demon- strating consistent improvements over existing geometric baselines (with average performance gains of 8.4%-33.5% across tasks). To further probe the model's 's ability to capture nuanced 3D structural features, developed two new benchmark datasets for Solvent-Accessible Surface Area (SASA) Prediction and Torsion Angle Prediction, which are crucial for applications in drug discovery and protein engineering (66 68). DualEquiNet achieves improvements of 3. 3.1%-28.88%. and 2.5%, re- spectively, over the strongest prior methods on these benchmarks as well. Code and datasets will be released publicly upon paper acceptance.
To sum up, our main contributions are: (i) We introduce DualEquiNet, a novel dual-space hierarchi- cal equivariant network that explicitly constructs complementary Euclidean and spherical harmonics (SH) representations, enabling the simultancous capture of local atomic interactions and long-range structural dependencies; (ii) We propose bidirectional cross-space message passing and novel Cross-Spar Interaction Pooling (CSIP) mechanism, which together allow efficient and expressive hierarchical aggregation from atoms to residues, tailored large biomolecular structures; and (iii) We introduce two new 3D structural benchmark datasets (SASA and Torsion Angle prediction) and evaluate DualEquiNet across these and other diverse established biomolecular tasks, including RNA and protein property prediction, consistently achieving state-of-the-a results and outperforming prior baselines.
2 Related Works
Invariant GNNs. Cotation-invarian GNNs leverage geometric features such as interatomic dis- tances and angles to produce representations invariant under rigid body motions. SchNet [63] uses continuous- filter convolutions over pairwise distances. DimeNet (30) and GemNet (28) further incor- porate angular information, improving directional sensitivity. SphereNet (50) expands the invariant feature set to include distances, angles, and torsions. While effective, such models discard essential orientation-dependent information, motivating the need for equivariant approaches.
Scalarization Equivariant GNNs. Scalarization-basee based equivariant GNNs aim to preserve E(3) cquivariance by decoupling scalar and vector representations, EGNN [62] uses pairwise distances

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to compute scalar messages for equivariant updates of coordinates. PaiNN 64 and GVPGNN [39
similarly apply scalar gating and vector decoupling maintain equivariance efficiently. However. their reliance on scalar guidance can limit expressivity, particularly for modeling complex rotational dependencies 40)
Higher-order Equivariant GNNs. Tensor-pres puassr--roduutbbasedd methods achieve full SE(3) or SO(3) cquivariance via Clebsch-Gordan (CG) tensor algebra. TFN (69 and SE(3)-Transffrmer [27] construct equivariant convolutions over irreducible representations, enabling expressive geometric reasoning. MACE utilizes a complete basis from the Atomic Cluster Expansion for maximal expressivity. However, these methods often suffer from high computational cost due to CG operations scaling as (L))) with the degree L. Recent methods aim to balance expressivity and efficiency by avoiding explicit CG products. HEGNN [16] and GotenNet [6] incorporate high-degree spherical harmonics but use inner-product-baseee scalarization to model cross-degree interactions.
Long-range Dependencies. Modeling long-range interactions remains a core challenge in GNNs due the tension between limited receptive fields and over-smoothing [18 (58). Recent benchmarks [24] 70 86] evaluate GNN performance on long-range tasks. Solutions include implicit models (e.g.. IGNNs) and Graph Transformers, which relax locality constraints via attention-based architectures. These efforts are complementary to equivariant modeling and may be integrated with geometric priors for improved scalability and fidelity.
more comprehensive discussion of related work, including technical distinctions from our work, is
provided in Appendix A
3 Methodology
Notations. Each molecule can be represented 3D geometric graph, G (H, H contains
node features across hierarchical scales, specifically atom-level (e.g, atomic number) and residue-level (e.g. nucleotide type in RNA), denoted by H {Hatom R Natom x Fatom Hres E N.... Fres }. where Natom and Nyes are the number of atoms and residues (e.g. the nucleotides in RNA and amino acids in proteins), Fatom and Fres are the dimensions of atom and residue features. hatom E RR Facom and hyes E IR Free are the corresponding features of a node. X E RNNaom x3 denotes the 3D coordinates of atoms. From initial G. we aim construct Euclidean (EU) space neighbors NEU and spherical harmonics (SH) space neighbors NSH. We use uppercase letters to denote matrices and lowercase symbols to represent the corresponding vectors for a given node (e.g. h; and æ; for node i). To distinguish between invariant and equivariant features, we use regular letters for invariant features (e.g., h) and bold symbols for equivariant features (e.g., 2))
Overview of DualEquiNet. Fig. a) gives an illustration of the proposed DualEquiNet. It begins with SH feature initialization, followed by multiple DualEqui layers applied the atomic level. A Cross-Space Interaction Pooling (CSIP) module then aggregates atomic representations into residue- level features, which are further refined by additional DualEqui layers. Each DualEqui layer first infers EU and SH neighborhoods, then performs both inner-space and cross- space message passing update the EU coordinates æ. invariant features h, and SH features r (See Fig. [b). [)). EU neighbors are determined based on a distance cutoff, capturing local geometry. SH neighbors are selected based on SH feature similarity, enabling long-range structural connections beyond spatial proximity (See Fig. [d)). CSIP aggregates level features within each residue cluster. It performs pooling for h, x, and r. and incorporates cross-space projections to exchange complementary information across spaces (See Fig. ()). Next, we give details of each component.
3.1 Initialization of Dual-Space Features
Most equivariant graph neural networks rely on EU distance between nodes alone define edges, which struggles to capture long-range dependencies. This often requires increasing distance thresholds or adding more layers, both of which increase computational complexity. Moreover, EU-based message passing assumes that neighboring nodes are close in EU space and share similar features, but this assumption doesn't hold for large biomolecules like proteins and RNA. Nodes may be distant in EU space yet share similar structural features, such as bond angles and torsion, which are better captured in SH space. Thus, relying solely on EU proximity may miss key structural similarities crucial for molecular representation.
3

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(a) DualEquiNet
(b) DualEqui Layer
(c) CSIP
Feature
A
h
ala
Atows residue
Residue
Initialization
CSIP
des
sill
illa
all
pooling +
N
DualEqui Layer
pooling !
Nay within cutoff threskold
Nau distand bud structural similar
+
Projaer
DualEqui Layer
Prefew
EU Space Update
SH Space Update
r pooling
magu
Mgu-SH
msw
(d) Neighborhoods
DualEqui Layer
Lyes
AXEU
Ahgu
AhsH
ArSH
DualEqui Layer
crods space indlinge
Figure 1: Overview of the proposed DualEquiNet. (a) DualEquiNet: It begins with SH feature initialization, followed by multiple DualEqui layers applied at the atomic level. A CSIP module then aggregates atomic representations into residue-level features, which further refined by additional DualEqui layers. (b) DualEqui Layer: Each layer first infers EU and SH neighborhoods, then performs both inner-space and cross-space message passing to update the Euclidean coordinates æ, scalar features h, and SH features r. (c) CSIP: Cross-Space Interaction Pooling aggregates atom-level features within each residue cluster. It performs pooling for h, æ. and r, and incorporates cross-space projections to exchange complementary information across spaces. (d) Neighborhoods:
EU neighbors are determined based on distance cutoff, capturing local geometry. SH neighbors selected based on SH feature similarity, enabling long-range structural connections beyond spatial proximity.
To address these issues, we propose a dual-space initialization strategy that constructs two com- plementary spaces: EU space for capturing local geometric interactions and SH space for global structural relationships. The EU neighborhood of node is defined as:
NE (1) {j || æy dEu
where dew is the Euclidean distance threshold. Next, we initialize the SH space features by aggregat- ing the EU neighbors follows. For each SH degree 1....,.. Imuxs the 1-th order SH feature for node i is defined as:
do (lhs, Neg ||a (ll 8o(0 (b:)).
(E)
JEWED(
where æij 02 2. is the direction vector, and &ig zig/ | is the unit direction vector, do is a differentiable message function (c.g., an MLP), and (1) e 2 is the normalized SH function of degree 1, with unit norm (+)() 1. The SH basis and corresponding node features up to degree Imxx are jointly represented
ÿ (Z:z) (}(0) (ftg), (1) (a:)), (-bma))) (æ:y)), and r, [r
(maz))
(3)
The details spherical harmonics are introduced in Appendix B Eq. aaim learn an initialization of the SH space features based on the EU neighbors of the node, after which the SH features are
further refined through updates in the SH space. With the initial SH vectors, we define the SH neighborhoods of node based on the cosine similarity as
NSH (1) = ()
r:
[rr 2 dsH
(4)
Ir:
where Hsh is a threshold for SH space. These two neighborhoods- EU and SH-form the basis for equivariant dual-space message passing and cross-space interaction in subsequent model stages. In this way, the SH neighborhood overcomes the distance limitations of EU space and captures relationships between nodes that are distant EU space.
3.2 DualEqui Layer
The DualEqui Layer integrates geometric information from both EU and SH spaces through bidirectional message passing scheme. Each layer updates invariant features h. EU coordinates and

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SH features r using combination of within-space and cross space messages. This design enables the model simultaneously preserve local geometry and capture global structural patterns.
Euclidean Space Update. In EU space, both the atomic positions a and invariant features are updated using information from two sources: (1) within-space interactions based EU neighbors, and (2) cross-space interactions from the SH neighbors. Specifically, we compute messages from both the EU and SH neighbors and aggregate them via a multi-head attention mechanism.
Within-space message mEu captures local geometric interactions the EU space using invariant features and distances, while cross-space message MSH-EU models the influence of SH-spo SH-space structural similarity on the EU update, enabling long-range information flow guided by symmetry- aware features. For each node and its neighbor j. have
mEU,ij = ΦBU (lh,, h2, || II) mSH-EBU = UhSH (Ir; O Ty. |æ,,
(5)
where ||ز denotes the Euclidean distance between nodes i and j. The function ΦLU denotes differentiable functions applied within EU space (e.g., MLP), and ØSH-EU denotes cross-space transformations. The operator @ computes degree-wise wise inner products of SH features to form an invariant representation
17
(0) (0) r r: rj
(1)
(6)
To combine the messages, we use K-head attention. Each head learns to attend to different aspects of the neighhorhood, both in the EU and SH spaces. For the k-th head, the attention weights are computed as:
k BU. (deu (meut,))) PEU,, =a (VEU,
where o is the sigmoid function, and obu,an VEU,S are attention blocks. The invariant feature h; is updated using a permutation-invarian aggregation + (e.g., sum or mean) over messages from both sets of neighbors:
ANgUE Σ Qupd, EUU (1tss
a + JeNsn
+
(EA
(8)
where, dhuu: and Vheur are learnable message transformation networks per head. Similarly, the coordinate update is computed by aggregating directional messages scaled by attention:
ACEU,
a'SU, δEU,z (meu.)) 2n
(eu.. (9)
K
(eNgu(()
JENsH (8)
The EU space update integrates both local geometry and global SH information through learned, attention-we weightedd message passing. This allows the model to effectively capture rich interactions that span different geometric representations.
Spherical Harmonies Space Update Similarly, in SH space, for each node and its neighbor compute messages from both the EU and SH neighbors as
mss = ØSH ([h,, hy, r, @ ry)),
MEU-SSH = (rr
(10)
We then adopt K Khhd attention to aggregate the information as
AhSH, Σ Supd, (inss
asnss de (mSHH)))
jeNss(()
(EN
(11)
where a a ( and 3h 35 = (UEU. The SH feature is
affected by r, and Ý (Z:;)) together. Specifically, for the within SH space update, utilizing ry directly captures the structural similarity, including angle and torsion features, among SH neighbors. For
cross-space updates from EU to SH, the relative direction of the EU neighbors is crucial. ( explicitly incorporates this geometric directional information from its EU neighbors, potentially impacting SH features with the update given by the following.
K
Arsh,
h OSH.,jOSH, +
(
k=
JENSH
JENEU((
(12)
5

---

Dual Space Update. We employ residual connection, and the overall update is given by h = h4 + A + Ahgn.i.
2 = I; + A
= + Arsu. (13)
3.3 Cross-space Interaction Pooling
To create residue- and molecule-level representations, we use hierarchical pooling over atomic node clusters. The challenge is to maintain both local geometry and global structural context. Standard pooling methods, like average or intention-baseed aggregation, handle features from different geometric spaces separately. In biomolecular structures, local and global geometry are interconnected; for instance, an RNA nucleotide's 's torsional configuration (well-represented SH space) be affected by nearby atomic arrangements (in EU space), and vice versa.
To address the challenge, introduce CSIP, novel hicrarchical pooling strategy that facilitates bidi- rectional information exchange between EU (local) and SH (global) spaces during aggregation. This approach integrates multi-scale geometric information while preserving equivariance. Specifically, we first compute attention weights ag using only invariant quantities, then the weights are used to perform attention- based pooling
h'
C
E
LEC
ashi,
a, = ( d()),
(14)
where op is neural network applied to invariant features to calculate the attention score, d, denotes the average pairwise distance from node to other nodes, and C is the cluster of nodes over which pooling is performed. For example, nucleotide in RNA or amino acid in protein consists of a cluster of atoms. To enrich the EU representation with global structural cues, we inject projected SH features:
1 C _ ass [at + y (ro)): Proj (r:)
- bmaz 2im wi
(15)
where r!") is the SH component at degree I, and wy are learnable weights. Projsh converts SH space features r to EU space by producing an invariant signal summarizing the SH features' norms, which
modulates the geometric position via learnable coefficient y. Moreover, we enhance SH features with a projection of relative EU coordinates:
r' c -iEc as [r; - c pr prr = y (x: - Barg),
(16)
where Bavg TCT EiEc D; centers the input to maintain translation equivariance. The projection Proj (I.)) converts from EU space to SH space. Therefore, we choose to the SH function Y. embeds local relative geometry into the SH domain, controlled by learnable coefficient e. The pooled features h;, x', and r' serve as residue-level inputs for the subsequent DualEqui layers, enabling hierarchical modeling from atoms to residues.
3.4 QualEquiNet
As shown in Figure DualEquiNet captures the hierarchical nature of biomolecular structures through two sequential modules: an initial atom-level processing module and a subsequent residue- level module, separated by the CSIP layer. Each module consists of multiple DualEqui layers. Figure (b) shows that each DualEqui layer encompasses the operations defined in Equations to and the network-level update is expressed as:
1+ x: pdt1 DualFqqill.ayrr(h ar', rrr))
(17)
where the superseript/ / denotes the layer index, and is distinct from the SH order (1) with parenthesis. There are two types of layers: atom layers and residue layers. The model first applies Latom DualEqui layers to capture fine-grained atomic interactions in both EU and SH spaces. The CSIP module then aggregates these features into residue-level representations, which are further processed by Lres residue-level DualEqui layers to capture h dependencies. The input to the first atom layer is ho hatom- The first residue layer receives 11 = [h fatom, hyres) a concatenation of the final atom-layer output and the CSIP-derived residue features. This hierarchical design enhances both computational efficiency and representational capacity, aligning naturally with the multi-scale structure of biomolecules. The equivariance of the networks is proved in Appendix C
higher-order structural patterns and long- long-range

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Table 1: (a) Left: The N-chain task aims to differentiate two N-chain geometric graphs that differ only in the positions of their end nodes. (b) Right: The expressivity in the N-chain task with N = 10. Anomalous results are highlighted in red and expected in green DualEquiNet demonstrates stronger capability differentiate the two geometric graphs due to its expressive architecture.
Layers =
[ =
50,0000
50.0000
50,0s00
EGNN
50.0+00
50.0+0.0
30.0-0.0
50.0s0.0
50.0.00
50,05000
GVPGNN
500+00
52.0-98
51.0-7.0
53.0+119
51.3+6.5
54.5-14.3
TEN
50.0000
50.0+0.1
50.0:0.0
50.0
50.0e0.0
50.0s000
FaSEGNN
99.5:5.
100
91.0:19.2
99.525.0
100.0211.0
51,5s8.5
nodes
50,0000
50,0.00
GovenNet
50,020.0
50.0g0.0
50.0g0.0
50.0s0.0
50.040.0
50,020.0
DualEquiNet dey=00955
52.5e10.9
S1.5
$2.5e10.9
51 Sa128
99.525.0
52.5
DualEquiNet dsy-00.0
98.528.5
50.5:5.0
50.525.0
99.0s7.0
99.3x5.0
53.5x1288
DualEquiNet d
98.0293
54,0213.6
100.0000
100.0s0.0
100.0:11.0
100.020.0
DualEquiNet d,y000700
100.0g00
100.0s0.0
100.0+0.0
DualEquiNet dgy=00.60
100.0.00
100.0000
100000000
100.0:0:0
100.0:0.0
100.0:0.
4 Experiments and Results
We evaluate DualEquiNet across suite of tasks assess its expressivity, ability to capture long-range dependencies, and effectiveness on real-world biomolecular property prediction. Our experiments span synthetic benchmarks and challenging biomolecular tasks, including RNA property prediction, solvent-accessible surface area estimation, and torsion angle regression. The dataset details are in Appendix D.6
We benchmark DualEquiNet against baselines from various categories: (I)Invariant: SchNet 663 (2)Scalarization Equivariant: EGNN [62), GVPGNN [39]; (3)Higher-order Equivariant: TFN 699 FastEGNN 85), HEGNN [17), and GotenNet [6), covering a range of popular and recent baselines including invariant, equivariant and low- to high-order tensor methods. A detailed baseline description is provided in Appendix D.7
4.1 Model Expressivity on Synthetic Dataset
To evaluate DualEquiNer's ss ability to capture long-range dependencies, we use the N-chain discrimi- nation task from prior work [4). This task provides a synthetic yet rigorous benchmark for assessing the expressive power of geometric GNNs, particularly their capacity to propagate and integrate information across distant nodes. In this experiment, as illustrated in Table (left), there are pair of graphs and each graph consists of a linear chain with N nodes, identical except for their end-node to chiral molecules. The model is trained on one graph and evaluated on both to test its ability distinguish the chiral pair. These graphs are A 1)-hop distinguishable, meaning a GNN requires at least that many layers to differentiate them as proven in [40]. As N increases, the task becomes more challenging due to oversquashing 4. While the original Geometric Weisfeiler-Leman (GWL) test uses N = 4, we extend this to N = 10, requiring at least 6-hop challenges in modeling large biomolecules. Note that in real-world scenarios, large biomolecules can consist of thousands of nodes, making long-range information exchange even more challenging. We vary the SH-space cutoff distance dsh study its effect DualEquiNet performance.
aanfiurrations-nnllgee
information-
As shown in Table 1 most baselines fail to distinguish the N-chain graphs. performing better than random guessing even when stacking more than | + layers. FasEEGNN is an exception due to its global virtual nodes. In contrast, DualEquiNet achieves higher accuracy even with fewer layers by leveraging SH-space neighbors for global context. Performance improves
than 1 dsH decreases, reaching near- sear-perfect accuracy at dşH 0.7, demonstrating the effectiveness of SH-space communication in overcoming long-range limitations with lower SH cutoff distance signifying more SH-space neighbors. We also validate expressivity another rotational symmetry task (Appendix (D.4) showing similar results, confirming the benefit of high-order SH features, in agreement with HEGNN 17 and prior analyses 44
4.2 RNA Property Prediction
We evaluate DualEquiNer on RNA property prediction tasks because RNA molecules hold significant therapeutic potential, and accurate in-silico modeling can greatly reduce experimental costs and
7

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Table 2: Performance of RNA property prediction stanaad deviation (std.)). The best-
performing model is highlighted in bold. DualEquiNet outperforms other baselines on each metric.
Method
COVID
Ribonanza
Te-Ribo
Reactivity
pH10
Mg pH10
Avg.
DMS
2A3
Avg.
Factor
SchNet
0.45120.006
0.633a0.012
0.51810.010
0.534.00011
0.90920.018
0.966200001
0.93740.012
0.737
EGNN
0.45310.006
0.63410.013
0.520x0.010
0:535:0001
0.912:0017
0.968
0:940z0.012
0.720=0.022
GVPGNN
0.44840.006
0.631+0.012
0.515+0.009
0.531+0011
0,901+0015
0.963+0004
0.932=0.010
0.736=0.00
TFN
0.453#0.006
063440.012
0.520+0.010
0.535=0.011
0001100018
0,968
0,939=0.012
0.694200008
FasEEGNN
0.39020.019
0.58240.029
0.46340.027
0.478
0.80840.022
0.86920.015
0.839.0.000
0,731
HEGNN
0.45010.005
0.63210.012
0.51740.009
0.533100011
0,90410015
0,964
0,934,0:010
0.729200007
GotenNet
0.38840.002
0.582+0.011
0.455+0.008
0.475+0.009
0.832+0017
0.882+0.009
0.857+0.013
0.7 0.718=0.002
EGNN + Pooling
0.280+0.006
0.469+0.006
0 0.36240.010
0.370+0.009
0.534+0.014 0.688+0
0.611=0.00
0.699+0.016
SchNet Pooling
0.30240.006
0.495+0.010
0.384+0.000
0.39420.000
0.721=0019
0.642=00044
0.681=0011
0.702=0.077
FasEEGNN + Pooling
0.37940.002
0.56640.009
0.454220006
0.46720.007
0.77350002
0.81920.006
0.796.0.013
0.721
DualEquiNet
0.272:0.004
0.44810.010
0.340s9.005
0.353:0.006
0.611s0.025
0.505
0.558:0.012
0.63620.049
time 21 RNA molecules pose unique modeling challenges due to their large size, complex secondary and tertiary structures involving both short-range and long-range interactions [67], and the necessity for equivariant representations respecting rotations and translations [57]. An effective RNA modeling approach must efficiently integrate both local atomic interactions and global structural dependencies.
We three RNA datasets consisting of thousands of atoms: (i) Covid 774] which provides
nuclcotide-level reactivity and degradation labels relevant for vaccine stability; (ii) Ribonanza [37]. which includes nucleotide-level reactivity under different compounds to assess RNA folding and flexibility; and (ii) Tc-ribo B3 which contains RNA-level regulatory behavior in response to
tetracycline. 3D RNA structures are generated using RhoFold (65] following prior works [57].
We follow the experimental protocol in [57] to use eta-theta pseudotorsional backbone (33 as input and use Root Mean Squared Error (RMSE) as performance metric. All models are evaluated using consistent train-validation--est splits (8: 18:1:1), 1), with hyperparameters selected based on the best validation performance, optimized via Optuna 3 ensure fair comparisons.
As shown Table 2 DualFquiNer outperforms all baselines, reducing RMSE by 25.7%, 33.5%, and 8.4% on Covid Vaccine, Ribonanza, and Tc-ribo, respectively compared to best bascline. This highlights DualEquiNet's strength in modeling complex RNA interactions, especially long-range dependencies, making it a strong candidate for RNA property prediction tasks. Note that the pooling in Table is the standard hierarchical pooling (e.g., atom-to-nucleoti for RNA, for proteins).
acid
4.3 Protein and RNA Solvent-Accessible Surface Area (SASA) prediction
Solvent-accessible surface area (SASA) measures how exposed atoms or residues are to solvent, reflecting large biomolecule's s 3D conformation and influencing interactions and stability [66] 54]
Accurate SASA prediction is crucial for understanding and engineering molecular behavior 68
For protein SASA prediction, we use the existing mRFP protein dataset from (68], with 3D protein structures generated using AlphaFold[4]. This dataset contains ground-truth SASA val- ues, offering a reliable benchmark for model evaluation.
Table 3: Performance on SASA Prediction (RMSF+std).
RNASolo
mRFP Protein (x10)
SchNet EGNN
48.560+2.187
8.370=0.963
32.011+2.323
9.054#1.300
GVPGNN
37.704#1.050
20.351
TFN
51.003+1.214
10.176
Additionally, we introduce new SASA benchmark dataset for RNA, sourced from RNASolo (2), which includes ex- perimentally determined 3D structures of RNA molecules. During preprocess- ing, we choose RNA molecules with sequence lengths ranging from 10 to 400 and exclude RNA sequences with multiple chains or non-AUGC nucleotides. Ground-truth residue-level SASA values are computed using DSSR [51]. Together with the existing protein SASA benchmark, this new RNA dataset provides a comprehensive benchmark for assessing models' ability to capture detailed structural representations of both RNA and protein.
FastEGNN HEGNN 37.782+1.795
37.251
12.282+1.310
12.382+3.525
GotenNet
34.966+1.671
8.743+1.558
DualEquiNet
31.972+0.859
5.959+1.160
8

---

Table Performance on Togle dataset with predicted angles. (Cireular MAE std.).
Avg.
<
X
SchNet
18.54=0.86
31.38+1.44
17.54=0.20
14.4
7.68+0.25
15.4840.56
20.50=1.30
12.7840.63
EGNN
17.79=0661
29.62a0.98
17.25:0.26
22.96s0.79
7.54
15.17+0.67
19.72=1.11
12.29x0.58
GVPGNN
17.56.0.21
29.6940.41
16.9420.25
22.77a0.49
7.0240.26
15.02+0.57
19.6320.54
11.85x0.15
TFN
21.97=0.45
36.52=0.62
18. 18.1490.40
27.63++066
11.6320.43
17.64+0.79
26.59=0.77
15.57=0.74
FastEGNN
18,89=0.22
31.9620.58
17.4320.40
24.6420.60
8.47a0.45
15.6610.45
21.31=0.33
12.7840.13
HEGNN
18.41=0.22
31.1240.59
17.49=0.21
23.78a0.66
7.91
15.3540.46
20.54.0.188
12.67a0.26
GotenNet
16.27=1.19
26,88+1.57
15 15.90=0.60
21.06+1..4
6.7240.35
13,85+0064 64
17.74=1.63
10.9640.84
DualEquiNet
15.87=2.03
26.8923.00
13.48=1.17
21.12+2.68
6.30200.45
13.68s1.29
17.03=1884
10.60:1.32
The baselines and data splits are consistent with those Section 4.2 and we use Optuna for optimal hyperparameter search. RMSE to the ground truth SASA values is reported. Table 3 presents performance results on the SASA benchmarks. DualEquiNet consistently achieves the lowest prediction error across both RNA and protein datasets, demonstrating its superior ability to capture and refine both local and global structural features through its dual-space hierarchical representations.
4.4 RNA Torsion Angle Prediction
Torsion angles define rotations around specific bonds, shaping the 3D conformation of biomolccules and influencing their secondary and tertiary structures, interactions, and functions. Accurate prediction of these angles is essential for modeling structural geometry. While sequence modeling techniques have been proposed for RNA torsion angle prediction [1], they rely solely on nucleotide sequences and ignore the 3D structural context. In contrast, torsion angles are fundamentally determined by spatial geometry (more details in Appendix D.6]
Similar to SASA experiments, the TorsionAngle dataset uses ground-truth RNA sequences and structures from RNASolo and torsion angles are computed using DSSR. Consistent with (.. we adopt the circular Mcan Absolute Error (MAE) loss. To predict seven torsion angles (a,, 3, 7, 8, E, S, X), the model outputs 14 values-sin sin and cos for each angle. The angle 0 is recovered via 0 arctan(sin(O), cos())), and the circular MAE is computed as MAE(()
Nur EN min (AO,, 360 20,) where A0; |pprd Omne.al and Nut is the number of pre- dicted nucleotides.
Results presented in Table 4 illustrate the performance of various models across different torsion angles prediction. Our DualEquiNet achieves the lowest average error and performs best for out of 7 angles prediction. These findings highlight that DualEquiNet effectively learns 3D structural representations, enabling accurate prediction of torsion angles based on molecular geometry.
4.5 Ablation and Neighborhood Dynamics: Importance of Dual-Space Learning
To quantify the contribu- tions of key components in DualEquiNet, we perform an ablation study across large-scale RNA property datasets. We evaluate four configurations: (i) the full DualEquiNet model; (ii)
Table 5: Ablation Study on RNA Property datasets (RMSE++tt..).
Model
COVID
Ribonanza
Tc-Ribo
QualEquiNet
0.353+0.006
0.55820.123
0.63620.049
w/o CSIP
0.356#0.002
0.58610.016
0.68240.026
w/o CSIP/Cross
0.356#0.013
0.591+0.081
0.698+0.010
w/o CSIP/Cross/Dual
0.362+0.010
0.592+0.028
0.69940.010
w/o CSIP, which replaces contextual structure-informer pooling (CSIP) with standard mean pooling; (iii) w/o CSIP/Cross, which disables cross-space message passing but retains dual-space repre- sentations; and (iv) w/o CSIP/Cross/Dull, single-space Euclidean model. As shown in Table 5 each component enhances performance, with CSIP particularly benefiting the Ribonanza and Tc-- ribo datasets, and the dual-space mechanism notably improving performance COVID. These results confirm that combining local and global geometric contexts through dual-space learning and cross-space communication is essential for generalization.
More experiments about the neighborhood analysis, time complexity analysis, and expressivity on symmetric structures are in Appendix D

---

Conclusion
We introduce DualEquiNet, a novel framework for modeling large biomolecules by hierarchically in- tegrating Euclidean and spherical harmonics representations across atomic, residue (nucleotide/amino acid), and molecular levels. DualEquiNet is E(3) equivariant and effectively captures both local and global structural dependencies. Extensive experiments on real-world datasets demonstrate state-of-the trrt performance over existing geometric GNNs, highlighting the benefits dual-space representations and hierarchical learning for large biomolecular modeling- Although DualEquiNet is effective across diverse tasks, its performance may depend on the quality of input 3D structures, which vary across datasets and affect generalization. Future work will explore integrating structure prediction into the pipeline and extending the framework to broader molecular applications.
Acknowledgments
This material is based upon work supported by, or in part by, the Army Research Office (ARO) under grant number W91 199-11111- and the Cisco Faculty Research Award. The findings in this paper do not necessarily reflect the view of the funding agencies.
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A Related Work
A.1 Invariant GNNs
Invariant GNNs learn molecular representations using rotation-invariant geometric features, ensur- ing predictions remain unchanged under rigid transformations. SchNet employs continuous-filter convolutions over interatomic distances to model molecular properties effectively [63]. DimeNet extends SchNet by integrating angular features, explicitly considering bond angles to capture di-
rectional information [30] 29 42 GemNet further enhances expressiveness by utilizing multi-hop
distance-angle interactions, implicitly encoding higher-order geometric features [28]. SphereNet
incorporates comprehensive set of invariant scalars, including distances, angles, and torsion angles,
to robustly represent molecular structures [50].
Despite their effectiveness, invariant GNNs inherently lose crucial directional and orientational information, motivating the development of equivariant models capable of explicitly encoding full geometric contexts.
A.2 Scalarization Equivariant GNNs
Scalarization equivariant graph neural networks maintain E(3) equivariance by explicitly handling vector-valued features through scalarized updates. EGNN 662 leverages pairwise distances to compute scalar-valued messages, enabling direct equivariant updates to atomic positions without high- order tensor operations. GVPGNN (39 introduces geometric vector perceptrons, managing scalar and vector embeddings separately and using scalar gating to achieve SO(3)-equivariant transformations. PaiNN [64] similarly utilizes scalarized vector updates, applying distance-based filters to coordinate features, thus ensuring consistent equivariant message passing.
These models share the common technique of scalarization, employing invariant scalars to guide separate, equivariant updates of vector or positional features. However, scalarization methods inherently lack the expressive power provided by high-degree spherical tensor representations, potentially limiting their ability to capture complex rotational relationships and subtle structural differences in molecular systems [4).
A.3
Higher-degree Equivariant GNNs
Several equivariant GNNs achieve expressivity by using high-order ir- reducible representations combined via Clehsch-Gordan tensor products. Tensor Field Networks (69 (TFN) introduced this paradigm by building SE(3)-cquivarian convolutional layers from CG-based feature tensor products, enabling rotation-cquivarian learning with theoretically maximal expressive- ness at significant computational cost. The Transfrreee 227 extended TFN by incorporating self-attention mechanism irreducible feature spaces, further improving representation power. SEGNN [12] construct streeable MLPs by using the tensor product so that formalize an equivariant message aggregation MAAE [9] uses high-order tensor features to form complete basis of scalar in- variants through the Atomic Cluster Expansion, capturing arbitrary many-body interactions. However, it also incurs significant overhead due to numerous CG tensor product computations.
CG-based Tensor Product
Efficient Implementation Notably, the reliance on full CG tensor products leads to steep scaling in complexity (e.g. O(L)) for degree features). Due the high computation cost of CG-based tensor product, more recent approaches aim to retain high-degree expressivity in more efficient way. HEGNN [16] incorporate tigher-degres spherical harmonics features but using EGNN- like inner- product scalarization trick to handle cross-degree interactions. Similarly, GotenNet (6) introduces high-order spherical tensor in unified attention architecture that eschews explicit CG transforms; instead, it updates features via inner products on steerable tensors. Our DualEqui also follows this paradigm: combines spherical harmonic feature channels with dual scalarization steps to capture rich 3D geometric information while keeping computations tractable.
A.4 Long-Range Dependencies in GNNs
Learning long-range dependencies is a key challenge in graph machine learning, arising from the fundamental trade-off between under-reaching (i.c., insufficient GNN layers leading to small
16

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receptive fields) and over-smoothing (i.e., excessive layers causing feature homogenization) [ 58]
variety of benchmarks [24] 70 86 79 have been proposed to study this problem. along with
several representative solutions, including Implicit Graph Neural Networks (IGNNs) and Graph Transformers (GTs). IGNNs define message-passinng
through fixed-point equations R4 49 or
unfolded optimization steps 20 [) allowing stable convergence to equilibrium state even with infinite layers. Efforts to improve their scalability have focused on accelerating convergence [19 and efficient neighbor sampling [8] A6], making IGNNs more practical for large-scale applications. GTs leverage global attention mechanisms [77
84 60 to capture long-range dependencies while incorporating structural encodings to preserve graph properties. Recent works have improved their scalability [75] 76 [2] and extended their applicability to directed graphs [3 53,, broadening their impact in graph learning.
B Basics of Group Theory, Equivariance, and Spherical Harmonies
In this section, we provide an introduction to the mathematical background used in this paper. For readers interested in more details, we refer them to the Guide to Geometric GNNs 23 and the e3nn documentation
31
B.1 Groups and Symmetry
A group is an algebraic structure consisting of set 6 together with binary operation (composition) that satisfies four fundamental axioms:
Closure: For any a, b E B, the composite a b is also in G.
Identity: There is an identity element E 6 such that e = a = for every a E 6.
Inverse: For each E G, there exists an inverse element a E @ with a a-a-- a-1.
Associativity: For all a, b, 6, have (a b) a (b.c).
Groups are often used to describe symmetries objects or spaces. For example, the set of all rotations about a fixed point forms group, does the set of all translations in the plane. The symmetries of an object form a group under composition of transformations.
The Euclidean Group E(3): In geometric deep learning, an important example is the Euclidean group in three dimensions, E(3), which captures the symmetries of 3D Euclidean space (i.e., all rigid motions in RR.). An element of E(3) can be represented as pair (R, t) where Rt is a 3 x 3 rotation matrix and E R is a translation vector. Composition in E(3) done by composing rotations and adding translations: (R1,t) (R2, t2) (R, R2, R t2 + t.). This group is essentially the cemi-direct product of translations and rotations. Intuitively, applying an E(3) transformation to a point x E R yields R.z + t. which is just rotated and translated version of the point. E(3) is the symmetry group of any rigid 3D object, mcaning that an object (or a dataset) is moved or rotated in space, it remains essentially the "same" object- concept captured by invariance equivariance (discussed in B.2). Many physical or geometric tasks exhibit E(3) symmetry. Hence, incorporating E(3) into machine learning models is powerful inductive bias [14 43)
The Rotation Group SO(3): The special orthogonal group SO(3) is the subgroup of E(3) consisting
of rotations about the origin (with no translations). Formally, SO(3) is the set of all 3 x 3 orthogonal matrices with determinant 1. Every element R E SO(3) represents a rotation in 3D space. SO(3) plays a central role in describing angular symmetries. For example, the orientation of an object in space can be changed by an element of SO(3). SO(3) is non-abelian (the order of rotations matters), unlike the translation sub-group (which is abelian). In many applications (like molecular modeling or 3D vision). only rotations need to be considered (c.g., when comparing shapes regardless of orientation), making SO(3) crucial symmetry group. Models that respect SO(3) symmetry avoid to learn the same feature in all rotated forms and group- group-theoretic approaches provide a unified framework incorporate such symmetries into neural networks.
B.2 Equivariance and Invariance in Geometric Learning
Equivariance to Fuclidean transformations a desirable property geometric learning tasks involving 3D molecular structures. Specifically, a function f X + y is said to be equivariant with respect to
17

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group & (e.g., the Euclidean group E(3)) if, for any transformation E 6 and any input a E A, there exists corresponding transformation g acting on the output space such that
fg x) = f(:).
(18)
For E(3), this means that if the input (e.g., atomic coordinates or features) is rotated, translated, or reflected, the output of the model transforms a consistent and predictable manner. This property ensures that the model's predictions are physically meaningful and invariant to arbitrary choices of coordinate systems-a a key requirement for tasks involving 3D biomolecular data.
To achieve rotational equivariance in 3D, spherical harmonics are commonly used to represent directional features on the unit sphere. Spherical harmonics form an orthonormal basis for square- integrable functions defined on the sphere and serve as the irreducible representations of the rotation group SO(3). key property of these representations their equivariant transformation under rotations. For unit direction vector è and a rotation matrix R, the spherical harmonics basis of degree transforms according to:
Y (R)) = (R) (z).
(19)
where (-) is the spherical harmonics of degree 1, and (R) D-matrix corresponding to rotation R. This property enables the construction of rotation-equivariant functions by combining SH features with these bases, making them particularly suitable for encoding angular dependencies molecular systems. We introduce more background in Appendix
(R) E R(2++1x(2I++1 is the Wigner
Invariance vs. Equivariance: Invariant functions are a special case of equivariant functions. We say f: X' + y' invariant to if [(g x) transformation. This corresponds to equation 18 where the 6-action on y is trivial (identity for all g). Invariance discards all information about the transformation (losing equivariant detail), whereas equivariance preserves information in structured way. Generally, equivariance is more powerful than invariance when building deep representations because it allows higher layers to reason about what transformation occurred, not just that something is symmetric.
(z) for all g 6, i.e. the output is unchanged by the
B.3 SH Space Captures Global Information
Spherical Harmonics (SH) provide orthonormal basis for functions defined sphere, similar to how the Fourier series represents periodic functions. In this section, we draw parallel between Fourier decomposition in one dimension and SH expansion on the sphere, highlighting how increasing the expansion order refines function approximation and why SH captures global structural information.
B.3.1 Fourier Series vs. Spherical Harmonics
well-known tool for analyzing periodic functions the Fourier series, which decomposes function into a sum of sinusoidal components. Given periodic function f(x) defined on E [- r.. its Fourier series expansion is:
f(z) = do +
Σ (an cos(na) + b,, sin(nz)),
(20)
where the coefficients are computed as:
an = _ [
T (() cos(nx) dx, bn - - (x) sin(nx) da.
(21)
By truncating the expansion order N, we obtain an approximation:
N
fapprox (x) = ao +
Σ (an cos(nz) + b, sin(nz)).
(22)
As N + oo, the approximation converges to the true function fzr) In this representation:
Low-frequency terms (n 1,2) capture smooth, large-scale structures.
High-frequene terms (n > 2) refine the function by encoding local variations.
18

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Analogously and Intuitively, SH expansion generalizes this idea to functions defined on the sphere. For function f(8, Φ) defined the unit sphere, the SH decomposition
f(0,
y m
(23)
1= m=-
where the coefficients are obtained via:
- | f(0, (0) (0, .) sin d0 dø.
(24)
Truncating the expansion at order L provides an approximation:
L Σ
fupprox (0, Φ)
Σ
y Φ).
(25)
I= m2
As L oo, this converges the true function f(0, The role of L in SH is similar N in Fourier analysis:
Low-degree terms (1 0, 1, capture smooth, global variations.
High-degree terms (1 >> 2) encode fine-grained local details.
B.3.2 Why Spherical Harmonics Capture Global Information
Spherical harmonics enable long-range information encoding because they provide global basis rather than relying on local neighborhoods. Unlike pesssag-ppassing GNNs, which aggregates information only from spatially close neighbors, SH representations allow direct interactions between distant nodes that share similar SH coefficients.
10.011-110
I..m
(8.00
Figure 2: Visualization of real Spherical Harmonics table
Figure provides visualization of the first few real-valued Spherical Harmonics. The figure illustrates how the functions oscillate across different degrees. Lower-degree harmonics (?) exhibit slower variations, while higher-degree harmonics demonstrate more rapid oscillations along the axes, reflecting their ability capture different scales of geometric features. To be more specific, when
1 = 0 (Monopole): Captures the global molecular shape.
'Image source:
Wikipedia
19

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I = 1 (Dipole): Represents molecular orientation.
1 = 2 (Quadrupole): Encodes bond angles and torsional interactions.
Higher values:
Capture localized atomic-scale interactions.
Since low-degree SH functions change gradually, functions dominated by low-degree SH terms exhibit long-range correlations, as information propagates smoothly across extended regions of the domain.
Mathematical Justification.
The SH functions are eigenfunctions of the Laplace- Beltrami operator:
A Y - 1)
(26)
Since the eigenvalues -[(1+)) control how rapidly the functions oscillate, low-degree SH components (1 0, 1, vary slowly and thus retain long-range structural features.
SH-Based Neighborhoods Enable Long-Range Dependencies. In our framework, nodes interact based on the similarity of their spherical harmonics (SH) feature representations, independent of their Euclidean spatial distance. This enables two spatially distant nodes to have strong interactions if they share similar local geometric environments (e.g., comparable bond angles and torsions with their respective neighbors), as captured by their SH representations.
Traditional message passing in molecular graphs is inherently local, where information propagates only between spatially adjacent nodes. To capture long-range dependencies, these methods require multiple message passing steps, which can be computationally expensive and may lead information dilution.
Our SH-based approach naturally encodes global structural dependencies while maintaining computa- tional efficiency. This is possible because low-degree spherical harmonics have broad spatial support, creating smooth, global basis functions that span large regions of the molecular structure. As result, nodes with similar SH coefficients maintain strong correlations regardless of their spatial separation.
This framework effectively captures long-range molecular interactions without requiring explicit long- range message passing, the similarity in SH representations automatically establishes meaningful connections between distant nodes that share similar local geometric environments.
C Proof of Equivariance
In this section, we formally demonstrate the E(3)-oquivariance property of our proposed DualEqui model. Specifically, we begin by proving the invariance property of the message functions, showing that they are built upon fundamental invariant quantities. Next, we analyze the forward computation of the DualEqui model and establish the equivariance property.
Notations. In this proof, we let X E N represent the set of all input 3D atomic coordinates, where each row æ, E R3 (for [1,.... (l, corresponds to one specific atomic position within X. We represent an arbitrary E(3) transformation using rotation matrix R E 3x333 and a translation vector t R
.N)
For convenience, we let be an arbitrary intermediate variable in our model computation (e.g.. message mgu.i Eq. 55). Occasionally, we adopt functional notation, expressing of all imput 3D coordinates, i.e., x(X). This allows us to denote the model output under E(3) transformations as 2 =RRX + t) and verify whether equivariance or invariance holds. By default, we consider the original non-functions form of intermediate variables z as shorthand notation for >(RXX t), where R I and 0.
function
C.1 Basic Invariant Quantities
The equivariance of the DualEqui model is grounded in the invariance of several fundamental quantities. Our message functions Eq. (5) and Eq. (T)) are computed using two basic invariant quantities, whose invariance is shown below.
20

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Euclidean Distance || ||-
Let 2;, 2, be any pair of 3D coordinate vectors. When an E(3)
transformation applied both æ; and zj the Euclidean distance invariant due the orthogonality of rotation matrices:
(Ræ;; + t) -R.z, + t)ll = || [ - zz)]
V (x: TR R(x; xz)
I - zzll
Spherical Harmonics Feature T2 @ r. To show the invariance of ry @ r. we first consider the
cquivariance an arbitrary r for node E {1, (L,",.,AV and order {0, -. [maw] under E(3)
transformations. Specifically, we have:
(RX +1) = (i) jENEu(()
Σ
Φo (lhi, bys [| Y (Rı:;)
E
do (14,, hi (|g)l))) (R) (diz)
WEU (i)
(eN
= DD)(R)
Σ (ENEu()
do (1h, he. |[ pd (Etg))
WEU (i)
D (R)
(27)
where (R) denotes the I-th order Wigner-D matrix, which denotes the irreducible repre- sentation of the rotation group SO(3). Specificallyy, all the Wigner-D matrices are unitary, i.c., D (R) D (R)=1 I.
Thus, since Eq. already establishes that
(1) [r. P. 5- r (maz))
We can then directly obtain
r: [RX +t) (D(0) (R)( (R)) Dmma)) RR)plmmus
Therefore, Since the Wigner-D matrices are unitary, the invariance of ri C r, follows:
r: (RX +t) Or, RXX +t)
(r! (()(()))))
(R))))) .... (I!/maz) (((mmaz)((lmaz)) D(lmax) (R)) " (D(lmax) (R)r (moa)))))) })
(r(()) (0) (1/ (fmme))) Ty
Orj
C.2 Invariance of Message Functions
Based on the invariance of the above quantities, the invariance of the message functions follows directly from their definitions. Recall Eq. (5) and Eq. (10):
mEU.i == ØEU ([hi, hy Il)))))
MSH,i δEU ([h,, h1, 3 rrl))
M EU-SSH = V'EU-SH ([r, @ ry, [[,,)]]]).
MSH-E. USH (Ir: O ry [æ,,)]))
Since h2 and hy are transformation-inaaiann scalars, and all the message functions use only invariant quantities ([ and r. ry)) we conclude that all the message functions are invariant.
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C.3 Equivariance of the DualEqui Model
We now prove that the entire DualEqui model is equivariant with respect to E(3) transformations.
In DualEqui, each node is represented by two types of coordinate vectors: æ; in Euclidean space and in Spherical Harmonics space. The equivariance of æ; is immediate, and, as shown in Eq. (27). each ri is equivariant. Hence, the initialized node representations are equivariant.
Initialization.
Euclidean Space Update. In Euclidean space, we update the scalar features h; and the position vector z; using updates Ahru, and AEru, The scalar update is given by Eq. (8)
K
(mee his
AlgU =
Bupd.EU
B & (mE,s( +
jENBu(()
JeNen(()
with attention scores defined as:
EU.A a
Buu = (VEu.o(msH-naa
Since these computations involve only invariant quantities, A is invariant. Similarly, the position update from Eq. (9) is:
K
1 K
AUEU = (e)
B'uu DEU, (mEV, mi +
Eu, (Ms
A=1
JENsn
Under an E(3) transformation, the scalar weights are invariant and the aggregation @ (i.e., sum or average) is linear, hence:
ATEU, RRX - t) = -RA
which proves that Aug is equivariant.
Spherical Harmonics Space Update. In Spherical Harmonics space, update the scalar features h; with Ahsh. and then update the Spherical Harmonics features T; with ArsH, The scalar update is defined in Eq. III as:
(To Ths
Ahgh, ==
Σ
Su
O'H USH (mSHH)))
+
jENsH
JENEO (1)
where the attention scores are defined as:
le OSH = (d (msH,)))
SSH. a
As these updates involve only invariant message functions, Ahsn. is invariant. For the Spherical Harmonies feature update, we have from Eq.
K Σ
ArsH. = K
a OSH,Y (msH.s))r +
B (ma
1=1 (eN
(eN
where both ry and (m.)) are equivariant, transforming under the block diagonal concatenation concatenation of all Wigner-D matrices [D(0) (R), (R), ((lma))
(R)). (.0ma) (R)). Therefore,
ArsH. (RX + t) (R)) D
(R), (R), D(mmo)) (R))ArSH.
which shows that Arsh, is equivariant.
22

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Cross-space Interaction Pooling, In the cross-space interaction pooling stage, we use biomolec- ular structure-aware hierarchical pooling strategy aggregate nodes within cluster in both Euclidean space (æ') and Spherical Harmonics space (r'), along with scalar features h.. Since the scalar features are invariant, we focus on the pooling in both vector spaces. For the Fuclidean space pooling in Eq. [5 we have:
x' RRX +
Σ IEC
(Ræ; + Proisu (rx + +t))
aj
Ræ; 7
wi |D(0 (R)rl Rar
L
Ræ; +
Σ
wi |r Rax:
IEC
1=1
wi || )
R
Oii
2i
+ 7
C
Re'
where D (R) denotes the l-order Wigner-D matrix.
Similarly, for the Spherical Harmonics space pooling in Eq. (16
r' 'RRX +t)
a;
(r. + +t) e Projeu (Ræ; + t))
C
iEC
Σ
(r((XX + t) + V()()) Favg))
C
05
IEC
Σ IEC
(D(R)r: e D(R) -aave)
as
D(R)
1
Σ as (rite Y(o sove))
C
= D(R)r',
where (RR) denotes the block diagonal concatenation of all Wigner-D matrices, i.c., D(R) == [D()) (R), D(1) (R),.... D(mmex) (R)). Specifically, Favg : [ Eiec 3,; in the equation above denotes the center of all input 3D coordinates, and preserves the translation invariance.
To summarize, we have shown that all node position vectors æ; and r;, and their updates, are equiv- ariant, and that the pooling operations preserve this equivariance. Thus, traversing the entire model structure of DualEqui, we conclude that DualEqui is equivariant with respect to EE3) transformations.
D
More Experiments and Details
D.1 rnaglib Datasets Performance
We conduct additional experiments on the RNAGo and RNAProt datasets from maglib (78), shown in Table DualEquiNet achieves state-of-the-art performance the datasets, further validating its effectiveness.
D.2 Comparison with Long-range Baselines
To separate the performance gains and provide stronger evidence of the effectiveness of the proposed method, we compare the performance with models that are made for long-range tasks and also use dual-space design. Specifically, we select Neural Atoms 47] and Ewald-hased message passing and show their performance in Table 7
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Table 6: Performance comparison on RNAGo and RNAProt datasets.
Model
RNAGo (Jaccard Index 1)
RNAProt (Aceuracy
SchNet
0.61==0.020
0.7091100005
EGNN
0.646=0.060
0.720=00009
GVPGNN
0.632=0.048
0.710=0.005
TFN
0.651=0.185
0.697=0.005
FastEGNN
0.527=0.159
0.718+00000
HEGNN
=0.038
0.706
GotenNet
0.555=0.170
0.709+00006
DualEquiNet
0.041
0.749
Table 7: Performance comparison with long-range baselines. (RMSE+std.)
Method
COVID
Ribonanza
Te-Ribo
Reactivity
pH11
Mg pH10
Avrg
Avg.
SchNNet + Ewald
0.392:0.004
0.583-20.010
0.459-0.0008
0.478-20.009
0.845:00002
0.88420.002
0.865=000114
0.736-0.0014
SchNet Neuna Atom
0.400-00.004
0.500-00.009
0.481-00:066
0955+0.008
0.544-0.515
0.919-0.009
0.886-0.012
0.726-0.002
Dimner Ewald
0.403-00.005
0.594=0.006
0.480+00.004
0.492:50.05
0.828=00.01
0.896=0.011
0.862=0.003
0.676-0.044
0.602-10.016
0.489:50.013
0.859=0.0005
0.679-0.006
DualEquiNet
0.272-0.000
0.448-0.001
10.40-0.1005
0.353-00006
0.611-
0.505 0.004
0.558-0.0112
0.636-0U049
D.3 Neighborhood Analysis
To better understand how DualEqui captures both local and global geometric contexts, we analyze how neighborhoods evolve across layers in the EU and SH spaces. We use the experiment of DualEqui with 4 atom layers on the Covid Vaccine dataset case study. At each layer, EU and SH neighborhoods are dynamically redefined based on updated 3D coordinates and SH features, respectively. We compute 4 statistics:
(1) the average distance of coordinates between each node and its EU neighbors;
Natam
disgu
1
Σ
||, xyll
(28)
Natom
(i)
jENNo (1)
(2) the average coordinates distance between each node and its SH neighbors;
Natum
dissg
1
Σ
|z,- 1e.-ap xyy
(29)
Notom
(i)(
JENsn
(3) the average cosine similarity of SH features between each node and its EU neighbors;
Natoo
COSEU
N Natom
1
WEU
Σ
[xll
T: T
|ryll
(30)
JENEo
(4) the average cosine similarity of SH features between each node and its SH neighbors.
Natam
1
Σ
T TE Ty
COSSH
Natom
[WsH (i)
||r/l| ||r
(31)
jENsi(
The distances of coordinates are plotted in Fig. 3(a). and the cosine similarity of SH features are plotted in Fig. 3). Results reveal that the average distance of EU neighbors (dis gu) and the SH features cosine similarity of SH neighbors (cossH) remain stable across layers. This behavior is expected since EU neighborhoods
constructed using fixed cutoff dgu based spatial proximity, while SH neighborhoods are selected using a threshold dsw on SH feature cosine similarity. this experiment, use d 4.8 and dsn 0.97.
In contrast, we observe clear trends the other two metrics. The average coordinates distance of SH neighbors (dissH) decreases significantly over layers (Fig 3 a), orange line), indicating that nodes

---

1.0
Cosine Similarity
EU Neighbors
20
0.8
SH Neighbors
Distance
EU Neighbors
0.6
SH Neighbors
10
0.4
Init
2
Init
2
Layers
Layers
Figure 3: (a) Left: Average distance variation of EU and SH neighborhoods across layers. (b) Right: Average similarity variation of EU and SH neighborhoods across layers.
with similar SH (eatures-whice initially may be far apart in EU space geredaall become closer in Euclidean distance. This trend reflects the impact of the cross-space update in and When update the SH features r, we also consider the EU coordinates, which brings distant SH neighbors closer in EU space. In practice, residues can interact through folding.
imagine the scenario in biomolecules where spatially distant
Meanwhile, the SH features cosine similarity of EU neighbors (co8gg;) drops across layers (Fig b)), green line), implying that while EU neighbors may remain close in EU space, their SH features increasingly diverge. According to Eq 2 neighbors at first and therefore have a higher similarity score. But after several layers, the SH features updated to differentiate local nodes based on their broader geometric context or functional roles. This is relevant in structured biomolecules where nearby atoms may belong to different residues or to distinct structural regions.
the SH neighbors are initialized according to the EU
Thus, the observed dynamics validate the model's dual-space architecture: SH space helps bridge distant but functionally related parts of the molecule, while EU space preserves local geometry but allows for semantic diversity.
D.4 Expressivity on Rotationally Symmetric Structures
An L-fold symmetric structure is one that remains unchanged under rotation by 360% exhibiting identical appearance at each of L evenly spaced angles around a central axis. Joshi et al. [4]] demonstrate that spherical tensors of order L cannot discern the orientation of structures with rotational symmetry greater than L-fold.
Table 8 presents the performance of various models the I.-fold symmetry classification task for = 2,3, 5, and 10. As expected, SchNet, EGNN, and FastEGNN fail to solve the task across all settings due their lack of higher-order spherical tensors. TFN and HEGNN exhibit correct behavior when equipped with sufficiently high spherical harmonic order (e.g., mmax 2 2), aligning with theoretical expectations from GWL test.
DualEqui achieves similarly expressive results. When Imax 2 2, it consistently attains 100% accuracy when L Imaz, matching the expressive power of TFN/HEGNN.
D.5 Computational Efficiency Analysis
To assess the computational efficiency of DualEquiNet, we compare the training time epoch and memory usage across several baseline models, shown in Table While DualEquiNet introduces additional overhead due to its dual-space message passing and spherical harmonics computations, remains significantly more efficient than high-order tensor-based models like TFN, with training time comparable to FastEGNN and GotenNet. This demonstrates that our design strikes a favorable balance between computational cost and model expressivity, enabling scalable training on large biomolecular systems.
25

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Table The expressivity in the L-fold task with 2,3,5, 10. Anomalous results are highlighted in red and expected in green DualEqui using the high-order spherical harmonic features as expressive as TEN and HEGNN.
Folds
2
3
10
SchNet
50.0=0.0
50.0+0.0
50.0+0.0
50.0=0.0
EGNN
50.0=0.0
50.0+0.0
50.0+0.0
50.0=0.0
GVPGNN
91.5=18.8
50.5+5.0
66.5+23.5
50.0=0.0
FastEGNN
50.0=0.0
50.0+0.0
50.0+0.0
50.0=0.0
TEN/HEGNN Lynxx
50.0=0.0
50.0=0.0
50.0+0.0
50.0=0.0
TEN/HEGNN Imax 2
100.0=0.0
50.0+0.0
50.0+0.0
50.0=0.0
TFN/HEGNN Ima 3
100.0=0.0
100.0#0.0
50.0=0.0
50.0=0.0
TFN/HEGNN Imaz 5
100.0=0.0
100.0=0.0
100.0=0.0
50.0=0.0
TFN/HEGNN Imax 10
100.0=0.0
100.0#0.0
100.0=0.0
100.0=0.0
QuilEqui baa
50.5+5.0
50.0=0.0
50.0=0.0
50,5=5.0
DualEqui baa
100.0=0.0
50.0+0.0
63.5+22.2
15.5
DualEqui Imx 3
100.0=0.0
100.0+0.0
55.0#15.0
60.0=20.0
DualEqui mma
100.0=0.0
100.0+0.0
100.0=0.0
50.0=0.0
DualEqui Lmox = 10
100.0=0.0
100.0+0.0
100.0=0.0
100.0=0.0
Table 9: Training time and memory usage of each model.
Model
Train Time (slepoch)
Memory (MB)
SchNet 4.361
628
EGNN
6.240
608
GVPGNN
11.172
10160
TFN
56,235
43970
FastEGNN
11.755
1954
HEGNN
6.320
1232
GotenNet
10.332
5894
DualEqui
10.990
2436
D.6 Datasets Details
74 consists of 4,082 synthetic RNA sequences, each 107-130 nucleotides long. It was developed to support the predictive modeling of RNA stability for mRNA vaccine development. Each nucleotide annotated with three quantitative properties: reactivity, degradation under pH10, and degradation under Mg pH10 conditions. These measurements reflect how susceptible each nucleotide to chemical degradation and serve as surrogates for RNA structural stability. Sequences with low signal-to-noise ratio (SNR < 1) are filtered out to ensure label reliability, following prior best practices. This dataset is especially suited for testing fine-grained, residue-level predictions. In experiments, we infer the 3D atomic structure for each sequence using RhoFold [65].
COVID Vaccine The Vaccine COVID-19 dataset
The Ribonanza-2k dataset 37 includes 2,260 RNA sequences, each around 177 nucleotides in length, annotated at the nucleotide level with reactivity values obtained from two chemical probing agents: DMS and 2A3. These chemical modifications reflect the accessibility and flexibility of each nucleotide, thereby encoding aspects of RNA's secondary and tertiary structures. The labels thus capture nuanced differences in base-pairing and folding landscapes across diverse RNA sequences, making the dataset benchmark for modeling structural flexibility.
Ribonanza
Tc- Ribbswiitthess The Te-Riboswitthes dataset 33 consists of 355 mRNA sequences, each 67-73 nucleotides long, paired with graph-level regression labels reflecting gene regulatory behavior in response to tetracycline binding. Specifically, the label represents the expression shift induced by the presence of a tetracycline switching factor, an RNA-based synthetic regulatory system widely
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Table 10: Dataset Statistics.
COVID
Ribonanza
Te-Riboswitches
RNASolo SASA
Protein SASA
TorsionAngle
Task Level
residue level
residue level
graph level
residue level
graph level
residue level
Target
reactivity & degradation 4082 107
reactivity
switching factor
SASA Value
SASA Value
torsion angles
# Sequences
2260
355
1018
718
1018 10
Sequence Length
66
216
# Labels
2
7
Avg. Atoms
363
531
216
249
649
249
studied in gene therapy and synthetic biology. Unlike COVIDVaccine and Ribonanza, which have residue-level labels, Teribo focuses sequence-level prediction, allowing evaluation of how local structure aggregates into global function.
RNASolo SASA The SASA dataset consists of RNA structures annotated with Solvent-Accesiibee Surface Area (SASA) values for each nucleotide. SASA quantifies the extent to which each nu- cleotide is exposed to solvent, providing important insights into RNA folding, stability, and interaction potential. We compute per-nucleotide SASA values by aggregating atomic-level SASA from experi- mentally resolved PDB structures in the RNASolo database (2]. These values are extracted using DSSR, which internally implements the Shrake and Rupley algorithm [66]. This method models solvent accessibility by rolling a spherical probe over the van der Waals surface each atom, and then estimating the exposed area based point sampling. The resulting SASA values reflect the 3D packing and surface topology of RNA molecules. Unlike earlier tasks focused on sequence-derived stability, RNASolo-SASA emphasizes the role of 3D structural precision and surface geometry. The dataset includes diverse set of RNA structures with variable lengths and topologies.
mRFP Protein SASA We utilize the mRFP protein dataset introduced by Stanton et al. al.[68] for the task of SASA prediction. This dataset was designed as part of realistic large-molecule sequence design benchmark, aiming to simulate practical protein engineering objectives, consisting of approximately 200-residue red fluorescent proteins (RFPs), where each protein sequence is annotated with ground-truth SASA values. We randomly select 718 proteins and the ground-truth 3D structures predicted by AlphaFold2 [41).
TorsionAngle The TorsionAngle dataset is designed to assess model's ability to recover fine- grained structural geometry by predicting residue-level torsion angles, including backbone and glycosidic dihedrals. These angles- sugles-such as a, B8 y, 6, c, Ç, and X-are crucial descriptors of RNA conformation and govern higher-order folding. We extract these angles from RNA structures in the RNASolo database also using the DSSR, which computes torsions directly from atomic coordinates. Each sample the dataset contains the full 3D atomic structure and a set of seven torsion angles per nucleotide. Because RNA torsions are sensitive to both local backbone rigidity and global conformation, this dataset provides challenging test for models to capture both short-range and long-range structural dependencies. All angles normalized into the range [-180%, 180%). and training uses a circular MAE loss described in Section 4.4
Here, we summarize all the datasets in Table 10
D.7 Comparison of Baselines
SchNet is invariant GNN using continuous filter convolutional layers based pairwise distances expanded in radial bases. It employs only scalar features, making predictions rotation- and translation- invariant. It lacks higher-order directional features and handles long-range interactions primarily through stacking layers or increasing cutoff distances.
EGNN achieves equivaauuve by scalarizing messages computed from pairwise distances and features to scale relative position vectors, updating Cartesian coordinates. Unlike SchNet, explicitly uses first-order geometric (Cartesian) features, capturing directional information. Long- range interactions are possible via dense connections, though it can become computationally expensive for large systems.
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GVPGNN maintains separate scalar and Cartesian vector features, updating them with an equivariant vector perceptron that couples vector updates with scalar invariants. This ensures -quiivaanre and improves geometric expressiveness over EGNN. It remains constrained to first-order Cartesian vectors and, like EGNN, relies on standard message passing for long-range information.
TEN utilizes spherical harmonics and Clebsch-Gordan tensor products to explicitly represent higher- order geometric correlations, achieving rigorous SE(3)-qquvarriane. While highly expressive, TFN's complexity is significantly greater than scalar-based methods. It primarily captures local interactions, with long-range anngrannge interactions requiring deep stacking or additional global modules.
FastEGNN extends EGNN by incorporating virtual nodes that aggregate and distribute global information, enhancing long-range communication without substantially increasing computational complexity. It preserves EGNN's E(3)-qquivariance and Cartesian features, improving efficiency on larger or distant interactions while maintaining EGNN's directional limitations.
HEGNN enhances EGNN's framework by integrating higher-order spherical harmonic features (beyond first-order vectors). It scalarizes tensor interactions for computational efficiency, achieving improved geometric expressiveness. While computationally manageable, it remains primarily focused on local interactions unless combined with specialized long-range mechanisms.
GotenNet employs geometry- attention with learned higher-order Cartesian tensors, avoiding explicit spherical harmonic computations while maintaining -(3--ivvivari Its attention mecha- nism efficiently captures long-range interactions and high-order geometric patterns, combining the benefits of expressive tensors with scalable global communication.
Table 11 summarizes key properties of baseline models in terms of equivariance, feature represen- tation, spherical harmonic usage, and ability to model long-range interactions, While many recent models achieve [(3)-qquivariance, they differ in expressivity and scalability. Invariant models like SchNet rely solely on scalar features and lack the capacity to model directional or long-range depen- dencies. Scalarization- based equivariant models such as EGNN and GVP-GNN use Cartesian vectors but are limited to local neighborhoods. High-order methods like TFN offer expressive SH-based tensors but suffer from inefficient tensor product calculations. In contrast, HEGNN, GotenNet, and DualEqui provide efficient SH updates, with DualEqui and FastEGNN uniquely supporting long-range modeling. DualEqui stands out for combining spherical representations with long-range communication, enabled by its cross-space design.
Table 11: Comparison of baseline geometric GNN models. "Long-Range indicates whether the model has a built-in mechanism to capture long-range interactions efficiently. Designs with higher expressivity or efficiency highlighted in green
Model
Equivariance
Feature Type Efficient SH Update
Long-Range
SchNet
Invariant
Scalar
X
EGNN
Equivariant
Cartesian
GVP-GNN
Equivariant
Cartesian
X
TFN
Equivariant
Spherical
FastEGNN
Equivariant
Cartesian
V
HEGNN
Equivariant
Spherical
GotenNet
Equivariant
Spherical
X
DualEqui
Equivariant
Spherical
D.8 Reproducibility
All experiments this work are conducted under unified and controlled conditions to ensure fair comparison and reproducibility. For each dataset, we adopt an 8:1:1 split for training, validation, and testing. All reported results are averaged over 5 independent random splits, with standard deviations included. For each run, the best-performing model is selected based validation set performance and evaluated on the test set. Our implementation is built on Py Torch and Py PoTcch Geometric [25]. All code and datasets will be publicly available upon acceptance.
We apply consistent hyperparameter tuning for all models-includee baselines and our DualE- qui-using the Optuna [3] framework with 100 trials per method. Search spaces and best config-
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Table 12: Optimal hyperparameters DualEqui on each dataset.
Dataset
Covid
Ribonanza
Teribo
RNASolo SASA
mRFP Protein SASA
TorsionAngle
Latom
4
3
Lest
3
4
4
heads
2
8
5
hidden
72
128
84
150
174
140
alom dgu
4.82
3.93
3.03
10.44
3.68
8.93
residue dgu
26.4
30.61
122.64
47.02
116.39
42.45
Ir
0.0030
0,0015
0,0002
0,0002
0,0005
0,0013
weight decay
1.37e-6
5.65e-8
1.17e-7
1.37e-5
3.00c-7
2.10e-7
Imaz atom dss
2
3
0.97
1.0
0.95
1.00
0.96
0.99
residue des
0.97
0.95
0.92
0.95
0.95
0.91
batch size
32
25
32
16
16
16
epochs 1000
1000
1000
1000
1000
1000
urations are detailed in Appendix D.5. To ensure consistency, we fix random seeds for all runs (PyTorch, NumPy, CUDA) and repeat each experiment across 5 different seeds. All experiments were conducted on single NVIDIA A6000 GPU with a number of parameters limit up to x 10 In Table 12] we provide our searched optimal hyperparameters of DualEqui on each dataset. More detialed training scripts will be released with the code.
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Reviews:
[{"review_id": "vhol6DzHin", "reviewer": "Reviewer_Yoot", "summary": "DualEquiNet integrates both euclidean and spherical harmonic features into a message passing network and proposes a novel pooling mechanism to integrate the euclidean and spherical harmonic spaces.", "questions": "1. Could you explicitly define the architecture and design choices of the “fully euclidean” model used in your ablations? Specifically, does it include multi-head attention and the same network depth/width as DualEquiNet? Additionally, how do its parameter count and training hyperparameters compare to the other baseline models such as EGNN, TFN, and GotenNet?\n\n2. Given that the fully euclidean variant already significantly outperforms established baselines, could you disentangle the performance gains attributable solely to the spherical harmonic integration? For example, can you provide ablations across tasks (especially SASA and torsion angle prediction) that isolate the effect of the cross-space interaction pooling (CISP) and the spherical message passing components?\n\n3. Have you run the ablations with a larger number of seeds to confirm the statistical significance of these improvements? If so, could you provide those results, including confidence intervals or standard errors, across all relevant tasks?", "limitations": "yes", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "STRENGTHS \n\nThe paper is well structured and easy to follow. They clearly explain how they define the euclidean and spherical harmonic neighborhoods, the node feature and coordinate updates which utilize both euclidean and spherical harmonic features, and the novel (CISP) cross space interaction pooling mechanism which mixes the euclidean and spherical spaces through projection functions. They run experiments on a synthetic dataset N-chain task, an RNA property prediction task on three real datasets, and a surface area prediction task where they show significant improvements to many current baselines. They also demonstrate similar performance to GotenNet on an angle prediction task.   \n\nWEAKNESSES\n\nThe paper runs an ablation study on the RNA property datasets and report the performance of the model when removing various components. A current outstanding flaw is that \\textit{it is not clear where the substantial improvements against current baselines are coming from}. For context, the proposed DualEquiNet achieves $0.353 \\pm 0.006$, $0.558 \\pm 0.123$, $0.636 \\pm 0.049$ on the COVID, Ribonanza, Tc-Ribo datasets respectively, whereas the single space Euclidean model achieves $0.362 \\pm 0.010$, $0.592 \\pm 0.028$, and $0.699 \\pm 0.010$. By integrating both euclidean and spherical harmonic features there is a mean performance improvement of $\\Delta_1 = -0.009$, $\\Delta_2 = -0.034$, $\\Delta_3 = -0.063$; normalizing by the DualEquiNet standard deviations on the datasets we get $\\frac{-0.009}{0.006}=-1.5\\sigma$, $\\frac{-0.034}{0.123}=-0.27\\sigma$, $\\frac{-0.063}{0.049}=-1.28\\sigma$ which shows some improvement to the fully euclidean model. (These numbers are calculated with 5 seeds—it would be great if the authors could post ablation results with more runs if computationally feasible.) \n\nHowever, to put these numbers in context, the fully euclidean model outperforms all the current reported baselines by a much larger delta (with the exception of TC-ribo; but even GotenNet performs worse than the fully euclidean model on TC-ribo). The fully euclidean model gets $0.362 \\pm 0.010$, $0.592 \\pm 0.028$, and $0.699 \\pm 0.010$ whereas the next best benchmarked baseline gets $0.475 \\pm 0.009$ (GotenNet), $0.839 \\pm 0.020$ (FastEGNN), $0.694 \\pm 0.008$ (TFN), which lead to deltas of $\\Delta_1 = -0.113$, $\\Delta_2 = -0.247$, $\\Delta_3 = 0.005$. It is unclear why the fully euclidean model would outperform the other baselines to this extent and without further clarification it suggests that the lion's share of the improvement of the model is due to factors beyond the integration of spherical harmonic features and euclidean features. \n\nWhile it is not explicit what the fully euclidean model entails (it would be helpful if the authors could clarify this), looking at the various sections we assume that it means that we have a messaging passing framework where \n\n • The neighborhoods are defined by a threshold $N_{(i)} = \\{j | ||x_i - x_j|| \\leq d_{EU} \\}$\n\n • The message has the typical definition $m_{ij} = \\phi([h_i, h_j, ||x_i - x_j ||])$ \n\n • The update calculation uses a network which aggregates the calculation of K heads, where each head has a different message \nfunction and moreover utilizes attention to weight the messages $$\\Delta h_{EU} = \\frac{1}{K} \\sum_{k=1}^{K} \\phi_{upd} ([h_i, \\Sigma_{j \\in N(i)} \\alpha^k_{ij} \\phi^k_{h}(m_{ij})])$$ and similarly for the coordinate update $$\\Delta x_{EU} = \\frac{1}{K} \\sum_{k=1}^{K} \\sum_{j \\in N(i)} \\alpha^k_{ij} \\phi_{x}(m_{ij}) x_{ij}])$$\n\nFrom traditional message passing networks, it seems like the major differences is that attention is additionally calculated to weight the various messages and moreover that there are K many heads which each define a message network. \n\nGiven this, we do not see a clear reason why this version of the euclidean only message passing network would lead to a substantial performance improvement to current baselines (ex: a simple graph network like EGNN) beyond the additional attention calculation and the K-many heads.   \n(Thus, it would help if the authors could report the size of the different models—ablated versions and baselines—to see if there is a fair comparison in terms of model size. It also would be helpful to have the ablation experiments run for the SASA and torsion angle prediction to see if there is a similar behavior going on.)\n\nWithout further clarification, it is difficult to measure how much the integration of spherical harmonic features with euclidean features is adding to the existing body of work. Currently, the recommendation stands at 3—borderline reject—but this rating can and will change if the authors provide evidence that their integration of euclidean and spherical harmonic features leads to substantive improvement to existing work.", "was_revised": true, "final_justification": "Based on the new data and clarification, I have increased the score.", "rebuttal": "We thank the reviewer for their valuable feedback, and we address the concern below.   \n\n&nbsp;\n\nQ1: Ablation explanations and difference of our euclidean only message passing version with other euclidean baselines (such as EGNN) : \n\nWe appreciate the reviewer’s observation and clarify a key source of confusion: our fully Euclidean model (DualEquiNet w/o CSIP/Cross/Dual in Ablation Table 5) is not equivalent to standard Euclidean GNN baselines. Even in this ablation, we preserve the biological organization of large molecules (e.g., atom-to-nucleotide for RNA, atom-to-amino acid for proteins) using hierarchical mean pooling. Specifically, “w/o CSIP” replaces cross-space interaction pooling with standard mean pooling, but the pooling is still applied over biologically defined hierarchical units rather than only individual atoms. This explicit incorporation of biologically meaningful structure is one of our key contributions, ensuring that even our Euclidean-only variant captures context beyond atom-level interactions used by standard atom-level Euclidean GNNs, which explains its stronger performance, and hence it already outperforms those baselines. We will make it more explicit in camera-ready version.\n  \nOriginally, we did not apply hierarchical pooling to the baselines to respect their original designs, and some are not directly compatible with such pooling. However, we have now run selected Euclidean baselines augmented with hierarchical mean pooling. Their performance improves over their original versions (reported in Table 1) and matches our Euclidean-only variant but still lags significantly behind the full DualEquiNet with cross-space interaction pooling (CSIP) and dual-space message passing (e.g., 4.6%, 8.7%, and 9.0% gains over best EGNN Pooling on COVID, Ribonanza, and Tc-Ribo datasets respectively). \n\n|                    |             |             |             | COVID       |              |             | Ribonanza   | Tc-Ribo     |\n|--------------------|-------------|-------------|-------------|-------------|--------------|-------------|-------------|-------------|\n|                    | Reactivity  | pH10        | Mg pH10     | MCRMSE      | DMS          | 2A3         | MCRMSE      | Factor      |\n| EGNN + Pooling     | 0.280±0.006 | 0.469±0.006 | 0.362±0.010 | 0.370±0.009 | 0.688±0.028  | 0.534±0.014 | 0.611±0.010 | 0.699±0.016 |\n| SchNet + Pooling   | 0.302±0.006 | 0.495±0.010 | 0.384±0.010 | 0.394±0.010 |  0.721±0.019 | 0.642±0.004 | 0.681±0.011 | 0.702±0.007 |\n| FastEGNN + Pooling | 0.379±0.002 | 0.566±0.009 | 0.454±0.008 | 0.467±0.007 | 0.773±0.024  | 0.819±0.006 | 0.796±0.013 | 0.721±0.013 |\n| DualEquiNet        | 0.272±0.004 | 0.448±0.010 | 0.340±0.005 | 0.353±0.006 | 0.611±0.025  | 0.505±0.024 | 0.558±0.012 | 0.636±0.049 |\n\n**Integration of euclidean and spherical harmonic features leads to substantive improvement to existing work:** Beyond the explanation provided above, integrating the euclidean and spherical harmonic features indeed provide consistent percentage improvements across datasets. For example, removing CSIP increases error by ~8% on Tc-Ribo and ~7% on Ribonanza, while disabling cross-space message passing further increases error by ~9% on Tc-Ribo. The combined removal of all dual-space components yields the largest degradation, with ~11% higher error on Tc-Ribo and ~7% on Ribonanza. Overall, these results indicate that DualEquiNet’s architectural contributions—particularly contextual pooling and cross-space learning—are synergistic and yield up to ~10% improvements on different tasks.\n\n&nbsp;\n\n\nQ2. Architecture and design choices of our “fully euclidean” model/parameter count and training hyperparameters compared to other baselines:   \n\nWe believe this question likely arises from the confusion addressed in the point above but we provide further details for completeness. DualEquiNet w/o CSIP/Cross/Dual uses the same design as the full-version DualEquiNet, except the ablated part, i.e., they use the same multi-head attention and the same network depth/width. For DualEquiNet and all other baselines, we constrained the number of parameters to at most 4 million and ensured that all models fit within the memory of a single NVIDIA A6000 GPU. Within this resource budget, we used Optuna to perform hyperparameter optimization and select the best configuration for each model. This ensures that our comparisons are fair and conducted under consistent computational and hyperparameter constraints for all models and baselines.   \n\n\n&nbsp;\n\n\n\nQ3. Ablation study with 10 random seeds: \n\nAs suggested by the reviewer, we have run ablations with 10 random seeds. The results are consistent with those reported in Table 5 of the paper, with similar performance and comparable standard deviations. Due to rebuttal time constraints, we were able to run it on four tasks (see table below). We found a typo in the reported standard deviation for DualEquiNet on the Ribonanza dataset in Table 5: it should be “0.558 ± 0.012” instead of “0.558 ± 0.123”. The correct value is also reflected in Table 1 of the paper and only the ablation Table 5 has this typo. With this correction, the normalized improvement on Ribonanza becomes 0.038/0.012, which is approximately 3.17x standard deviations using 10 seeds, or 0.034/0.012, which is about 2.83x standard deviations using 5 seeds. We will correct this typo in the camera-ready version.  \n\n\n|                     | COVID ↓     | Ribonanza ↓ | TcRibo ↓    | TorsionAngle ↓ |\n|---------------------|-------------|-------------|-------------|----------------|\n| DualEquiNet         | 0.353±0.006 | 0.558±0.012 | 0.636±0.049 | 15.872±2.032   |\n| w/o CSIP            | 0.364±0.012 | 0.585±0.013 | 0.687±0.020 | 16.921±1.375   |\n| w/o CSIP/Cross      | 0.363±0.012 | 0.588±0.020 | 0.694±0.015 | 17.413±1.404   |\n| w/o CSIP/Cross/Dual | 0.368±0.011 | 0.596±0.028 | 0.696±0.008 | 17.459±1.331   |\n\n\n&nbsp;\n\n\nPlease let us know if there are any remaining concerns. If we have addressed your questions, we kindly request you to consider updating your score. Thank you very much for your time and thoughtful review!"}, {"review_id": "Q7QlaMLfVp", "reviewer": "Reviewer_cWyf", "summary": "This submission presents DualEquiNet, a graph neural network (GNN) for the simulation of macromolecules such as RNA and proteins.   This submission proposes a dual-space representation method that concurrently functions in both Euclidean space (for local atomic interactions) and Spherical Harmonics space (for global structural relationships).  The authors suggest bidirectional cross-space message transfer between these complementary representations and a Cross-Space Interaction Pooling (CSIP) method that facilitates hierarchical aggregation from atoms to residues.  These improvements enable DualEquiNet to proficiently capture both intricate atomic details and extensive connections essential for biomolecular modeling.  The authors assess their methodology across several RNA and protein property prediction tasks, using two newly established 3D structural benchmarks (SASA and torsion angle prediction), demonstrating consistent improvements over current geometric GNN baselines.", "questions": "1. How would DualEquiNet work with more complex biomolecular systems, like protein-RNA interactions? \n\n2. The SH-space neighborhood is implemented by a threshold $d_{SH}$ and the cosine similarity between points. Could you provide some insights on how to implement an adaptive threshold mechanism to prevent tuning hyperparameters?\n\n3. Could you provide empirical results or discussion with models specifically designed for long-range tasks, such as [1,2,3], as they also share a similar design idea?\n\nReferences:\n\n[1] UniIF: Unified Molecule Inverse Folding. In NeurIPS, 2024.\n\n[2] Neural atoms: Propagating long-range interaction in molecular graphs through efficient communication channel. In ICLR, 2024.\n\n[3] Ewald-based Long-Range Message Passing for Molecular Graphs. In ICML, 2023.", "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:**\n\n1. The submission shows a cool way to make complementary representations in both Euclidean and Spherical Harmonics spaces. This fixes a problem with the current GNNs that use geometry. With two spaces, this design lets the model see both atomic interactions in a small area and structural patterns in a large area at the same time.\n2. The Cross-Space Interaction Pooling mechanism that was suggested is a great way to combine features at the atomic level into representations, without losing the natural hierarchical structure of biomolecules.\n3. Extensive experiments are conducted to verify the effectiveness of the proposed DualEquiNet. The authors also perform ablation studies to provide further insight into the method designed.\n\n**Weaknesses:**\n\n1. The submission states that the model works well, but it doesn't go into much detail about how hard it is to compute in the main text, instead putting it in the appendix. There needs to be more upfront discussion about the resources it needs compared to baselines, especially for the \"large biomolecules\" it targets.\n\n2. The main point of this submission is that it can better model long-range dependencies. But most of the experimental comparisons are only between other equivariant GNNs. To support the claims about long-range modeling, it would be more convincing to compare it to models that are made for long-range tasks and also use a dual-space representation [1,2,3]. This kind of comparison would help separate the performance gains, provide stronger evidence of the effectiveness of the proposed method.\n\n3. The proposed architecture adds several new modules (dual-neighborhood construction, bidirectional cross-space attention, CSIP projections) and new hyperparameters without clear instructions on how to set them. Without design heuristics or simpler versions, it is hard for practitioners to adapt, tune, or extend the method to new datasets. This could make it harder to reproduce and use it more widely.\n\nReferences:\n\n[1] UniIF: Unified Molecule Inverse Folding. In NeurIPS, 2024.\n\n[2] Neural atoms: Propagating long-range interaction in molecular graphs through efficient communication channel. In ICLR, 2024.\n\n[3] Ewald-based Long-Range Message Passing for Molecular Graphs. In ICML, 2023.", "was_revised": true, "final_justification": "Based on the rebuttal comments and the paper quality, I propose to accept this submission.", "rebuttal": "W1: The submission states that the model works well, but it doesn't go into much detail about how hard it is to compute in the main text, instead putting it in the appendix. There needs to be more upfront discussion about the resources it needs compared to baselines, especially for the \"large biomolecules\" it targets.    \n\nRW1:   \nFollowing reviewer's recommendation, we will move this discussion from the Appendix to the main text in the camera-ready version. To emphasize, our DualEquiNet has substantially lower compute compared to tensor-product-based models such as TFN or SE(3)-Transformer (complexity $O(L^6)$) because DualEquiNet relies only on inner products over spherical harmonic features ($O(L)$). In practice, on the COVID dataset with identical hyperparameter settings, DualEquiNet achieves 3–5× faster training on large biomolecules compared to these tensor-product-based models, while maintaining strong predictive performance (see also table below for time per epoch and GPU memory requirements).    \n\n|          | Time (s/epoch) | Memory (MB) |\n|:--------:|:--------------:|:-----------:|\n|  SchNet  |      4.361     |     628     |\n|   EGNN   |      6.240     |     608     |\n|  GVPGNN  |     11.172     |    10160    |\n|    TFN   |     56.235     |    43970    |\n| FastEGNN |     11.755     |     1954    |\n|   HEGNN  |      6.320     |     1232    |\n| GotenNet |     10.332     |     5894    |\n| DualEquiNet |     10.990     |     2436    |\n\nAdditionally, for all baselines and our method, we constrained the number of parameters to at most 4 million and ensured that all models fit within the memory of a single NVIDIA A6000 GPU. Hyperparameters for each model were optimized using Optuna under the same computational budget, ensuring fair comparisons.      \n\n\n&nbsp;\n\n\n\nW2: Comparison with other long-range baselines.  \n\nRW2: We appreciate the reviewer’s suggestion to include additional baseline comparisons. Our original submission **already compared against a recent long-range model (FastEGNN)**. FastEGNN passes global information by introducing a small set of virtual nodes that connect to all real nodes and each other, enabling global message passing on graphs.   \n\nFollowing your recommendation, we further considered three additional baselines (UnIF, Neural atoms and EWALD). Unfortunately, to the best of our knowledge, the implementation of UniF is not publicly available; therefore, we conducted experiments with Neural atoms and EWALD. Specifically, we employed SchNet and DimeNet enhanced with Ewald-based message passing and Neural Atom interactions, respectively as reported in their original works. The results, summarized below, show that our proposed **DualEquiNet consistently outperforms these strong baselines** across all evaluated datasets (RMSE reported below).     \n\n\n|                      |             |             |             | COVID       |             |             | Ribonanza   | Tc-Ribo     |\n|----------------------|-------------|-------------|-------------|-------------|-------------|-------------|-------------|-------------|\n|                      | Reactivity  | pH10        | Mg pH10     | Avg.        | DMS         | 2A3         | Avg.        | Factor      |\n| SchNet + Ewald       | 0.392±0.004 | 0.583±0.010 | 0.459±0.008 | 0.478±0.009 | 0.845±0.020 | 0.884±0.002 | 0.865±0.014 | 0.736±0.004 |\n| SchNet + Neural Atom | 0.409±0.004 | 0.596±0.009 | 0.481±0.006 | 0.495±0.008 | 0.854±0.015 | 0.919±0.009 | 0.886±0.012 | 0.726±0.002 |\n| Dimnet + Ewald       | 0.403±0.005 | 0.594±0.006 | 0.480±0.004 | 0.492±0.005 | 0.828±0.011 | 0.896±0.011 | 0.862±0.003 | 0.676±0.044 |\n| Dimnet + Neural Atom | 0.411±0.009 | 0.602±0.016 | 0.489±0.013 | 0.501±0.015 | 0.825±0.010 | 0.892±0.013 | 0.859±0.005 | 0.679±0.006 |\n| DualEquiNet          | **0.272±0.004** | **0.448±0.010** | **0.340±0.005** | **0.353±0.006** | **0.611±0.025** | **0.505±0.024** | **0.558±0.012** | **0.636±0.049** |\n\n\n\n&nbsp;\n\n\nW3: The proposed architecture adds several new modules (dual-neighborhood construction, bidirectional cross-space attention, CSIP projections) and new hyperparameters without clear instructions on how to set them. Without design heuristics or simpler versions, it is hard for practitioners to adapt, tune, or extend the method to new datasets. This could make it harder to reproduce and use it more widely.     \n\nRW3: Thank you for pointing this out. We argue that we did not introduce many new hyperparameters. **The only new hyperparameter we added is the threshold used to construct SH-based neighborhoods ($d_{SH}$)**. In other modules, such as CSIP, we make the coefficients learnable by design—specifically to reduce manual tuning and avoid introducing additional hyperparameters, as the reviewer noted.    \nFor $d_{SH}$, we acknowledge it is a unique component of our method and have discussed possible adaptive thresholding mechanisms in our response to RQ2 in this thread below. Additionally, if one prefers not to search over $d_{SH}$, a practical heuristic can be applied:\nWe can compute the distribution of SH similarities (i.e., cosine similarity between SH vectors) across all node pairs in the dataset, and then **select a threshold such that the number of SH-based edges is approximately equal to the number of Euclidean edges**. This balances the neighborhood density between Euclidean and SH spaces and provides a data-driven way to set $d_{SH}$ without tuning.    \nAdditionally, DualEquiNet has two depth-related parameters—$L_{atom}$ and $L_{res}$—which specify the number of layers in the atom- and residue-level modules. These are analogous to layer counts in standard GNNs, which are typically shallow. We searched over a small space {1,2,3,4} making this tuning lightweight and practical, so we don’t count this as a new hyperparameter.\nOverall, we have deliberately designed the architecture to minimize new hyperparameters and make it accessible for adaptation to new datasets. We will clarify this in the revised version.    \n\n\n&nbsp; \n\n\nQ1: How would DualEquiNet work with more complex biomolecular systems, like protein-RNA interactions?   \n\nRQ1: Following the reviewer’s suggestion, we evaluated DualEquiNet on the RNAProt dataset from RNAGlib [1], which focuses on predicting protein–RNA interactions. The results (shown below) demonstrate that DualEquiNet achieves the highest accuracy among existing baselines, thereby validating its effectiveness for protein–RNA interaction tasks.     \n\n\n| Dataset     | RNAProt     |\n|-------------|-------------|\n| Metrics     | Accuracy (↑)   |\n| SchNet      | 0.709±0.005 |\n| EGNN        | 0.720±0.009 |\n| GVPGNN      | 0.710±0.005 |\n| TFN         | 0.697±0.005 |\n| FastEGNN    | 0.718±0.010 |\n| HEGNN       | 0.735±0.016 |\n| GotenNet    | 0.709±0.006 |\n| DualEquiNet | **0.749±0.015** |\n\n\n&nbsp;\n\n\nQ2: The SH-space neighborhood is implemented by a threshold and the cosine similarity between points. Could you provide some insights on how to implement an adaptive threshold mechanism to prevent tuning hyperparameters?    \n\nRQ2: We appreciate the reviewer’s insightful suggestion. To reduce reliance on a fixed threshold d_{SH}, we can consider the following alternatives in future work.\n(1) **Percentile-based thresholding**:\nEach node selects neighbors whose SH cosine similarity falls within the top-p percentile of all pairwise similarities to that node:\n$cos(r_i, r_j) ≥ Percentile_p({ cos(r_i, r_k) }_k)$.\nWhile $p$ remains a hyperparameter, it is scale-invariant and often easier to generalize across datasets than an absolute similarity threshold. It also adapts the number of neighbors per node based on local similarity distribution.   \n\n(2) **Learnable neighborhood gating**:  \nWe can further eliminate manual thresholding by introducing a learnable, soft connectivity mechanism. Specifically, for each node pair (i, j), we define: $η_{ij} = σ(w · cos(r_i, r_j) + b)$,\nwhere $\\sigma(\\cdot)$ is the sigmoid function, and $w, b$ are learnable parameters. $\\alpha_{ij}$ can be used directly as edge weights or to probabilistically construct neighborhoods. This approach is fully adaptive and differentiable, enabling end-to-end learning of neighborhood structure in SH space.\n\nThese strategies offer practical and principled alternatives to manual threshold tuning. We plan to explore and benchmark them in future work.\n\n\n\n&nbsp;\n\n\n\n\nQ3: Could you provide empirical results or discussion with models specifically designed for long-range tasks, such as [1,2,3], as they also share a similar design idea?  \nRQ3: We have now provided the comparison to the other long-range methods that you recommended. Please see RW2 earlier in this thread for the results demonstrating our DualEquiNet outperforms other long-range methods.\n\n&nbsp;\n\nPlease let us know if there are any remaining concerns. Thanks a lot!     \n\n\n&nbsp;\n&nbsp;  \n\n\nReference:     \n[1] A Comprehensive Benchmark for RNA 3D Structure-Function Modeling."}, {"review_id": "zmlfBCTttj", "reviewer": "Reviewer_DmGV", "summary": "The authors present DualEquiNet, a novel geometric graph neural network operating in joint Euclidean-spherical harmonics spaces for biomolecular representation learning. The authors' methodology is interesting, and their experimental results are promising. Nonetheless, a few concerns remain regarding the evaluation of their proposed methodology.", "questions": "1. What do the authors think about simpler approaches to modeling 3D biomolecules that have emerged over the last 12 months, such as vanilla Transformer encoders trained with rotation data augmentation (e.g., AlphaFold 3)? Do they think such simpler methods may hold some promise in future work as generic biomolecular representation learners, or do the authors think there are fundamental limitations to such (generic) approaches that methods such as DualEquiNet address?", "limitations": "The authors haven't thoroughly discussed the computational complexity (time and memory) of their proposed method, so it's currently unclear how scalable it truly is.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Points of strength:**\n1. The authors' methodology is well-motivated, sound, and interesting. I particularly appreciated the authors' depth and clarity in discussing the importance of local-to-global (and vice versa) representation learning on biomolecular graphs. Using spherical harmonics to bridge this gap is a compelling idea.\n2. The authors evaluate their new method on a variety of protein and RNA datasets, and in all cases, their results are positive.\n3. The authors' writing style is clear and consistent, making it easy for readers to grasp the main takeaways from each section of the manuscript.\n\n**Points for improvement:**\n1. My main concern is that the authors' empirical results with DualEquiNet are preliminary. Tables 1-2 are compelling experiments demonstrating the method's expressiveness. However, Tables 3-4 struck me as a bit unrealistic from the perspective of how this method might be applied in a drug discovery setting. To thoroughly demonstrate how DualEquiNet compares to leading e.g., protein representation learning methods, I would highly recommend the authors report the results of their method for standardized protein and RNA representation learning benchmarks such as ProteinWorkshop [1] and rnaglib [2] in their supplementary materials.\n2. In theory, it's great to see the authors' method working well in Tables 2-4, but after considering the authors' ablation studies in Table 5, I'm left with the feeling that DualEquiNet is outperforming the baselines for reasons beyond its key architectural components, as ablating several of them doesn't change the method's average RNA performance much. I'd like the authors to discuss why the components aren't as necessary as readers may initially be led to believe and what other reasons may be leading DualEquiNet to achieve top performance for RNA property prediction (in the ablations).\n3. The authors highlight the importance of E(3) equivariance for 3D biomolecules, but they don't discuss the relationship between E(3) and SE(3) equivariance for modeling molecular chirality. As such, it's currently unclear if the authors' proposed method is truly E(3)-equivariant and therefore can't detect chirality or whether it is actually SE(3)-equivariant and can detect chirality. A discussion of this point would be helpful.\n\n**References:**\n\n[1] Jamasb, A. R., Morehead, A., Joshi, C. K., Zhang, Z., Didi, K., Mathis, S. V., ... & Blundell, T. L. Evaluating Representation Learning on the Protein Structure Universe. In The Twelfth International Conference on Learning Representations.\n\n[2] Wyss, L., Mallet, V., Karroucha, W., Borgwardt, K., & Oliver, C. (2025). A Comprehensive Benchmark for RNA 3D Structure-Function Modeling. arXiv preprint arXiv:2503.21681.", "was_revised": true, "final_justification": "The authors have addressed my concerns regarding RNA benchmarking, ablations, and methodological novelty.", "rebuttal": "We thank the reviewer for the valuable feedback, and we address the concern below.  \n\nW1: Significance of the evaluation datasets in Tables 3 and 4, and new results on RNAGlib. \n\nRW1:   \nSignificance of Tables 3 and 4 : Tables 3 and 4 focus on torsion angle prediction and solvent-accessible surface area (SASA) estimation. These tasks were chosen specifically because they capture fine-grained 3D geometric features that are highly relevant in drug discovery. **Torsion angles** play a central role in determining local conformations of biomolecules, directly influencing active site geometry and ligand binding modes. Accurate torsion angle prediction is critical for modeling flexible regions such as RNA loops and protein side chains, which are often key determinants of druggability. Experimental techniques such as NMR, X-ray crystallography, and cryo-EM can provide this information but are costly and time-consuming, motivating accurate learning-based approaches.  \n**SASA** is a well-established property used in structure-based drug design to identify hydrophobic and hydrophilic regions, estimate ligand accessibility, and correlate with binding affinities. Models capable of reliably predicting SASA provide valuable insight for guiding ligand optimization and binding site characterization. \n\nThese two tasks are widely used in molecular modeling literature [1][2][3] to benchmark a model’s ability to learn detailed structural properties in drug discovery applications.   \n\nWe thank the reviewer for this valuable suggestion. In response, we have conducted additional experiments on the RNAGo and RNAProt datasets from **RNAGLib** and will include the results in the camera-ready version. DualEquiNet achieves state-of-the-art performance on these as well, further validating its effectiveness (see Table below). We also note that the RNA property prediction benchmarks reported in the main paper (e.g., Ribonanza, COVID and Tc-Ribo) are themselves widely used and established in works [4][5][6], ensuring that our evaluation is based on community-accepted standards and datasets. Due to time constraints, experiments on the ProteinWorkshop datasets, which are much larger, are not feasible now. We plan to include these results in the camera-ready version. We will update all references accordingly.  \n\n|             |     RNAGo     |   RNAProt   |\n|:-----------:|:-------------:|:-----------:|\n|             | Jaccard Index |   Accuracy  |\n|    SchNet   |  0.613±0.020  | 0.709±0.005 |\n|     EGNN    |  0.646±0.060  | 0.720±0.009 |\n|    GVPGNN   |  0.632±0.048  | 0.710±0.005 |\n|     TFN     |  0.651±0.185  | 0.697±0.005 |\n|   FastEGNN  |  0.527±0.159  | 0.718±0.010 |\n|    HEGNN    |  0.661±0.038  | 0.735±0.016 |\n|   GotenNet  |  0.555±0.170  | 0.709±0.006 |\n| DualEquiNet |  **0.681±0.047**  | **0.749±0.015** |\n\n\n&nbsp;\n\n\nW2: Significance of architectural components through ablation.  \nRW2:   \n(1) **Ablating components in DualEquiNet indeed changes the performance.**\nThe absolute RMSE differences in Table 5 translate to consistent percentage improvements and are data-dependent. For example, removing CSIP increases error by ~8% on Tc-Ribo and ~7% on Ribonanza, while disabling cross-space message passing further increases error by ~9% on Tc-Ribo. The combined removal of all dual-space components yields the largest degradation, with ~11% higher error on Tc-Ribo and ~7% on Ribonanza. Overall, these results indicate that DualEquiNet’s architectural contributions—particularly contextual pooling and cross-space learning—are synergistic and yield up to ~10% improvements on different tasks.  \n\n(2) **Explanation of results in Table 5.**   \nDualEquiNet w/o CSIP/Cross/Dual in Ablation Table 5 is **not equivalent to standard Euclidean GNN baselines**. Even in this ablation, we preserve the **biological organization of large molecules** (e.g., atom-to-nucleotide for RNA, atom-to-amino acid for proteins) using hierarchical mean pooling. We also mentioned this in Line 346 in our submission. Specifically, “w/o CSIP” replaces cross-space interaction pooling with standard mean pooling, but the pooling is still applied over biologically defined hierarchical units rather than only individual atoms. This explicit incorporation of biologically meaningful structure is **one of our key contributions**, ensuring that even our Euclidean-only variant captures context beyond atom-level interactions used by standard atom-level Euclidean GNNs, which explains its stronger performance, which explains why it already outperforms those baselines. We will make it more explicit in the camera-ready version.  \nOriginally, we did not apply hierarchical pooling to the baselines to respect their original designs, and some are not directly compatible with such pooling. However, we have now run selected Euclidean baselines augmented with hierarchical mean pooling. Their performance improves over their original versions (reported in Table 1) and matches our Euclidean-only variant but still lags significantly behind the full DualEquiNet with cross-space interaction pooling (CSIP) and dual-space message passing (e.g., 4.6%, 8.7%, and 9.0% gains over best EGNN Pooling on COVID, Ribonanza, and Tc-Ribo datasets respectively).  \n\n|                    |             |             |             | COVID       |              |             | Ribonanza   | Tc-Ribo     |\n|--------------------|-------------|-------------|-------------|-------------|--------------|-------------|-------------|-------------|\n|                    | Reactivity  | pH10        | Mg pH10     | MCRMSE      | DMS          | 2A3         | MCRMSE      | Factor      |\n| EGNN + Pooling     | 0.280±0.006 | 0.469±0.006 | 0.362±0.010 | 0.370±0.009 | 0.688±0.028  | 0.534±0.014 | 0.611±0.010 | 0.699±0.016 |\n| SchNet + Pooling   | 0.302±0.006 | 0.495±0.010 | 0.384±0.010 | 0.394±0.010 |  0.721±0.019 | 0.642±0.004 | 0.681±0.011 | 0.702±0.007 |\n| FastEGNN + Pooling | 0.379±0.002 | 0.566±0.009 | 0.454±0.008 | 0.467±0.007 | 0.773±0.024  | 0.819±0.006 | 0.796±0.013 | 0.721±0.013 |\n| DualEquiNet        | 0.272±0.004 | 0.448±0.010 | 0.340±0.005 | 0.353±0.006 | 0.611±0.025  | 0.505±0.024 | 0.558±0.012 | 0.636±0.049 |\n\n\n&nbsp;\n\n\nW3: The authors highlight the importance of E(3) equivariance for 3D biomolecules, but they don't discuss the relationship between E(3) and SE(3) equivariance for modeling molecular chirality. As such, it's currently unclear if the authors' proposed method is truly E(3)-equivariant and therefore can't detect chirality or whether it is actually SE(3)-equivariant and can detect chirality. A discussion of this point would be helpful.  \n\nRW3: **SE(3)-equivariant** models are equivariant to translations and rotations but not to reflections. This allows them to distinguish chirality because mirror images are treated as distinct configurations. Most fully SE(3)-equivariant models—such as Tensor Field Networks (TFN), SE(3)-Transformer, and SEGNN—achieve this by using Clebsch–Gordan tensor products over spherical harmonics, which provide full orientation-sensitive tensor features but introduce significant computational overhead (up to $O(L^6)$ for maximum tensor degree $L$). To avoid this cost, our DualEquiNet, like recent HEGNN and GotenNet models, uses spherical harmonic–based **inner products** instead of full tensor products. This design enables rich geometric reasoning at scale while remaining computationally efficient, but it also makes the model reflection-equivariant (E(3) rather than SE(3)). \nExtending DualEquiNet to an explicitly SE(3)-equivariant variant to handle chirality more directly is an exciting direction for future work.  \n\n\n&nbsp;\n\n\nQ1: Thoughts on potential of simpler transformer-Based 3D Biomolecular Models.  \nRQ1: We thank the reviewer for raising this timely question. We agree that recent developments, such as AlphaFold3, have shown that large-scale Transformer-based architectures with rotation data augmentation can implicitly learn geometric symmetries and achieve excellent results. However, as highlighted in [7] learning equivariance can provide greater data efficiency and generalization in many applications such as molecular property prediction and dynamics simulation. Similarly, [8] shows that explicit equivariance offers substantially improved **data efficiency and better compute efficiency following power-law scaling**.   \nDualEquiNet is designed precisely for such settings: it combines explicit geometric inductive biases with hierarchical biomolecular organization to achieve strong generalization even when labeled data is limited, while also supporting dual-space reasoning required for cross-scale biomolecular interactions. We view Transformer-based approaches trained with augmentation and DualEquiNet as complementary: Transformers excel for large-scale foundation models, while architectures like DualEquiNet remain crucial for data-limited and physics-sensitive problems.   \n\n&nbsp;\n\nPlease let us know if there are any remaining concerns. If we have addressed your questions, we kindly request you to consider updating your score. Thank you very much for your time and thoughtful review! \n\n&nbsp;\n\nReference:   \n[1] Accelerating Bayesian Optimization for Biological Sequence Design with Denoising Autoencoders.  \n[2] Deeprna-twist: Language model guided rna torsion angle prediction with attention-inception network.   \n[3] Rna-torsionbert: leveraging language models for rna 3d torsion angles prediction.  \n[4] Deep learning models for predicting rna degradation via dual crowdsourcing.  \n[5] Ribonanza: deep learning of rna structure through dual crowdsourcing.  \n[6] Tuning the performance of synthetic riboswitches using machine learning.  \n[7] Equivariance is dead, long live equivariance? (a blog post on Molecular Modelling).  \n[8] Does equivariance matter at scale?"}, {"review_id": "hL24x1euaD", "reviewer": "Reviewer_U3ua", "summary": "In this paper, the authors propose the DualEquiNet model to combined invariant and equivariant-based representation together. The key idea is to develop  a dual-space hierarchical equivariant network that constructs complementary representations in both Euclidean and spherical harmonics spaces to capture both invariant and equivariant properties. More importantly, the model employs bidirectional cross-space message passing and cross-space interaction pooling mechanism to hierarchically aggregate atomic features. The model has been validated based on several existing benchmarks and shows promising results.", "questions": "1. The definition for SH neighborhood is confusing! From the definition, it is not the distance between the two atoms that matter, instead if they have similar local structures (with similar orientation), they will be treated as neighbors. This is very weird. For instance, if we have two benenze ring in the structure (with similar orientations), then they will be regarded as neighbors. However, if two benzene rings are oriented differently, they are no longer neighbors. Why the authors choose to do it this way? Why not just use the standard SH-based neighborhood definition? Is there any ablation study to show that the specially-design neighborhood will have better performance?\n\n2. The ablation study part is not clear. Normally SH will contain more information that standard invariant-based graph models. State differently, SH-based models, such as E3NN, will have better performance than general GNNs, as they are more complicated and consider equivariant information. For the ablation study, the authors should consider just using SH-based model (w/o CSIP/Cross/Dual) instead of removing SH-based models.\n\n3. The benchmarks are not very widely used. More interesting dataset from MolecularNet should be considered. In particular, the prediction of SARA and torsion angles seems to be not proper for this case. Note that the model input has all the geometric information, and SARA and torsion angles are just special properties that can be analytically calculated.", "limitations": "N.A.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength: The developed Dual-Space-based learning architecture is novel and promising.\n\nWeakness: The message-passing for spherical harmonic features is not explained clearly. The definition of neighborhood is very confusing! The importance of two difference components is unclear!", "was_revised": true, "final_justification": "All my concerns are well addressed. I keep my score 4!", "rebuttal": "RQ1 & RQ2: About the difference between our SH neighborhood and spherical tensor-based baselines.    \n\nThank you for raising this important question. We believe there may be some misunderstandings, and we clarify the differences below.   \n\n(1) There are two distinct classes of methods that utilize spherical harmonics:    \n\n**Type 1**: **Methods based on spherical tensor products** using Clebsch–Gordan coefficients and Wigner-D matrices. These methods, such as TFN, SE(3)-Transformer, and SEGNN, perform full tensor contractions, which lead to high computational cost up to $O(L^6)$ [4], where L is the maximum tensor degree. This is because they compute tensor products across all combinations of degrees up to L. A detailed explanation can be found in Section 5.3 of [2]. While e3nn is a Python library and not a specific method, the above baselines rely on it for implementation.  \n\n**Type 2**: Even though the calculation of spherical tensor products is too expensive, the high-order spherical tensors are proved to contain more information [2]. To address the complexity  issue, some recent baselines still use spherical tensors but  try to replace the “tensor product process” with **“vector inner product”** (Inner product itself is $O(L)$). In this way, we incorporate the spherical tensor information while avoiding the complexity.  \nThis kind of methods includes HEGNN [5], GotenNet, and our DualEquiNet. Similar discussions can also be found in Sections 1 and 2 of HEGNN [5] and Sections 1 and 2 of GotenNet [6].   \nBecause **these two categories differ fundamentally in their computation and architecture**, it is not feasible to use a Type 1 method as an ablation for our Type 2-based approach.  \n\n(2) On our SH-based neighborhood definition:  \nThe SH-based neighborhood is a core contribution of our method and is not present in any existing baseline. It enables long-range, symmetry-aware interactions by linking nodes with similar local geometric environments, even if they are spatially distant—complementing standard Euclidean neighborhoods.  \nRegarding the reviewer’s concern: SH similarity is invariant to global rotations due to our equivariant design (see Appendix C.1 and B.2). Therefore, two benzene rings with similar local structures will still be linked regardless of their global orientation. Additionally, in both Type 1 and Type 2 methods, spherical tensors are computed based on directional vectors (i.e., relative positions), and are independent of absolute distances.  \n\nWe will revise the paper to make these distinctions and design choices more explicit.   \n\n\n&nbsp;\n&nbsp;\n\n\nQ3: The benchmarks are not very widely used. More interesting dataset from MolecularNet should be considered. In particular, the prediction of SARA and torsion angles seems to be not proper for this case. Note that the model input has all the geometric information, and SARA and torsion angles are just special properties that can be analytically calculated.    \n  \n**RQ3**:  Since there is no reference attached, we guess you mean the “MoleculeNet” instead of “MolecularNet” in [1]. The datasets in MoleculeNet, like QM9, are about small molecules and have been widely used in papers of the baselines. However, our **DualEquiNet aims to deal with the long-range dependencies in the large biomolecules**. Tasks in MoleculeNet are not well-suited for evaluating this capability. Therefore, instead of using those datasets of small molecules, we develop new datasets of large biomolecules like protein and RNA. We identify this as a contribution instead of a weakness of our paper.      \n\nIndeed, the SASA and torsion angles can be calculated if we have ground-truth 3D structures. However, in real-world scenarios, we don’t have access to the ground-truth 3D structures. Especially for RNA, the 3D structures are highly noisy. So here our point is how accurately we can infer the SASA and torsion angle if we only have noisy 3D structures?   \nTo this end, we ensure that ground-truth structures are used only to generate labels, while all models, including ours and the baselines, are provided only with noisy 3D structures derived from sequence-based prediction methods. This better simulates real-world conditions and tests a model’s capabilities.    \n\n\n&nbsp;\n\nWe appreciate the reviewer’s comment and will include any proposed changes in camera-ready version. If we have addressed your questions, we kindly request you to consider updating your score. Thank you very much for your time and thoughtful review!\n\n\n&nbsp;\n&nbsp;\n\n\nReference:      \n[1] MoleculeNet: A Benchmark for Molecular Machine Learning.    \n[2] A Hitchhiker’s Guide to Geometric GNNs for 3D Atomic Systems.   \n[3] On the Expressive Power of Geometric Graph Neural Networks.   \n[4] From Peptides to Nanostructures: A Euclidean Transformer for Fast and Stable Machine Learned Force Fields.   \n[5] Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks?    \n[6] GotenNet: Rethinking Efficient 3D Equivariant Graph Neural Networks."}]

Author rebuttal:

Meta-review: Summary:
This paper introduces DualEquiNet, which combines Euclidean distance–based neighborhoods with neighborhoods defined by cosine similarity of local environment spherical harmonic (SH) projections, and pools atom-level features into residue-level representations. The approach is novel and delivers strong empirical results on RNA/protein benchmarks, including new datasets introduced by the authors. The rebuttal added helpful comparisons, ablations, and significance tests.

Strengths:
- Strong empirical gains across RNA/protein tasks.
- Sensible architectural design: hierarchical pooling and dual neighborhood definitions that incorporate biomolecular structure.
- Careful additional experiments during the rebuttal (e.g., pooling baselines, long-range baselines, new datasets).
- Practical complexity advantages compared to tensor-product equivariant networks.

Concerns:
- Terminology: The term “dual space” has well-established meanings in mathematics (dual vector space) and physics (real vs. reciprocal space duality, defined only after fixing a group). One could in principle frame Euclidean vs. SH neighborhoods as translation vs. rotation representations of the Euclidean group, but the paper does not present it this way. Instead, “dual space” is used simply to mean “two neighborhood definitions,” which risks misleading readers.
- Local environment SH neighborhoods: The cutoff based on cosine similarity of SH projection vectors couples motif similarity with relative orientation similarity. This is distinct from invariant approaches (power spectrum, bispectrum) that capture geometric similarity independent of orientation. The paper should make this trade-off explicit and note invariant cutoffs as a natural future direction.
- Hierarchy and pooling: The Euclidean-only ablation of the authors’ model does include atom→residue pooling, but external baselines (e.g., SE(3)-Transformer, TorchMD-Net) were left atom-only. Thus, part of the observed gains may reflect hierarchical pooling in addition to the dual-neighborhood design. This asymmetry should be clearly acknowledged.

Decision:
On balance, the ideas and results justify acceptance. The work would be significantly strengthened by clearer framing, especially regarding terminology, the SH neighborhood design, and the role of pooling in comparisons. The camera-ready must include explicit clarifications on these points. I note that with a stronger conceptual framing, this contribution could rise from a solid poster to a spotlight or even oral-level paper.
Decision: Accept (poster)