REBIND: Enhancing Ground-state Molecular Conformation Prediction via Force-Based Graph Rewiring
Authors: Taewon Kim
Hyunjin Seo
Sungsoo Ahn
Eunho Yang
Venue: ICLR 2025 Poster (2025)
Keywords: molecular conformation prediction
molecule modeling
graph neural network
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: Force-aware edge rewiring technique to enhance conformation prediction of moleucles

Abstract: Predicting the ground-state 3D molecular conformations from 2D molecular graphs is critical in computational chemistry due to its profound impact on molecular properties. Deep learning (DL) approaches have recently emerged as promising alternatives to computationally-heavy classical methods such as density functional theory (DFT). However, we discover that existing DL methods inadequately model inter-atomic forces, particularly for non-bonded atomic pairs, due to their naive usage of bonds and pairwise distances. Consequently, significant prediction errors occur for atoms with low degree (ie., low coordination numbers) whose conformations are primarily influenced by non-bonded interactions. To address this, we propose ReBIND, a novel framework that rewires molecular graphs by adding edges based on the Lennard-Jones potential to capture non-bonded interactions for low-degree atoms. Experimental results demonstrate that ReBIND significantly outperforms state-of-the-art methods across various molecular sizes, achieving up to a 20% reduction in prediction error. The code is available in: https://github.com/holymollyhao/ReBIND

Published as a conference paper at ICLR 2025
REBIND: ENHANCING GROUND-STATE MOLECU-
LAR CONFORMATION PREDICTION VIA FORCE-BASED GRAPH REWIRING
Taewon 12= Hyunjin Seoti2,, SungSoo Ahn', Eunho Yang
Korea Advanced Institute of Science and Technology (KAIST)', Polymerize?, AITRICS3 (maxkim139, bella72, sungsoo. ahn, eunhoy ) @kkasst ac.kr
ABSTRACT
Predicting the ground-state 3D molecular conformations from 2D molecular graphs is critical in computational chemistry due to its profound impact molec- ular properties. Deep learning (DL) approaches have recently emerged as promis- ing alternatives to computationally -hevy classical methods such as density func- tional theory (DFT). However, we discover that existing DL methods inadequately model inter-atomic forces, particularly for non-bonded atomic pairs, due to their naive usage of bonds and pairwise distances. Consequently, significant prediction errors occur for atoms with low degree (i.e., low coordination numbers) whose conformations are primarily influenced by non-bonded interactions. To address this, we propose REBIND, novel framework that rewires molecular graphs by adding edges based on the Lennard-Jones potential to capture non-bonded inter- actions for low-degree atoms. Experimental results demonstrate that REBIND significantly outperforms state-of-the-ar methods across various molecular sizes, achieving up to 20% reduction in prediction error. The code available in: how.yooolllohhlelaD ReBIND
https: / //g/tttuuu com hwte///oollollllloeeeeee
INTRODUCTION
The ground-state conformation of molecule represents the lowest energy state on the potential en- ergy surface, where the inter-atomic forces are balanced at equilibrium. This 3D conformation of the molecule plays crucial role in determining the molecule's ''s physical, chemical, and biological properties. As a result, it is utilized in various applications such as molecular property predic- tion (Satorras et al., 2021; Schütt et al., 2021; Liu al., 2022; Thölke & De Fabritiis, 2022; Zhou et al., 2023a; Zaidi et al., 2023; Ni et al., 2024), drug discovery (Luo et al., 2021; Ganea et al., 2021; Jing al., 2022; Xu et al., 2022; Zhou al., 2023b; Tang et al., 2024), and protein-ligand interactions (Pei et al., 2024; Wang et al., 2024).
Recently, deep learning approaches (Hu* al., 2020; Xu al., 2021e; Brody al., 2022; Ram- pasek et al., 2022; Xu et al., 2024) have emerged as promising alternatives to reduce the computa- tional costs of ab initio calculations such as density functional theory (DFT; Kohn & Sham, 1965; Parr et al., 1979). These approaches focus on predicting 3D molecular conformations by leveraging graph neural networks (GNNs) with 2D molecular graphs as input. Central to most of these models the assumption that a 2D molecular graph, where atoms are represented as nodes and covalent bonds as edges, can effectively capture atomic interactions. Building on this assumption, they em- ploy iterative message- phessage-passing updates to predict molecular conformations, which are parallel the iterative force-bare based updates used for conformer optimization. Specifically, GNN-based methods rely on covalent bonds (Hu* et al., 2020; Brody al., 2022: Rampasek et al., 2022) and pairwise distances (Xu et al., 2024) during the message-passing phase to represent inter-atomic forces, en- couraging bonded and proximate atomic pairs be close in the representational space.
In this work, we challenge the assumption of existing approaches that bonds and inter-atomic dis- tances are sufficient capture the complex behaviors of atomic interactions. In reality, forces are
"Equal Contribution.
1 "oorrsponndig Author.

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influenced by factors beyond bonds and monotonic distance properties, especially for non-bonded atomic pairs that are typically dominated by van der Waals potentials (Lu & Chen, 2020: Lii & Allinger, 1989). Consequently, significant prediction emrors observed for existing GNNs on atoms with low degree, i.e., low coordination numbers, whose conformations can be more sensi- tive non-bonded interactions, as illustrated in Figure/ 1 in Section 3. Therefore, a new model to accurately account for the atomic interactions is warranted.
In response to this challenge, we present REBIND, novel framework that rewires molecular graphs by adding edges between non-bonded atomic pairs exhibiting high inter-atomic forces, often mod- eled with the Lennard-Jones (LJ) potential (Jones, 1924). To this end, we propose force-award aware self-guidance graph transformer, whereupon the initial conformation predicted from the encoder, a IJ potential-based adjacency matrix constructed, then utilized in the decoder to predict the final conformation.
To be specific, utilizing the encoder's prediction of inter-atomic distances from the initial confor- mation, REBIND calculates the absolute forces acting between all non-bonded atomic pairs through the derivative of the LJ potential. Subsequently, REBIND augments the graph by adding edges to non-bonded pairs with the largest computed forces. Our edge augmentation is performed in engrree-compesaating manner, ensuring that atoms with fewer connections receive additional edges to enhance the modeling of its non-bonded interactions. Furthermore, to differentiate the nature of these interactions, we distinguish the augmented edges into distinct adjacency matrices each specif- ically modeling repulsive and attractive forces. Incorporating these force-based adjacency matrices, the decoder refines the initial conformation prediction by generating residual adjustments, leading more precise molecular geometry.
The versatility of our proposed framework is demonstrated through benchmarks on both small-scale datasets, i.e., QM9 (Ramakrishnan al., 2014) 4) and Molecule3D (Xu et al., 2021c), and a large-scale GEOM-DRUGG (Axelrod & based rewiring brings universal improvements to GNNs even outside the proposed architecture, i.c. GINE (Hu* et al., 2020), GATv2 (Brody et al., 2022), and GraphGPS (Rampasek et al., 2022).
2022) dataset. We also show that our idea of force-
Our contributions are summarized follows:
We reveal that current approaches, which solely rely on bonds and predicted pairwise distances, are insufficient for accurately modeling inter-atomic forces, especially resulting in significant errors for nodes with low-degree atoms.
We introduce REBIND, a novel graph rewiring framework that adds edges between non-bonded atomic pairs with high inter-atomic forces guided by the Lennard-Jones potential. The number of augmented edges for each atom is determined in a degree-compensating fashion, improving the modeling of non-bonded interactions for low-degree atoms.
Extensive evaluation on diverse molecular sizes demonstrates the effectiveness of REBIND in enhancing ground-state molecular conformation prediction. Notably, our framework achieves improvements of up to 20% on the QM9 dataset.
2 PRELIMINARIES
Problem definition. We focus predicting the 3D ground-state molecular conformation from its corresponding 2D molecular graph G = (V,E), where V denotes the set of N = =|>] atoms (nodes) and È represents the set of M = [] bonds (undirected edges) between atomic pairs. Nodes are characterized by a feature matrix X = [x] ....... xx]" E Nx xd where each feature encodes atomic properties such atom types and chirality. Edges are described by binary adjacency matrix A E R N where A[i,5] /1 = a bond exists between atoms and j. and A [] = otherwise. Additionally, an edge feature matrix E [ei - em] T M may be utilized to represent bond-specific attributes, including bond types. For each node in single graph, its degree is notated as deg, = Ejev [i = 1] and relative degree is defined as deg.' = deg,/ max,e( deg,- The ground-state molecular conformation is represented C |C1, C..... Cv] e NNx3 where each C, E R corresponds to the 3D coordinate of atom i. The atomic pairwise distance is denoted
D E RNXN where Dij C; Cjll2-
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Multi-head self-attention. the core of transformer (Vaswani, 2017) is self-attention mecha- nism, which allows each instance the input data to attend to every other instances, thereby enabling the model to capture the instance- wise relationships. Formally, given input feature matrix X, the self-attention computes query (Q), key (K), and value (V) matrices through linear transformations:
Q = xWw K= K == VXWW
where Wº, W W Rdxds are learnable weight matrices. The attention scores at are then computed as follows:
[QKT
Attention (actiion(Q.AA (Q. K. V) = Jsm
V.
dd
where Jmm a softmax function and the scaling factor vdz stabilizes the gradients during training.
To enhance the model's ability to capture diverse patterns, multi-head attention employs multiple attention heads in parallel. Each head independently performs self-attention, and their outputs are concatenated and linearly transformed as:
MultiHead Q, K, V) = Concat Oi.,a.,,.,..""""""""" OH)Wo,
(3)
where each O Attention Qh: K₁, V₁. and 8 E Rlldy xdmads is a learnable weight matrix.
Inter-atomic interaction modeling in prior works. Traditional messagg-passing-- based architec- tures (Hu et al., 2020; Brody et al., 2022; Rampasek et al., 2022) leverage covalent bonds model interactions between connected atomic pairs. For a given node at the 1-th layer, the node repre- sentation h, is updated via message aggregation from set of its bonded neighboring nodes Mt, as formulated below:
h (1+1) v/
v (n.h) o(th Ii (
(4)
where @ is an aggregation function that combines the representations of neighboring nodes, and & is an update function that integrates this aggregated information with the node's current hidden state. (1) Pij denotes the hidden representation of the edge feature corresponding connected node pairs.
Recently, a new encoder-decoder based graph transformer (Xu et al., 2024) was proposed to capture inter-atomic forces by utilizing both bonds and atomic pairwise distances. In this framework, the pairwise distance matrix D E N is computed from the initial conformation predicted by the encoder. For each hith head in the decoder, the adjacency matrix A representing bond existence and row-subtracted Euclidean distance matrix D sow defined Dij Drowwsshh max,e(1,N) D.n Daj,
are integrated as residuals in the multi-head attention score Sp. formulated as follows:
5. = S, + S, O (3) x A) + S, O (3) Drow-suh)
S K. K.
( TT
where S₁. E N is the original self-attention score computed by the outer product between key and query matrices Qn, K₁, Nxd, and 33 3P R are learnable parameters that determine the influence of bonds and distance factors, respectively.
3 LIMITATION OF INTER-ATOMIC FORCE MODELING IN PRIOR STUDIES
In this section, we highlight the shortfalls of current approaches in using covalent bonds (Hu al., 2020; Brody et al., 2022; Rampasek et al., 2022) or pairwise distances (Xu et al., 2024) to model inter-atomic forces, followed by experimental results supporting our claim.
Limitations of prior works for non-bonded interactions. In ground-state molecular conforma- tion prediction, accurately modeling inter-atomic forces is essential, as the net force on each atom defined by the gradient of molecular energy E, approaches zero at equilibrium (Shi et al., 2021a). This equilibrium condition primarily determines the spatial arrangement of atoms. Meanwhile, a classic model of the energy E divides the term into bonded and non-bonded interactions (Leach,
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(a) Average Atom-wise RMSD (Low vs High Degree)
(b) Average Atom-wise MAE (Low vs High Degree)
Degree
Degree
0.5
Averagen Overage RMSD
A= 0.410
0.8
0.362
Average
0.6
4= 0.343
0.3
== 0.121
= =
0:110
0.1
GINE
GATV2
GraphGPS GTMGC
Ours
GINE
GATv2
GraphGPS GTMGC
Ours
Figure 1: Atom-wise error analysis with respect to the relative atom degree varing architectures in the QM9 dataset. (a) represents the average atom-wise RMSD while (b) shows the average atom- wise MAE for each bin of low-dege low-degree and high-degree atoms. A denotes the gap between the errors of the low-degree and high-degree groups.
2001; Luo al., 2021), where bonded interactions can be further divided into bond stretching, angle bending, and torsion as shown in Equation 6.
E Emonded + Emon-bonded = Ebond + Euggle + Exsssion + Emon-bonded bonded
(6)
We find that previous works inadequately account for non-bonded interactions. As shown in Sec- tion 2, these methods incorporate the bond adjacency matrix A Abond capture the bonded interactions Eponded (Hu* ct al., 2020; Brody et al.. 2022: Rampasek et al., 2022), or the distance-basee matrix Dow-suml (row-subtracted Euclidean distance matrix) approximate non-bonded interactions En bonded (Xu et al., 2024). However, even the distance- proximity matrix D row-sub of GT- MGC, which explicitly aims model non-bonded interactions, oversimplifies non-bonded forces by assuming monotonic relationship between attention weights and D row-subb In reality, non-bonded interactions are governed by complex, non-linear functions of distance with atom type-specific co- efficients (Leach, 2001).
proximity
Consequently, this abstraction leads to performance degradation in atoms dominated by non-bonded potential Enon-tondod- and furthermore, E. We believe that atoms with fewer covalent bonds (i.e., low-degree atoms) correspond to such cases, due to being associated with fewer bonded potentials and hence being empirically verify this assertion in the subsequent paragraph.
sensitive inaccuracies in the modeling of non-bonded interactions. We
Experimental verification. To validate assertion, conduct analysis how prediction errors vary with respect to their node degree. Specifically, we calculated an atom-wise Root Mean Square Deviation (RMSD) and Mean Absolute Error (MAE) of conformation predictions the QM9 dataset. The calculation of atom-wise RMSD and MAE for each atom are as follows:
RMSD(i) I C,lla, MAE(i) N
Σ D.g Diyl1 JEVI(3
Subsequently, we categorize nodes into low-degree (deg. rel E (.3) and high-degree (deg." E [0.7, IJ) groups based on their relative degree deg.el deg(")) maxne[L,N deg("n) then visualize the average errors for each group in Figure 1. We evaluate upon GINE, GATv2 and GraphGPS which utilize covalent bond adjacency hond and GTMGC which additionally incorporates inter- atomic distance sub-sub As illustrated, the low-degree atoms exhibit significantly higher errors compared to high-degree atoms, with deviations up 0.488 for RMSD(1) and 0. 123 for MAE(i). This demonstrates that previous works fail to sufficiently model inter-atomic forces between non- bonded atomic pairs, hindering the effective modeling of L'non-bonded-
REBIND: ENHANCING GROUND-STATE MOLECULAR CONFORMATION VIA FORCE-BASED GRAPH REWIRING
Given the limitation of previous studies, we introduce REBIND, a novel graph rewiring framework that selectively adds edges in force-aware manner, prioritizing atoms with low-degree. In Sec-
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Encoder
Decoder
X
Contormddos Head
Block
Corfommtion
Head
K
Sectforae
Add
enc
A AP?
Encoder Attention Mechanium
Decoder Attention Mechanism
Lpplaiin Pustiooll
8,
0%
organt Sei-Ammttion
hrnd
Figure 2: Overview of the REBIND framework.
tion 4.1, we provide an overview of the REBIND architecture. Section 4.2 details the force-aware graph rewiring component, and Section 4.3 describes the integration of the augmented edges into the multi-head self-attention. Finally, the learning objective of REBIND and comparison with existing studies are detailed in Section 4.4.
4.1
OVERVIEW
The overall architecture of REBIND is illustrated in Figure 2. Our framework receives 2D molec- ular graph as input, characterized by its bonded adjacency matrix bond and node feature or em- beddings augmented with Laplacian positional encoding L. The framework outputs predicted molecular conformation, denoted as C. We employ standard encoder-decoder architecture with multi-head self-attention, following the approaches presented in (Vaswani, 2017; Cai & Lam, 2020; Xu al., 2024). In this setup, the encoder processes the input graph generate hidden represen- tations gene task-specific prediction head then produces an initial conformation prediction Cene from H From this intermediate prediction, we derive 1) an inter-atomic distance matrix Hene and force-aware adjacency matrices foorce {Aattr ) where A and TP denote adja- cency matrices identifying attractive and repulsive atomic pairs. The hidden representation Heme along with A bond and A are subsequently fed into the decoder generate the refined, residual molecular representation.
4.2 FORCE-AWARE GRAPII REWIRING
Here, we outline the construction of force-aware adjacency matrix foree which connects non- bonded atomic pairs exerting significant forces to one another, mitigating the shortfalls of prior works identified in Section 3.
Force modeling with Lennard- Jones potential. To capture the interactions between non-bonded atomic pairs, we leverage the Lennard-Joner (LJJ) potential (Jones, 1924), a well-established model for non-bonded interactions such as der Waals forces. The LJ potential is defined
((r) = 4s (-)
where r is the distance between two atoms, E is the depth of the potential well (representing the strength of the interaction), and a is the finite distance at which the inter-atomic potential is zero. The first term (*) accounts for the repulsion when atoms
close, while the second term
(e) accounts for the attraction moderate distances. By using the derivative of this potential, compute the inter-atomic forces acting upon non-bonded atom pairs, which is critical for modeling the equilibrium conformation of the molecule. This force model provides tation of how spatial positions are influenced, especially for low-degree atoms, where non-covalent interactions dominate.
realistic represen-
The values of and E are assigned based on the specific pair of atom types being modeled (e.g., carbon, hydrogen, etc.). In line with conventional force-field modeling approaches used in com-

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Force based Adjacency Matrix Construction
Repulsive Force
Attractive Force
V(r) 4c [(,) (gy)
Di
+1
ghond
A ,
Figure 3: An illustration of edge augmentation REBIND using the potential. Non-bonded atomic pairs with the largest force magnitudes are added as edges to the molecular graph in degree- compensating manner. The augmented edges are treated according to the nature of the forces, dis- tinguishing between attractive and repulsive interactions.
putational chemistry, adopt the predefined parameter values from the Universal Force Field (UFF) (Rappé et al., 1992). For interactions between different atom types, the parameters are deter- mixing rules. Specifically, the interaction parameters dzj and Eij
mined using the Lorentz-Berthell for atom pair and j are defined as follows:
oi t J
TU 2
Egg VEL
(8)
Thus, the full inter-atomic force between atom i and j can be written as:
dV(r) 24g
F(r) Here, the sign of Frr) denotes the directionality of the force, where positive values correspond to repulsion, and negative values indicate attraction.
2
01
dr
Construction of force adjacency matrix. Given the derived force function from the LJ potential, we construct set of force adjacency matricesdj Aforce which introduces additional edges atoms that significantly impact its spatial conformation, while prioritizing those of low-degree. Utilizing the predicted distance matrix obtained from the encoder Emc where D | Cy
V(C, (C, Cj). we compute the inter-atomic force FIJ for non-bonded atomic pairs using the Equation 9. We then augment the graph by connecting each node to the top K; maX deg." degi' non-bonded atoms that exert the largest forces on i as follows:
e eforce = i E Topk. (Gi (i.iil) eij gbood
(10)
Here, for each atom i, we select non-bonded neighbors j that exert the largest forces on i, adding up to the maximum possible degree maxne[L.N]] deg,, deg," without thresholding or introducing any hyperparameters. To differentiate the nature of these forces, we decompose Elonce into separate adjacency matrices A attr and 'TP which capture attraction and repulsion, respectively. With as the indicator function, we formalize A a and rep as follows:
A. [(i,]) Efones A [i.jj
A. 1 [(i,5) E Eforce FU [i 0]
FU
Note that the edge weights of aater are positive ones (i.e. +1) while weights of A rep are negative ones (i.e. -1), ensuring distinetiveness between the directionality of forces. The complete process is outlined in Figure 3.
4.3 INTEGRATION WITH MULTI-HEAD SELF-ATTENTION
Leveraging the augmented edges, REBIND enhance the multi-head self-attention by incorporating both bond-based and force-based adjacency matrices. Specifically, the augmented force-based adja- cency matrices flace = A ATr) and original bond-based adjacency matrix A bend are incorpo- rated as residual components in the attention scores for each head This integration formalized as follows:
S, = S,, S₁. @ (B A beod) + Sh @ (A x A + Sn. @ (Bftep x A ATP,
(11)
Here, Z = where S, KT (ZW,)) represents the global self-attentin
H L E Xxd the input the decoder, combining the encoder's hidden representations Hrec

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with Laplacian positional encoding L. 3/ S,ttt, , _ are learnable parameters that modulate the influence 1) bonds, 2) attraction and 3)repulsion adjacency matrices, respectively. Subsequently, the output of each attention head, On, is computed by applying softmax normalization to the scaled attention scores and then multiplying by the linear projection of the decoder input Vn, E RRV xxx:;
Sh
VV. O 5.m VTTE
Vn.
V, ZW
(12)
where Jam is a softmax function applied to the attention score scaled by vdg. The final confor- mation C is obtained by refining the initial encoder representation H with the residual decoder representation Hds which achieved via channel-wise attention:
Č FFN((Y) E RRX3
Y
ORi O High RRNxx.
(13)
Hene W,
H W
Nxdxx2 = H
H E N
Jsm
dde
where (-N is a task-speeific head layer, W, E Rostade is a linear transformation applied to the hidden representations, and || denotes concatenation along the last dimension. The channel- wise attention weights a determine the contribution of the encoder and decoder representations in producing the final conformation.
Throughout this attention mechanism, REBIND effectively models inter-atomic forces by learning from both bonded and non-bonded atomic pairs. By incorporating the structural supervision from A hond and the force-aware supervision from force' into the global attention scores S, REBIND facilitates comprehensive force modeling, enhancing the accuracy of conformation predictions.
4.4 OBJECTIVE FUNCTION
To ensure that the predicted molecular conformation remains invariant to rotation and translation, adopt a loss function based the difference between the predicted and ground-truth pairwise Euclidean atomic distances, denoted D and D, respectively, which is also utilized in prior works (Xu et al., 2021c; 2024). Additionally, to achieve precise force modeling within the decoder, we apply the same loss objective to the initial conformation predictions Dene and D from the encoder. The overall loss objective L is formulated as follows:
N
Σ
D Dig
Σ
ΣΣ
D Dial
(14)
N
N2
Comparison with existing graph transformers. Several existing works integrate inter-atomic relationships as residuals within multi-head self-attention mechanisms. For instance, the Geometric Transformer (Choukroun & Wolf, 2021) utilizes the pairwise Euclidean distances D, computed from ground-truth atomic coordinates, to perform molecular property prediction. Similarly, MAT (Maziarka et al., 2020) incorporates both the pairwise distances D and the bond adjacency matrix A hund for the same task. In contrast, GTMGC (Xu et al., 2024) employs predicted inter-atomic distances roow-sb to forecast ground-state molecular conformations.
While our work also leverages pairwise atomic relationships as residuals in the computation of atten- tion scores, REBIND distinguishes itself by enabling fine-grate mme-grained force modeling. This is achieved through the joint utilization of both bonded interactions and non-bonded interactions characterized by the largest forces. Consequently, our framework attains more precise ground-state molecular geometries, which is substantiated in the subsequent section.
5
EXPERIMENTS
EXPERIMENTAL SETUP
Datasets. We evaluated REBIND well-established benchmark datasets, including QM9 (Ra- makrishnan et al., 2014), Molecule3D (Xu et al., 2021c), and GEOM-DRUGS (Axelrod & Gomez- Bombarelli, 2022). QM9 consists of small organic molecules and is widely utilized for quantum

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Table 1: Conformer prediction performance and percentage reduction (%) of REBIND compared to
the best baseline performance on the QM9 and Molecule3D datasets.
Splits
Validation
Test
Datasets
Methods
D-MAE
D-RMSE [ C-RMSD
E-RMSD
D-MAE
D-RMSE
C-RMSD
E-RMSD
RDKI-DO
0,328
0.570
0.302
1.044
0.330
0.573
0.504
1.266
RDKit ETKDG
0.324
0.458
GINE
0.605
0.950
0.865
1.703
0.606
0,946
0.867
1.696
GATv2
0,382
0.695
0.711
1.371
0.382
0.690
0.712
1.358
QM9
GraphGPS (RW)
0.328
0.629
0.628
1.196
0.327
0.624
0.628
1.193
GraphGPS (LP)
0.283
0,500
0.544
1.049
0.283
0.499
0.546
1.064
GTMGC
0,280
0.470
0.415
0.792
0.281
0,800
REBIND
0.252
0.442
0,320
0
0.254
0,446
0.321
0.610
Reduction
10.00
5,96
22.89
24.12
9.61
23.75
RDKi-DG
0.581
0,930
1,033
1,864
0.582
0,932
1.044
1.872
RDKit ETKDG
0.575
0,942
0.981
1,700
0.576
0.943
0,983
1.710
DeeperGCN--DAGN
0,509
N/A
N/A
0.571
GINE
0.591
1.016
1.103
2.227
0.592
1.019
1.104
2.230
Molecule3D
2.163
(random)
GraphGPS (RW)
0.512
0.900
1,006
2,089
0,513
0,903
1,008
2.094
GraphGPS (LP)
0,440
0,730
0.854
1.687
0.441
0.732
0.854
1.689
GTMGC
0.429
0.713
0.708
1.347
0,430
0.715
0.709
1.350
REBIND
0.418
0.706
0.698
1.314
0.419
0.708
0.699
1.317
Riduuction
2.56
1.41
2.45
2.44
RDKIt-DO
0.542
0.872
0.993
1.751
0.524
0.857
0,970
1,750
RDKit- ETKDG
0,531
0.874
0.916
1,565
0.511
0.858
0.892
1.595
DeeperGCNN DAGNN
0.617
0,930
N/A
N/A
0.763
1.176
NIA
N/A
GINE
0.889
1.517
1.398
3.007
1,388
2.200
1,928
4.142
Molecule3D
GATv2
0.791
1,402
1,264
2.675
1.248
2.082
3.832
(scaffold)
GraphGPS (RW)
0.503
0.853
0.978
2.047
0.595
1.024
1,065
2.203
GraphGPS (LP)
0,417
0.690
0.827
1.639
0.411
0.827
GTMGC
0.406
0.670
0.682
1.318
0.397
0.671
0.692
1.275
REBIND
0,391
0.661
0.640
1.174
0.386
0.663
0.667
1.182
Reslicttin
3:88
2,200
6.16
100.53
2277
1.199
33.61
7.25
chemistry applications. Molecule3D is a large-scale dataset of molecular structures, for which we employed two distinct splitting strategies: random split and scaffold split. The scaffold split groups molecules based on their core substructures, enabling a more realistic evaluation. Additionally, GEOM-DRUGS comprises large- size molecules relevant drug discovery, providing a challenging benchmark for assessing the scalability of our framework on complex molecular structures. Since the original dataset includes multiple stable conformations, we choose the most stable conformation with respect the Boltzmann energy for each molecule. Detailed descriptions of each dataset provided in Appendix C.
Metrics. Following previous work (Xu et al., 2024), we evaluate the performance of our model using Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Root Mean Square Deviation (RMSD). Additionally, we introduce a new metric, Energy-weighted RMSD (E-RMSD), which accounts for the chemical feasibility of predicted conformations. By denoting G; and G as the aligned predicted and ground-state conformation via the Kabsch algorithm (Kabsch (1978)), E-RMSD is calculated as:
P E-RMSD(@, G)
Σ wi w GGll2 LEV
(15)
$1
Here, g denotes the Boltzmann factor denoted exp
B=E while atom-wise normalized force
wi is defined as w; F. E and È denotes the total energy of ground-truth and predicted conformations, and F, denotes the force calculated via the force-field defined from the predicted conformation, both computed using the Merck Molecular Force Field (Halgren, 1996). The con- stants & and T each the Boltzmann constant and thermodynamic temperature, respectively. This approach penalizes 1) errors of molecules that are energetically unstable via Y. and 2) errors of atoms that are off- off-equilibrium with respect to net force via wy. Detailed descriptions of other metrics are provided in Appendix C.2
Following Xu al. (2024), compared REBIND against traditional cheminfor- matic methods, DG and ETKDG algorithms, from RDKit (Landrum et al., 2013) and five repre- sentative GNNs. The GNNs considered in our experiments can be broadly broadly categorized as
Baselines.
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(1) traditional GNNs such as GINE (Hu* et al., 2020), GATv2 (Brody et al., 2022), DeeperGCN- DAGNN adapted from Xu et al. (2021c). (2) graph transformers including GraphGPS (Rampasek et al., 2022) and GTMGC (Xu et al., 2024). For GraphGPS, conducted experiments using both random walk (RW) and Laplacian (I.P) positional encoding strategies. To ensure a fair comparison, trained the MoleBERT (Xia al., 2023) tokenizer in GTMGC separately for each dataset, which differs from the original setting where the tokenizer trained on Molecule3D random split is used universally across all benchmarks. To further validate the efficacy of our framework, we included comparison with the liffusion-bbsee conformer generation model, Torsional Diffusion (Jing et al., 2022). Detailed experimental configurations are specified Appendix C.
5.2
MAIN RESULTS
We present the performance results with of REBIND in Table where the percentage reduction is calculated as the ratio of the performance improvement over the best baseline, expressed as per- centage. As demonstrated, our method achieves consistent superiority over baseline methods across all datasets and evaluation splits. On the QM9 dataset, REBIND achieves notable improvements across all evaluation metrics, with gains of up to 24% in both C-RMSD and E-RMSD. Additionally, method exhibits robust generalization capabilities on the larger Molecule3D dataset. Notably, REBIND further excels in the scaffold split, where training and test sets contain structurally diverse molecules, achieving up to 10% reduction in E-RMSD. This highlights REBIND'S ability gen- eralize effectively to novel molecular scaffolds. Furthermore, it is worth emphasizing that higher performance gains are observed in E-RMSD compared C-RMSD. This suggests that REBIND generates molecular conformations that are not only geometrically accurate but also more realistic terms of energetic stability, resulting in more physically plausible molecular structures.
5.3 SCALABILITY TO LARGE MOLECULES
We further validate REBIND on the GEOM-DRUGS dataset with large molecules by including Torsional Diffu- sion (Jing et al., 2022)as an additional baseline, which adopts the same dataset as its benchmark. Since Torsional Diffusion was originally designed to generate mul- tiple conformers, we adapt it to produce a single conformation per molecule for evaluation.
Table Conformer prediction performance and per- centage reduction (%) of REBIND compared to the best baseline performance on the GEOM-DRUGS dataset.
Splits
Datasets
Methods
D-MAR,
D-RMSE L C-RMSD
F.RMSD
RDKin-DG
1.181
2.132
2.097
3.623
1.120
2.055
3,330
GINE
1.125
1.777
2.033
3.925
GATv2
1.042
1.662
1,901
3.728
GEOM-DRUGS
GraphGPS (RW)
0.879
1.399
1,368
3.472
GraphCPPS
1.300
3.171
GTMGC
0.823
1.319
1.458
2,830
As shown in Table 2. REBIND signifi- cantly outperforms the baselines, achiev- ing improvements of up to 4% in C-RMSD
Tersional Diffusion
0.959
1.648
1.751
2.992
REBIND
0.776
1,283
2.602
Percentage Reducction
4.79
1:31
4:23
K.06
and 8% in E-RMSD. Diffusion-based models like Torsional Diffusion are designed generate mul- tiple stable conformations and are not directly suited for predicting the single most stable conforma- tion of molecule, resulting in suboptimal performance. In contrast, REBIND demonstrates strong generalizability to drug-like molecules, surpassing diffusion- based models with lower computational costs and eliminating the need for multiple inference steps.
5.4 ABLATIVE STUDY
Ablation on components. We conduct an ablation study on the components of
Table 3: Ablative study of REBIND.
REBIND on the QM9 dataset, with the re-
L
A bosa
A 140
C-RMSD
E-RMSD
sults presented in Table 3. Using a fully- connected transformer with Laplacian po- sitional embedding L results in limited performance, highlighting that uniformly connecting edges without accounting for inter-atomic forces is suboptimal. Integrating the bond- based adjacency matrix A band with L leads to substantial performance improvements. Moreover, evaluate the impact of incorporating force-aware edges (i.e., Afore) compared to edge augmen-
0.5294
1.0323
0.3524
0.6920
0.3522
0.6807
0.3209
0.6103

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tation based proximity (i.e., A the strategy adopted in Luo al. (2021). The distance-based edges A near are constructed using the intermediate distance matrix Dome, connecting atoms i and when De ij 8. The threshold 8 is set as 10 Å, following the protocol in Luo et al. (2021). Our results demonstrate that Af which accounts for significant inter-atomic forces influencing spatial conformation, consistently outperforms pear, showing up 9% improvements in C-RMSD and larger gains up to 10% in E-RMSD. This demonstrates the effectiveness of A in aiding the model generate conformations that are both closer the ground state and energetically stable
Application to GNN-based architectures. The core concept of REBIND is adaptable to any encoder-decoder framework. To demonstrate its versatility, we integrate REBIND with three GNN- based architectures: GINE, GATv2, and GraphGPS (LP). We implement REBIND by dividing the L layers of original GNNs into two halves, where the top half operates as the encoder and the bottom half operates as the decoder. To compute foorc we follow the same procedure as REBIND, wherein the encoder's output passed through a task-specific head layer to generate Čene and Den:
The results are shown in Table of Appendix B. Results demonstrate that incorporating REBIND significantly improves performance across all backbone architectures and metrics, with gains of at least 5%. Notably, when applied to GraphGPS, we achieve the lowest prediction errors, with improvements of 5.1 11 in C-RMSD and 8.74% in E-RMSD. These results underscore the versatility of REBIND in boosting performance regardless of the underlying architecture. Moreover, REBIND'S 's integration requires minimal implementation effort, making it compatible with any existing GNN- based architecture through the proposed mechanism.
5.5 QUALITATIVE ANALYSIS
We present qualitative comparison of REBIND against baseline architectures based on atom-wise RMSD on the QM9 dataset, shown blue bars in Figure 1. As illustrated, REBIND achieves sig- nificant егтоr reductions for both groups Specifically, when compared against GTMGC, REBIND further reduces the atom-wise RMSD for lower-degree groups by 0.11, compared to a reduction of 0.05 for high-degree groups, thereby verifying the efficacy of our degree-compensatee tion. Moreover, our method greatly reduces the gap between the atom-wise errors of low-degree and high-degree groups, achieving 15.16% and 20.91 1% percentage reduction in the gap for atom- wise RMSD and MAE, respectively, compared to GTMGC. These results substantiates that REBIND comprchensively captures bondodded interactions through force-awar edge augmentation.
augmenta-
6 RELATED WORKS
There has been a growing interest in generating molecular conformations from 2D molecular graphs Traditional cheminformatics methods, such as DG and ETKDG from RDKit (Landrum et al., 2013), widely used for their efficiency, relying chemical heuristics for fast generation. However, their performance is limited due to insufficient handling of non-bonded interactions and minimal energy optimization. To address this limitation, deep learning models have emerged as powerful alternatives, producing multiple conformations for molecules using generative models (Mansimov al., 2019; Xu al., 2021b;a; Luo et al., 2021; Shi et al., 2021b; Xu al., 2022; Jing al., 2022). Recently, the focus has shifted to predicting ground-state molecular conformations, which is critical for practical applications requiring stability and feasibility. A benchmark and training pipeline were introduced in Xu et al. (2021c) for this task, followed by the graph transformer (Xu et al., 2024), achieving state-of-the-ae results. Further details on related works are provided in Appendix A.
7 CONCLUSION
Given the limitations of previous studies modeling inter-atomic forces, we introduced REBIND, an innovative framework that incorporates force-aware edge augmentation to accurately capture inter- atomic interactions. We anticipate that REBIND can be extended to other critical applications such as drug discovery or property prediction, particularly in scenarios where ground-truth coordinates are unavailable. As a future work, we plan to integrate our framework with these applications to enhance their predictive accuracy.
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ACKNOWLEDGMENTS
This work was supported by Institute for Information & communications Technology Planning & Evaluation(IIIP) grant funded by the Korea (ove) RRS-2019-I1190099 Artificial Intel- ligence Graduate School Program(KAIST). We sincerely appreciate Polymerize for their generous support. Special thanks also go Juneyong Yang and Changhun Kim for invaluable comments in improving this manuscript.
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SUPPLEMENTARY MATERIALS
Table 4: Conformer prediction performance and percentage reduction (%) of GNN-based backbones integrated with REBIND on the QM9 dataset.
Method
Test
Backbone
Methods
D-MAE & D-RMSE
CRRMSD
E E-RMSD
Vanilla
0.606
0.946
0.867
1.696
GINE
+ REBIND
0.505
0.761
0.801
1.605
Reduction 16.65
19.55
7.61
5.36
Vanilla 0.382
0.690
0.712
1.358
GATv2
+ REBiND 0.319
0.523
0.611
1.198
Reduction
16.62
24.14
14.14
11.77
Vanilla
0.283
0.499
0,546
1.064
GraphGPS (LP)
+ REBIND
0.211
0.361
0,518
0.971
Reduction f
23.35
27.38
5.T1
'8:74
A RELATED WORKS
Molecular Conformer Generation. In recent years, there has been significant progress in the field of molecular conformation generation from 21) molecular graphs. Traditional cheminformatics approaches, such as Distance Geometry (DG) and its extension, ETKDG (Landrum et al., 2013), have been widely employed due to their computational efficiency. These methods rely chemical heuristics and geometric rules to generate conformations efficiently, which makes them attractive for large-scale applications. However, their reliance approximations and minimal energy opti- mization limits their ability to accurately capture non-bon mondeddd interactions, which play a crucial role in determining the stability and realism of molecular conformations.
To overcome the limitations of traditional approaches, deep learning models have emerged as promising choices. Generative models, in particular, have been extensively explored for molecu- lar conformation generation, offering more accurate and realistic predictions. Early works (Mansi- mov al., 2019; Xu al., 2021b) leverage variational autoencoders (VAEs) encode molecular information into latent spaces, where each molecule's structure is represented as a probabilistic dis- tribution. From this latent space, multiple plausible 3D conformations are sampled by decoding the latent variables. Similarly, flow-based models (Xu et al., 2021a) generate molecular conformations by transforming latent variables sampled from a Gaussian distribution into atomic distance matrices, capturing long-range dependencies between atoms. The 3D coordinates are then derived from the generated distances, followed by refinement using energy-based models. Score-based models have function
also gained attention for molecular conformation generation. These models learn representing the gradient of the log probability density of atomic coordinates (Luo et al., 2021; Shi al., 2021b). By perturbing the molecular data with Gaussian noise multiple levels, these models iteratively denoise the data using the learned score function to guide the generation of valid confor- mations. Building the success of diffusion models (Ho al., 2020) in the vision domain, their application to molecular conformation generation has become another emerging area of interest (Xu al., 2022: Jing al., 2022: Zhang al., 2023; Fan et al., 2024). They have shown remarkable success in generating diverse molecular geometries by learning the desired geometric distribution from noise distribution through reverse diffusion process.
Ground-state molecular conformer prediction. While the one-to me-to-many task of generating mul- tiple conformations has been thoroughly studied, recent research has shifted toward the prediction of molecule's 's ground-state conformation. This focus reflects the practical importance of identify- ing the most stable and energetically favorable molecular structure, which is essential for real-world applications such as drug design and material discovery. Ground-state conformation prediction quires models to identify the most stable configuration on the molecule's potential energy surface, which is challenging for one-to-many methods address.
To facilitate progress in this area, the Molecule3D benchmark and training pipeline are introduced in Xu al. (2021c). They provide standardized dataset and evaluation protocols for predict- ing a molecule's s ground-state geometry. Building on this, the graph transformer architecture (GT- MGC) (Xu et al., 2024) was recently proposed, tailored to predict ground-state molecular confor-
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mations. GTMGC incorporates bonds and pairwise distances within the self-attention mechanism to capture both local and global molecular interactions, achieving state-of-the-an performance on the benchmarks with ground-state conformations.
Table 5: Examples of visualizations and RMSD predictions from RDKit-ETKDG, GTMGC, and REBIND on the QM9 and GEOM-DRUGS datasets.
Datasets
QN9
GEOM-DRUGS
Ground-truth
RDKi-ETKKD
1.33
0.844
1621
3,033
2515
3,400
GTMGC
228
2521
REBIND
0,450 0
0.376
0,844
0,8%
1,174
B ADDITIONAL EXPERIMENTS
B. MOLECULAR VISUALIZATIONS
In this section, we provide 3D visulaizations and corresponding RMSD predictions from our method, RDKit-ETKDG, and GTMGC on the QM9 and mations predicted by REBIND shows significantly closer alignment with the ground-truth conforma- tions. This superiority is consistent across molecules of varying sizes, demonstrating the robustness of our approach.
dataset. As illustrated, the confor-
B.2 FURTHER QUALITATIVE ANALYSIS
We present an additional qualitative compari- son of REBIND against baseline architectures based on atom-wise E-RMSD, as shown in Figure 1. Analogous to previous analyses, we calculated the atom-wise E-RMSD between the ground-truth 3D coordinates and predic- tions on the QM9 dataset. For an atom i in the m-th molecule, the atom-wise E-RMSD, E-RMSD(1), is formulated as:
Degres
High Degrea
2.00
4
0.420
1.75
A
0.762
0.915
Average E.RMSD
0.543
0.503
0.75
0.50
(m)) ((m) Wi C Cill2. (16)
0.55
E-RMSD(1)
GINE
GATV2 GraphGPS GTMGC
Ours
where C and C. denotes the predicted and ground-truth coordinates of atom i. pimn is Boltzmann factor of the m-th molecule and of the low-degree and high-degree bins.
Atom-wise E-RMSD analysis with re- spect to the relative atom degree on REBIND and baselines. A denotes the gap between the errors
Figure 4:
(m)
denotes the normalized atom-wise force, as de- tailed in Section 3. Following the same procedure, categorized nodes into the low-degree and high-degree groups based on the relative degree, and computed the average atom-wise RMSD for each group, depicted in Figure 4.
Consistent with the results in Section 5.5, REBIND achieves significant error reductions for both low-degree and nigh-degree groups. Specifically, when compared against GTMGC, REBIND further reduces the atom-wis. wise E-RMSD for lower-degree groups by 0.18, compared to reduction 0.14 for high-degree groups Furthermore, method greatly reduces the gap between the atom-wise
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Table 6: Comparison of predicted molccules' energy from models across datasets, denoted in KJ mol Best results are shown in bold, while last column denotes percentage reduction with respect the second best performance.
Datasets
GAT
GINE
GraphGPS
GTMGC
Ours
% Reduction
QM9
3.28
9.26
0.38
0.39
0.26
33.3
Scalfold
91.20
131.30
0.73
0.56
0.32
42.6
Random
10.40
5.50
1.38
0.56
0.27
51.8
GEOM-Dre -Drugs
271.86
270.00
195.17
1.75
1.34
23.4
(a)
Average Atom-wise RMSD (Low vs High Degree)
(b) Average Atom-wiie RMSD (Low vs High Degree)
Low
High Degree
Low Degree
Degree
4
1.368
0.622
2.5
1.446
0.661
Average RMSD
4 = 1.325 325
Average RMMSO
1.5
2.0
4 0.790
1.5
1.0
- 0.587
GINE
GATV2 GraphGPS GTMGC
Ours
GINE
GATv2 GraphGPS GTMGC
Ours
Figure Further analysis of Atom-wise RMSD on (a) GEOM-DRUGS and (b) Molecule3D
datasets. A denotes the gap between the emrors of the low-degree and high-degree groups
RMSD of low-degree and high-degree groups, achieving a 7.37% percentage reduction the gap compared to GTMGC, demonstrating the efficacy of REBIND in accurately modeling interatomic interactions.
B.3 IMPORTANCE OF LOW-DEGREE ATOMS
To highlight the performance disparities be- tween low-degree and high-degree atoms dis- cussed in Section 3, we further visualize the proportion of low-degree atoms within each molecule in Figure 6, using the Molecule3I
Proportion of Low Degree Atoms by Molecule Size
0.40
Proportion of Low-Degree stoms
0.50
dataset. The visualization reveals that the pro- portion of low-degree atoms (deg." (0, 0.3))
0.25
0.30
increases significantly as molecular size grows. Combined with our findings in Section 3, this suggests that inaccuracies in predicting the po- sitions low-degree atoms are a major con- tributor to performance degradation in confor- mation prediction for large molecules. These observations emphasize the necessity of novel methods, such as REBIND, to address the chal- lenges associated with low-degree atoms effec- tively.
0.35
0.30
805
8.00
(Number
Figure 6: Proportion of low-degree (deg rel (0, 0.3)) within each atom, binned with respect to their molecular sizes. Molecule3D is used during visualization.
B.4 PREDICTION OF L.WWEENERGY CONFORMATIONS
To highlight REBIND'S capability predicting low-energy (i.e., stable) conformations, we conduct an energy-based validation experiment. Specifically, we compute the average energy difference be- tween the predicted conformations and the ground-state conformations across all baseline methods. performed using the Merck Molecular Force Field (MMFF) in RD- Kit. The results are shown Table 6, demonstrating that REBIND consistently achieves the lowest energy differences ( (peed Eggounal) compared to baseline methods. This finding underscores RE-
The energy calculations
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Published as a conference paper at ICLR 2025
Table 7: Comparison of time complexity among GTMGC and REBIND. The experiments are conducted on the machine with NVIDIA GeForce RTX 3090 GPU//ntel(R) Xeon(R) Gold 5215 CPU @ @2.00GHz.
Time (sec.)
QM9
Molecule3D
Geom-DRUGS
Forward
Backward
Total
Forward
Backward
Total
Forward
Backward
Total
A bond
0.029
0.038
0.067
0.030
0.037
0.067
0.094
0.105
0.199
GTMGC
0.031
0.039
0.070
0.031
0.047
0.077
0.129
0.144
0.273
hoood + Area
0.059
0.040
0.010
0.061
0.043
0.104
0.169
0.092
0,261
Ours
0.064
0.043
0.107
0.069
0.041
0.110
0.195
0.109
0,304
BIND'S ability reliably predict the most stable conformation, which aligns closely with our task objective of identifying the global energy minimum.
B.5 COMPUTATION BURDEN ASSESSMENT
For rewiring the edges incurs additional computational overhead, we provide a detailed comparison of both forward and backward computation times (wall-clock time) across all three datasets. Specif- ically, we compare the time complexity between 1, incorporating bond alone, 2. A bond + D Drow-sub (i.e., GTMGC (Xu et al., 2024)). 3. A bond + near (i.e., adjacency matrix constructed via close distance). and 4. REBIND. The results are shown in Table Note that, the results are averaged upon 50 gradient update steps for each dataset, for they have varying molecular sizes (8, 15, and 25 average molecular size for each dataset). As expected, the computation time increases from QM9 to GEOM-DRUGS due the larger average molecular sizes. On average, our method introduces an additional 0.03 seconds of computational overhead compared to GTMGC, due to the calculation of the Lennard-Jones (I.)) potential matrix.
B.6 ANALYSIS UPON VARIOUS REWIRING TECHNIQUES
To further provide insights upon edge-rewiring of edges, we conduct extensive analysis upon the combinations of possible construction of adjacency matrices. Specifically, (1) a comparison between ffoore and A uear in the absence of A bond and sow and (2) additional combinations of these four components. As shown in Table 8, incorporat- ing our proposed torce A hond and D mow-sub significantly reduces errors, outperforming those involving A near and Drow-sub + A bond setting.
Table 8: Comparison of possible combi- nations of rewiring techniques in QM9 dataset.
Combinations
C-RMSD
E-RMSD.
Vanilla Transformer force
0.529
1.032
0.483
0,951
Attar
0.454
0.903
important to note that RE- BIND is designed to integrate distance and force infor- mation a complementary manner. As such, employing A alone may yield limited performance improvements due to the absence of distance factors (e.g., atomic pairs with close distances but repulsive forces). However, when combining interatomic forces with A near or D now-suh it achieves substantial improvements in confor- mation prediction accuracy by effectively modeling both interatomic forces and spatial relationships.
nyon-suh f
0.328
0.646
bond foncr
0.359
0.712
Anear Aband
0.337
0.655
bbond rjum-uuh Aueu
0.349
0.694
bonnd rytm-uub,
ffrcc
0.326
0.617
C EXPERIMENTAL SETTINGS
C.I DATASET DESCRIPTION
We evaluate our method on three benchmark datasets: QM9, Molecule3D, and GEOM-Drus De- tails for each datasets are specified below.
QM9 is a widely-used quantum chemistry dataset containing molecular geometries, elec- tronic properties, and energy attributes for small organic molecules with up 9 heavy atoms. The 3D conformations are obtained using density functional theory (DFT). We adopt the data split proposed by (Liao & Smidt, 2023).
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Published as a conference paper at ICLR 2025
Molecule3D is large-scale dataset consisting of approximately million molecules. Each molecule is annotated with 2D molecular graphs, ground-state 3D conformations, and var- ious quantum properties. We follow the splits used in (Xu et al., 2024), including random split and a scaffold split. The scaffold split groups molecules based on their core substruc- tures, allowing for more realistic evaluation.
GEOM-Drugs is subset of the GEOM dataset, focusing on drug-like molecules. For each molecule, multiple conformation sets along with their chemical properties are provided. Thus, we choose the most stable conformation with respect to Boltzmann energy for each molecule for our experiments. For data splits, we adhere the splits used in (Ganea al., 2021).
C.2 METRICS DESCRIPTION
We consider the following metrics for evaluation.
MAE.
Mean absolute Error (MAE) quantify the accuracy of predicted interatomic dis- tances relative to the ground truth on a pairwise basis. Let Dij and Di, represent the predicted and ground truth distances between atoms i and j. respectively. The metric is formulated as follows:
MAB D)
D Dig
N
LJEVV
RMSE. Root Mean Square Error (RMSE), also quantifies the error in interatomic distances, on a pairwise basis. Specifically, the metric is formulated as follows:
RMSE D, D) = N
(b,
NJEV
RMSD. Root Mean Square Deviation measures the spatial deviation between two confor- mations that are aligned using the Kabsch algorithm (Kabsch, 1978). Only heavy atoms (i.e., excluding hydrogens) are considered during this calculation. Let G; and G; represent the aligned conformations of atom i. RMSD is defined as:
RMSD(G G)
Σ G.. G:ll2 LVV
E-RMSD. Energy- weighted RMSD, considers chemical feasibility of the predictions on top of the spatial deviations. The probability likelihood of the whole preicted confor- mation G with respect to the ground state G is accounted via the Boltzmann factor B-E RT
), while the feasibility at atom-level is considered via sum uum-normalized F. force (i.e.w,- Σ Fj acting upon each atom. Both the Boltzmann factor and force is calculated using Merck Molecular Force Field (Halgren, 1996). For all our experiments, we set 298.15 15 K and 0.001987 kcal mol mo-! reflecting standard laboratory conditions.
( exp
C.3 IMPLEMENTATION DETAILS
We adopted the evaluation protocols and train/validationntess splits from Xu al. (2024) for the QM9 and Molecule3D datasets and from Jing et al. (2022) for the GEOM-DRUGS dataset. Test results were derived from the best-performing model on the validation set based on the D-MAE metric. All GNN architectures were implemented using PyTorch Paszke et al. (2019) and Py PyTccc Geometric Fey & Lenssen (2019). The experiments were conducted on RTX Titan and RTX 3090 (24GB) GPU machines. Throughout all experiments except for Jing et al. (2022); Xu et al. (2024), we set the hidden dimension to 512 and the number of layers to 8: for these references, we adopted their optimal configurations. We employed the AdamW optimizer with batch size of 100 and no weight decay. Learning rates were initially warmed up from 0 and then fixed based on the best validation performance the QM9 dataset, within the range of [3e-5, 5e-5. 7 9c-5]. The number of attention heads was set to 8. We used a seed of 42 for all experiments and trained all models for 20 epochs, following the configuration in Xu et al. (2024).
18


Reviews:
[{"review_id": "AHOSscvYpZ", "reviewer": "Reviewer_5PhF", "summary": "This work introduces REBIND for ground-state molecular conformation prediction using force-based graph rewiring. By adding edges based on Lennard-Jones potential, REBIND captures non-bonded interactions between low-degree atoms. This framework includes an encoder-decoder graph transformer to first estimate atomic forces and then integrate those interactions into refined molecular graphs. Experimental results show substantial performance improvements, demonstrating REBIND’s effectiveness and scalability across multiple benchmarks and molecular sizes.", "questions": "Are the force-aware adjacency matrices only computed once for decoding? If so, have the authors tried recycling the predicted molecular conformation C several times, i.e. using the final predicted C to compute new force-aware adjacency matrices and doing the decoding again? The new force-aware adjacency matrices could capture more accurate atomic pairs, so as to enable even better conformation predictions.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This work presents a novel contribution to molecular conformation prediction, primarily through its force-based graph rewiring mechanism. By incorporating the Lennard-Jones potential for non-bonded interactions, REBIND effectively addresses the low-degree atoms, which often suffer from high prediction errors in conventional models. The proposed force-aware edges between atoms provide an extra solution for creating molecular graphs besides relying on bonds and pairwise distances. REBIND could inspire further research in both conformer generation and broader applications in molecular property prediction.\n\nThis work contains rigorous experimental design and comprehensive evaluation on well-established datasets (QM9, Molecule3D, GEOM-DRUGS). The quantitative results are compelling, showing substantial improvements over baseline methods across multiple metrics.\n\nIn terms of clarity, the paper is well-structured, with clear explanations of the related concepts.", "weaknesses": "The authors didn't provide an efficiency evaluation of the proposed method. It would be better to know about the computational burden by introducing the force-aware adjacency matrices when compared to the ablated settings in ablation study.\n\nWhen introducing the prior works, the authors didn't fully cover the previous GNN works [1-3] in this field that make efforts to model non-bonded interactions. Unlike GTMGC, those works already use non-linear functions of distance with atom type-specific coefficients when modeling non-bonded interactions. It is better to mention them to strengthen the coverage of prior related works.\n\n[1] Kosmala, Arthur, et al. \"Ewald-based long-range message passing for molecular graphs.\" International Conference on Machine Learning. PMLR, 2023.\n\n[2] Zhang, Shuo, Yang Liu, and Lei Xie. \"A universal framework for accurate and efficient geometric deep learning of molecular systems.\" Scientific Reports 13.1 (2023): 19171.\n\n[3] Li, Yunyang, et al. \"Long-Short-Range Message-Passing: A Physics-Informed Framework to Capture Non-Local Interaction for Scalable Molecular Dynamics Simulation.\" International Conference on Learning Representations. 2024.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "viYXKj7KgV", "reviewer": "Reviewer_P3Lq", "summary": "The paper proposes a new attention mechanism for molecular conformation prediction tasks. This additional attention bias is constructed by extracting the estimated vdw force from the conformation predicted at an intermediate point in the network. The new architecture is then applied to the task of ground state prediction of molecules.", "questions": "Are the RMSD/MAE/RMSE results corrected for molecular atom perturbations? If not, it is important to add such a correction.", "limitations": "", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "The addition of the force-based bias is well motivated. The usage of supervised predictions from the center of the model into the second half of the architecture seems somewhat novel. The paper is mostly clear (although certain parts could be summarized and simplified).", "weaknesses": "Although the paper does not have major flaws, the contribution seems somewhat incremental and the experimental validation unclear. \n\nRegarding the contribution, while the authors present it as force-based graph rewiring, to me it appears that the authors simply add to the second half of the network a hand-crafted pairwise feature that is mapped with a linear layer to the attention bias of the transformer. Is my understanding correct? Are there key contributions I'm missing? With this in mind, while this is an interesting idea, I’m not sure it needs a 9 pages paper, especially if the experimental validation is not very strong. \n\nRegarding the experimental validation, my main concern is the task definition and the way that it is evaluated. Molecular conformer generation methods do not aim to generate a single pose but multiple poses that represent the different low-energy conformation that the molecule can take. For this reason, the objective function with which they are typically trained are very different than the one used by the authors and the evaluation metrics are also significantly different and look not at the accuracy in capturing one pose but the whole set. A metric like the E-RMSE that the authors propose makes no sense to me: we are optimizing for the models to learn a weighted geometric average of the different conformations. For example, if we have two poses equally energetically favorable, this evaluation function would rank best the method that predicts the geometric average of these two poses, why would this be an interesting conformation? I believe the authors should adopt the metrics regularly used in the molecular conformer generation field that look at coverage & precision of the generated poses as well as their energetic properties.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "reRRvNUwqN", "reviewer": "Reviewer_8Kvn", "summary": "This paper proposes a method to enhance the conformation prediction for non-bonded edges, inspired by the Lennard-Jones potential. Experimental results show that their method can improve the performance across multiple datasets, especially for those atoms with lower ranks.", "questions": "(1) As mentioned in the \"Weaknesses\" section, it would be helpful if the authors could provide a detailed analysis of the large errors in non-bonded interactions. Specifically, clarifying whether these errors are primarily attributable to model limitations rather than to the incomplete representation of ground-truth conformations would strengthen the discussion.\n\n(2) The \"E-RMSD\" metric combines normalized force and Boltzmann factors, which makes it challenging to distinguish the contributions of each. It would be beneficial to disentangle these components, perhaps by introducing Boltzmann factors/force alone.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "In general, this paper is a well-written ML paper.\n\n1. The background and motivation are clarified clearly. The methodology is reasonable and well-aligned with the motivation.\n\n2. The empirical study shows a consistent improvement on multiple datasets.", "weaknesses": "My main concern is whether the \"large errors\" observed for low-degree atoms arise from methodological limitations or issues in the evaluation process. Additionally, it's essential to assess how significant these \"errors\" might be for practical applications.\n\nThere might be multiple conformations with similar energy levels, thus they would probably occur with comparable Boltzmann weights. If this is the case, the sampling bias in the training/evaluation set might be the reason for large errors for low-degree atoms.\nIf a predicted conformation has a large RMSD with one ground truth, maybe it is also in an energy-favored state, but not included in the evaluation. \n\nFurthermore, flexibility in non-bonded conformations may not adversely impact empirical applications, as real-world cases often involve ensembles of multiple conformations.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lh1CaCNtsj", "reviewer": "Reviewer_twMi", "summary": "This article proposes a method, REBIND, to enhance the performance of deep learning (DL)-based molecular ground-state conformation prediction. The authors begin by identifying a key limitation in current approaches, which primarily rely on chemical bond information and interatomic distances but fail to adequately model atomic interactions, particularly for non-bonded atom pairs influenced by van der Waals potentials. Through detailed observations and statistical analysis, the authors conclude that current methods tend to accumulate more significant errors in predicting conformations of low-degree (i.e., low coordination numbers) molecules with fewer chemical bonds. This supports their viewpoint on the limitations of existing methods. To address this shortcoming, the authors introduce an approach based on the Lennard-Jones potential, deriving a new connectivity framework (comprising two adjacency matrices, $A^{attr}$ for attractive forces and for $A^{rep}$ repulsive forces) to redefine the molecular graph. By adding these new edges, the model enhances the representation of non-bonded atomic interactions as a form of data augmentation. The overall model architecture builds upon the GMTGC [1] network, replacing its original interatomic relation modeling component, $D^{row-sub}$, with the newly proposed $A^{attr}$ and $A^{rep}$, thereby strengthening the native self-attention mechanism. The results are clear and well-documented, strongly supporting the authors’ assertions. Comparative experiments demonstrate that these improvements lead to significant performance gains across various datasets.\n\nFor me, a major strength of this work is its clear observations and straightforward improvements that lead to noticeable performance gains, with a well-organized structure throughout. However, the methodological contributions feel somewhat modest. Given that molecular ground-state conformation prediction is still an underexplored area, this work holds potential to drive further advancements in the field.\n\n[1] Xu, Guikun, et al. \"GTMGC: Using Graph Transformer to Predict Molecule’s Ground-State Conformation.\" *The Twelfth International Conference on Learning Representations*.", "questions": "1. I would like to know the results of the observations in **Figure 1** on large datasets like Molecule3D.\n2. I would like to understand the trade-off in terms of overall forward pass time cost after incorporating compared to GTMGC.\n3. **as Weaknesses.2**", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The article is well-structured, with a clear and logical flow.\n\n2.\tThe authors begin by hypothesizing and questioning the accuracy of existing methods for modeling atomic interactions, then validate their assumptions through experimental observations. Building on this, they propose an improved method, making the work both robust and effective.\n\n3. The newly proposed metric, Energy-weighted RMSD (E-RMSD), is a practical contribution, enhancing evaluation standards for this task within the research community.\n\n4. The extension of experiments to the GEOM-DRUGS dataset is also a valuable contribution, enriching the research standards for this field.\n\n5.\tThe comparison in the ablation study between $A^{near}$ and $A^{force}$ effectively demonstrates the advantage of the authors’ proposed edge-enhancement method over the traditional distance-threshold-based neighbor edge enhancement. This approach could potentially be applied as a new edge construction strategy in other molecular modeling fields.", "weaknesses": "1. Some claims in the paper are unclear:\n   1. In the Introduction, the statement “To this end, we propose a novel encoder-decoder graph transformer architecture…” is ambiguous. The process of using a Transformer encoder to predict the initial conformation, followed by a decoder to refine the final conformation, is actually a contribution from the GTMGC [1] architecture and should be properly cited. The main contribution of this paper is the improvement of how atomic interactions are modeled within the GTMGC framework.\n   2. The definition of ${deg}^{rel}_i$ should be included and thoroughly explained in the Notation section, as it is the key metric used to classify nodes into low-degree and high-degree categories.\n\n2. The ablation study would be more convincing if it included an analysis of the effects of removing $A^{bond}$ and$D^{row-sub}$ . Additionally, testing combinations of $A^{bond}$, $D^{row-sub}$ , $A^{near}$ and $A^{force}$ could reveal whether certain configurations lead to better performance.\n\n3. In my opinion, the subjective evaluation results (Table 5) should be included in the main body of the paper for better clarity.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)