ET-Flow: Equivariant Flow-Matching for Molecular Conformer Generation
Authors: Majdi Hassan
Nikhil Shenoy
Jungyoon Lee
Hannes Stark
Stephan Thaler
Dominique Beaini
Venue: NeurIPS 2024 poster (2024)
Keywords: Flow Matching
molecular conformers
equivariant graph networks
Primary area: machine_learning_for_physical_sciences
TL;DR: We leverage flow matching and equivariant graph networks to state-of-the-art performance on generating conformers.

Abstract: Predicting low-energy molecular conformations given a molecular graph is an 
important but challenging task in computational drug discovery. Existing state-
of-the-art approaches either resort to large scale transformer-based models that
diffuse over conformer fields, or use computationally expensive methods to gen-
erate initial structures and diffuse over torsion angles. In this work, we introduce
Equivariant Transformer Flow (ET-Flow). We showcase that a well-designed
flow matching approach with equivariance and harmonic prior alleviates the need
for complex internal geometry calculations and large architectures, contrary to
the prevailing methods in the field. Our approach results in a straightforward
and scalable method that directly operates on all-atom coordinates with minimal
assumptions. With the advantages of equivariance and flow matching, ET-Flow
significantly increases the precision and physical validity of the generated con-
formers, while being a lighter model and faster at inference. Code is available
https://github.com/shenoynikhil/ETFlow.

ET-Flow: Equivariant Flow-Matching for Molecular Conformer Generation
Majdi Hassan"
Nikhil Shenoy +2,4
Jungyoon Lee"l
Hannes Stärk
Stephan Thaler
Dominique Beaini
Mila & Université de Montréal
University of British-Columbia
3 Massachusetts Institute of Technology Valence Labs
Abstract
Predicting low-energy molecular conformations given a molecular graph an important but challenging task in computational drug discovery. Existing state- of-the-art approaches either resort to large scale transformer-based models that diffuse over conformer fields, or use computationally expensive methods to gen- erate initial structures and diffuse over torsion angles. In this work, we introduce Equivariant Transformer Flow (ET-Flow). We showcase that well-designed flow matching approach with equivariance and harmonic prior alleviates the need for complex internal geometry calculations and large architectures, contrary the prevailing methods in the field. Our approach results a straightforward and scalable method that directly operates all-atom coordinates with minimal assumptions. With the advantages of equivariance and flow matching, ET-Flow significantly increases the precision and physical validity of the generated con- formers, while being lighter model and faster at inference. Code is available https//ggiitulccomm///emmiiiiiif
Introduction
Generating low-energy 3D representations of molecules, called conformers, from the molecular graph is a fundamental task in computational chemistry as the 3D structure of molecule is responsible for several biological, chemical and physical properties (Guimarães et al., 2012; Schütt et al., 2018, 2021: Gasteiger et al., 2020; Axelrod and Gomez-Bombarelli 2023). Conventional approaches to molecular conformer generation consist of stochastic and systematic methods. While stochastic methods such Molecular Dynamics (MD) accurately generate conformations, they be slow, cost-intensive, and have low sample diversity (Shim and MacKerell Jr, 2011: Ballard et al., 2015; De Vivo et al., 2016; Hawkins, 2017; Pracht et al., 2020). Systematic crule-based) methods (Hawkins al., 2010; Bolton et al., 2011: 1; Li et al., 2007; Miteva et al., 2010; Cole et al., 2018; Lagorce et al., 2009) that rely torsional profiles and knowledge base of fragments are much faster but become less accurate
Equal contribution. Correspondence to: {majdi hassan, , jungyoon. leel Cumontreal.. ca and aelalvaleeeeee laba.com com
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Rotation Alignment
ET-Flow F,G.t
Flow
Eg [1vg(C,x;)) 20.
Embadding Leyer
Equiverient Attention Layer
X Equivarient Attintion Layer
Representation
Equivariant Vectorr Output
Pt
Po Harmonic
Pi-Data
(b)
Figure 1: (a) Overview of ET-Flow. The model predicts conditional vector field tig using interpolated positions (2t), molecular structure (G), and time-step (t). Samples are drawn from the harmonic prior (No Po) and then rotationally aligned with the samples from data (x1 P1). A conditional probability path is constructed hetween pairs of Zo and 21, and I2 is then sampled from this path random time f. (b) The ET-Flow architecture consists of representation module based on the TorchMD-NET architecture (Thölke and De Fabritiis, 2022) and an equivariant vector output module. For detailed architecture and input preprocessing information, see Section A.I.
with larger molecules. Therefore, there has been increasing interest in developing scalable and accurate generative modeling methods in molecular conformer generation.
Existing machine learning based approaches use diffusion models (Ho et al., 2020; Song and Ermon, 2019) to sample diverse and high quality samples given access to low-energy conformations. Prior methods typically fall into two categories: diffusing the atomic coordinates in the Cartesian space (Xu al., 2022: Wang et al., 2024) or diffusing along the internal geometry such as pairwise distances, bond angles, and torsion angles (Ganea et al., 2021; Jing et al., 2022).
Early approaches based diffusion (Shi al., 2021; Luo al., 2021; Xu et al., 2022) faced challenges such as lengthy inference and training times as well as having lower accuracy compared cheminformatics methods. Torsional Diffusion (Jing al., 2022) the first outperform cheminformatics methods by diffusing only on torsion angles after producing an initial conformer with the chemoinformatics tool RDKIT. This reliance on RDKiT structures instead of employing an end-to-end approach comes with several limitations, such as restricting the tool to applications where the local structures produced by RDKiT are of sufficient accuracy. Unlike prior approaches, the current state-of-the-art MCF (Wang et al., 2024) proposes a domain-agnostic approach by learning to diffuse over functions by scaling transformers and learning soft inductive bias from the data (Zhuang et al., 2022). Consequently, it comes with drawbacks such as high computational demands due to large number of parameters, limited sample efficiency from a lack of inductive biases like euclidean symmetries, and potential difficulties in scenarios with sparse data a common challenge in this field.
In this paper, we propose Equivariant Transformer Flow (FT-Flow), a simple yet powerful flow- matching model designed generate low-energy 3D structures of small molecules with minimal assumptions. We utilize flow matching (Lipman et al., 2022; Albergo et al., 2023; Liu et al., 2022), which enables the learning of arbitrary probability paths beyond diffusion paths, enhancing both training and inference efficiency compared to conventional diffusion generative models. Departing from traditional equivariant architectures like EGNN (Satorras et al., 2021), we adopt an Equivariant Transformer (Thölke and De Fabritiis, 2022) to better capture geometric features. Additionally, our method integrates a Harmonic Prior (Jing et al., 2023; Stark et al., 2023), leveraging the inductive bias that atoms connected by bond should be in close proximity. We further optimize our flow matching objective by initially conducting rotational alignment on the harmonic prior, thereby constructing shorter probability paths between source and target distributions at minimal computational cost.

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Our contributions can be summarized as follows:
1. We obtain state-of-the fi--the-art art precision for molecule conformer prediction, resulting more physically realistic and reliable molecules for practitioners. We improve upon the previous methods by large margin on ensemble property prediction.
2. We highlight the effectiveness of incorporating equivariance and more informed priors in generating physically-groundee
molecules in our simple yet well-engineered method.
3. Our parameter-effiieen model requires orders of magnitude fewer sampling steps than GeoDiff (Xu et al., 2022) and has significantly fewer parameters than MCF (Wang et al., 2024).
2 Background
Diffusion Generative Models. Diffusion models (Song and Ermon, 2019; Song et al., 2020; Ho et al., 2020) enables a high-quality and diverse sampling from an unknown data distribution by approximating the Stochastic Differential Fquation(SDE) that maps simple density i.e. Gaussian to the unknown data density. Concretely, it involves training neural network to learn the score, represented as Vx log Pe(x) of the diffused data. During inference, the model generates sample by iteratively solving the reverse SDE. However, diffusion models have inherent drawbacks, as they (i) require on longer training times (ii) are restricted to specific probability paths and (iii depend on the use of complicated tricks to speed up sampling (Song et al., 2020; Zhang and Chen, 2022).
Flow Matching. Flow Matching (Albergo et al., 2023; Lipman et al., 2022; Liu al., 2022) provides a general framework to learn Continuous normalizing flows (CNFs) while improving upon diffusion models in simplicity, generality, and inference speed in several applications. Through simple regression against the vector field reminiscent of the score-matching objective in diffusion models, Flow matching has enabled fast, simulation-free training of CNFs. Several subsequent studies have then expanded the scope of flow matching objective to manifolds (Chen and Lipman, 2024), arbitrary sources (Pooladian al., 2023), and conditional flow matching with arbitrary transport maps and optimal couplings between source and target samples (Tong et al., 2023).
Molecular Conformer Generation. Various machine learning (ML) based approaches (Kingma and Welling, 2013; Liberti et al., 2014; Dinh et al., 2016; Simm and Hernández- Lobatto 2019; Shi et al., 2021; Luo et al., 2021; Xu et al., 2021; Ganea al., 2021; Xu al., 2022: Jing et al., 2022; Wang et al., 2024) have been developed to improve upon the limitations which the most advanced TorfionDiff (Jing et al., 2022) and Molecular Conformer Fields (MCF) (Wang et al., 2024). TorsionDiff designs diffusion model on the torsion angles while incorporating the local structure from RDKiT ETKDG (Riniker and Landrum, 2015). MCF trains a diffusion model over functions that map elements from the molecular graph to points in 3D space.
conventional methods, among
Equivariant Architectures for Atomistic Systems. Inductive biases play an important role in generalization and sample efficiency. In the case of 3D atomistic modelling, one example of a useful inductive bias is the euclidean group SO(3) which represents rotation equivariance in 3D space. Recently, various equivariant architectures (Duval et al., 2023) have been developed that act on both Cartesian (Satorras al., 2021; Thölke and De Fabritiis, 2022; Simeon and De Fabritiis, 2024; Du et al., 2022; Frank et al., 2022) and spherical basis (Musaelian et al., 2023; Batatia et al., 2022; Fuchs et al., 2020; Liao al., 2023; Passaro and Zitnick, 2023; Anderson et al., 2019; Thomas et al., 2018). For molecular conformer generation, initial methods like ConIGF, DGSM utilize invariant networks they act upon inter-atomic distances, whereas the use of equivariant GNNs have been used in GeoDiff (Xu et al., 2022) and Torsional Diffusion (Jing et al., 2022). GeoDiff utilizes EGNN (Satorras al., 2021), Cartesian basis equivariant architecture while Torsional Diffusson uses Tensor Field Networks (Thomas et al., 2018) to output pseudoscalars.
3 Method
We design ET-Flow, a scalable equivariant model that generates energy-minimized conformers given molecular graph. In this section, we layout the framework to achieve this objective by detailing the generative process in flow matching, the rotation alignment between distributions, stochastic sampling, and finally the architecture details.
3

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Preliminaries We define notation that we use throughout this paper. Inputs are continuous atom positions X E RN*x where N is the number of atoms. We use the notation vx(x) interchangeably with v(t, x) for vector field.
3.1 Flow Matching
The aim to learn time-dependent vector field (x()) RNx3 [0, NR associated with the transport map X, RR x3 x [0, 1] R that pushes forward samples from a base distribution Po, often an easy-to-sample distribution, to samples from more complex target distribution P1, the low-energy conformations of molecule. This can be defined as an ordinary differential equation (ODE).
(1) where zo Po- We construct the U, via a dime-differentiable interpolation between samples from Po and 01 that gives rise to a probability path Pr that we can easily sample (Lipman et al., 2022: Liu et al., 2022; Albergo and Vanden-Eijnden 2023; Tong et al., 2023). The general interpolation between samples zo ~ Pu and 21 ~ P1 can be defined as:
(x(x) (x)(xx)),
X Xo+
(x0, x1) axx1 + 8,x0.
(2)
Given this interpolant that couples Xo and X1, can define the conditional probability path as (x(x[xo, x1) WXx(/e(xx x1). x1). I),), and the vector field can be computed as ve(x).
O.ptl (x(xx, x1) which has the following form
(x(x) 6; + Bexo + d,z
z ~0.1).
Here use á as shorthand notation for 0jar and similarly apply the same notation to / and In our work, use linear interpolation where 04 t. Br in the vector field
and ar a / t), resulting
1 2t
(x) = X X0
2v - z.
Now, we can define the objective function for learning a vector field vy (x))) that generates probability path Pr between base density Po and the target density PI as,
x) wo (t, x²
For training, sample Xo Po. X1 ~ Pt, and (10,) 1), (ii) interpolate according to Equation 2, (iii) add noise from a standard Gaussian, and (iv) minimize the loss defined in Equation 5. For sampling, we sample x ~ Po and integrate from each time-step, the Euler solver iteratively predicts the vector field for x; and updates its position Xt+A: = X, (t At. More details on the training and sampling algorithms are provided in Appendix B.
t 1 using the Euler's method. At
3.2 Alignment
Several previous works (Tong et al., 2023; Klein al., 2024; Jing al., 2024; Song et al., 2024) demonstrate that constructing a straighter path between base distribution po and target distribution P1 minimizes the transport costs and improves performance. In work, reduce the transport costs between samples from the harmonic prior Po and samples from the data distribution by rotationally aligning them using the Kabsch algorithm (Kabsch, 1976) similar (Klein al., 2024; Jing et al., 2024). This approach leads to faster convergence and reduces the path length between atoms by leveraging the similarity in "shape" of the samples as seen Figure without incurring high computational cost.
3.3 Stochastic Sampling
We employ a variant of the stochastic sampling technique inspired by (Karras et al., 2022). Specifically, we inject noise at each time step to construct an intermediate state, evaluate the vector field from the intermediate state, and then perform the deterministic ODE step from the noisy state. The original method utilizes a second-order integration, which averages the denoiser output at the noisy intermediate state and the state at the next time step after integration.

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In our experiment, we use the stochastic sampler without this second-order correction term, which empirically provided performance boost comparable to the second-order method. We apply stochastic sampling only during the final part of the inte- gration steps, specifically within the range t E [0.8, 1.0]. This helps prevent drifting towards overpopulated density regions and improves the quality of the samples (Karras et al., 2022).
posifam
Stochastic sampling has improved both diversity and accuracy of the generated conformers, measured by Coverage and Aver- age Minimum RMSD (AMR) respectively as shown in Table 1. Detailed information the stochastic sampling algorithm is provided in algorithm B.
uie
Figure 2: Stochastic sampling pro- cedure used in inference. Noise is added to the positions I; indicated by the purple line, resulting in øe- Then, the model predicts the vec- tor field ét from 2, instead of zt indicted by the yellow line and up- dates it using is get 2++11
3.4 Chirality Correction
While generating conformations, is necessary take account of the stereochemistry of atoms bonded to four distinct groups also referred to tetrahedral chiral centers. To generate con- formations with the correct chirality, we propose simple post hoc trick as done in GeoMol (Ganea al., 2021). ). We compare the oriented volume (OV) (Equation 6) of the generated conformation and the required orientation with the RDKit tags. the case of mismatch, we simply flip the conformation against the z-axis. This correction step be efficiently performed comparison with the required RDKit tags and an inversion of position if necessary.
batched operation since it involves a simple
1
1
1
(V P2, P2,
sign
42
T3
F4
5/
1/2
1/3
1/4
21
22
23
24
We also consider an alternative approach for chirality correction. Instead of using the post hoc correction with our 0(3) equivariant architecture, we slightly tweak our architecture to make it SO(3) equivariant by introducing cross product term in the update layers. We compare these methods on both the method (ET-Flow) corresponds to using the post hoc correction whereas the SO(3) variant referred by (33. We empirically observe that using additional chirality correction step is not only computationally efficient, but also performs better. We provide details on the architectural modification and proof of (3) equivariance in Section A.1 and Section C. respectively.
and GEOM-QM9 dataset in Table and Table 2. Our base
3.5 Architecture
ET-Flow (Figure lb) consists of two main components: (1) a representation module based on the equivariant transformer architecture from TorchMD-NET (Thölke and De Fabritiis, 2022) and (2) the equivariant vector output module. In the representation module, an embedding layer encodes the inputs (atomic positions, atomic numbers, atom features, bond features and the time-step) into set of invariant features. Initial equivariant features are constructed using normalized edge vectors where the edges are constructed using radius graph of 10 angstrom and the bonds from the 2D molecular graph. Then, series of equivariant attention-basee equivariant features using a multi-head attention mechanism. Finally, the vector field is produced by the output layer, which updates the equivariant features using gated equivariant blocks (Schütt et al., 2018). Given that TorchMD-NET was originally designed for modeling neural network potentials, we implement several modifications its architecture to better suit generative modeling, as detailed Section A.
layers update both the invariant and
Experiments
We empirically evaluate ET-Flow by comparing the generated and ground-truth conformers in terms of distance-based RMSD (Section 4.2) and chemical property based metrics (Section 4.4). We
5

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Table I: Molecule conformer generation results on GEOM-DRUGS (6 0.75A). ET-Flow SS ET-Flow with stochastic sampling and ET-Flow SS((3 is ET-Flow using the SO architecture for chirality correction. For ET-Flow, ET-Flow-SS and ET-Flow-50 (3),, we sample conformations over 50 time-steps.
Recall
Precision
Coverage
AMR
Coverage 1
AMR !
mean
median
mean
median
mean
median
mean
median
GeoDiff
42.10
37.80
0.835
0.809
24.90
14.50
1.136
1.090
GeoMol
44.60
41.40
0.875
0.834
43.00
36.40
0.928
0.841
Torsional Diff.
72.70
80.00
0,582
0.565
55,20
56.90
0.778
0.729
MCF- S (13M)
79.4
87.5
0.512
0.492
57.4
57.6
0.761
0.715
MCF B (62M)
84.0
91.5
0.427
0.402
64.0
66.2
0.667
0.605
MCF (242M)
84.7
92.2
0.390
0.247
66.8
71.3
0.618
0.530
ET-Flow (8.3M)
79.53
84.57
0.452
0.419
74.38
81.04
0.541
0.470
ET-Flow SS (8.3M)
79.62
84.63
0.439
0,406
75.19
81.66
0.517
0.442
ET-Flow (3(3) (9.IM)
78.18
83.33
0,480
0.459
67.27
71.15
0.637
0.567
present the general experimental setups in Section 4.1. The implementation details are provided in Appendix A.
4.1
Experimental Setup
Dataset:
We conduct our experiments on the GEOM dataset (Axelrod and Gooez-Bombareee 2022), which offers curated conformer ensembles produced through meta-dynamies in CREST (Pracht et al., 2024). Our primary focus is GEOM-DRUGS, the most extensive and pharmacologically relevant subset comprising 304k drug-like molecules, each with an average of atoms. We use train/validation/ees. /243473/30433/1000 split as provided in (Ganea et al., 2021) Additionally, train and test model on GEOM-QM9, a subset of smaller molecules with an average of atoms. Finally, in order to assess the model's 's ability to generalize to larger molecules, trained on DEOM-DRUGE on a GEOM-XX XL dataset, a subset large molecules with more than 100 atoms. The results for GEOM-QMA RMM and GEOM-XL can be found in the Appendix D.
evaluate the model
Our evaluation methodology similar that of (Jing al., 2022). First, we look at RMSD based metrics like Coverage and Average Minimum RMSD (AMR) between generated and ground truth conformer ensembles. For this, we generate 2K conformers for a molecule with K ground truth conformers. Second, we look at chemical similarity using properties like Energy (E), dipole moment (f), HOMO-LUMO gap (Ae) and the minimum energy (Emin) calculated using xTB (Bannwarth al., 2019).
Evaluation:
Baselines: We benchmark ET-Flow against leading upproaches outlined Section Specifically, we assess the performance of GeoMol (Ganea et al., 2021), GeoDiff (Xu et al., 2022), Torsional Diffusion (Jing al., 2022), and MCF (Wang et al., 2024). Notably, the most recent among these, MCF, has demonstrated superior performance across evaluation metrics compared to its predecessors. It's worth mentioning that GeoDiff initially utilized a limited subset of the GEOM-DRUGS dataset; thus, for fair comparison, we consider its re-evaluated performance presented in (Jing et al., 2022).
4.2 Ensemble RMSD
As shown Table I and Table ET-Flow outperforms all preceding methodologies and demonstrates competitive performance with the previous -fete--rtt,, MCF (Wang et al., 2024). Despite being significantly smaller with only 8.3M parameters, ET-Flow shows substantial improvement in the quality of generated conformers, as evidenced by superior Precision metrics across all MCF models, including the largest MCF-L. When compared to MCF-S, which is closer in size, ET-Flow achieves markedly better Precision while the impact Recall is less significant and limited to Recall Coverage. Notably, our Recall AMR remains competitive with much bigger MCF-B, underscoring the inherent advantage of our method in accurately predicting overall structures.

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Table 2: Molecule conformer generation results on GEOM-QM9 0.5Á), ET-Fllww SO(3) ET-
Flow using the SO(3) architecture for chirality correction. For both ET-Flow and ET-Flow-50(3). we sample conformations over 50 time-steps.
Recall
Precision
Coverage
AMR
Coverage
AMR
mean
median
mean
median
mean
median
mean
median
CGCF
69.47
96.15
0.425
0.374
38.20
33.33
0.711
0.695
GeoDiff
76.50
100.00
0.297
0.229
50.00
33.50
1.524
0.510
GeoMol
91.50
100.00
0.225
0.193
87.60
100.00
0.270
0.241
Torsional Diff.
92.80
100.00
0.178
0.147
92.70
100.00
0.221
0.195
MCF
95.0
100.00
0.103
0.044
93.7
100.00
0.119
0.055
ET-Flow
96.47
100.00
0.073
0.047
94.05
100.00
0.098
0.039
ET-Flow SO(33
95.98
100.00
0.076
0.030
92.10
100.00
0.110
0.047
4.3 Coverage Threshold Plots
We compare the coverage metrics of ET-Flow against Torsional diffusion (Jing et al., 2022) and MCF (Wang al., 2024) against wide range of thresholds the GEOM DRUGS dataset in Figure ET-Flow consistently outperforms previous methods in precision based metrics. In terms of recall, approach demonstrates better performance than Torsional Diffusion across all thresholds. Despite MCF performing better at higher thresholds, ET-Flow outperforms in the lower thresholds, underscoring its proficiency in generating accurate conformer predictions.
Cocil mean
Coos mean (T)
Model
FTF
Threshold
Threshhold
Cov-R Cow-Rmeddann treeian (7) threshuld
Coe-P median [T] vs. threshald
Coe-Pmeedien
714
Cousil medlen
Mozel ET Roo 3V
Model ET Flow 18 3NI
onii Diffucion
Tardonal Detusioo MCT -8 (825))
MCF 5 (827)
MCF (120))
MCF (.29)
0.25
1.30
1.2
3.50
3.75
3/6
3/6
Threshold
Threshold
Figure 3: Recall and Precision Coverage result on GEOM-DRUGS function of the threshold distance. ET-Flow outperforms TorsionDiff by a large margin especially in a lower threshold region. We emphasize the better performance of ET-Flow lower thresholds in both Recall and Precision metrics.
4.4 Ensemble Properties
RMSD provides geometric measure for assessing ensemble quality, but it also essential to consider the chemical similarity between generated and ground truth ensembles. For random 100-molecule subset of the test set GEOM-DRUGS, a molecule has K ground truth conformers, we generate minimum of 2K and maximum of 32 conformers per molecule. These conformers are then relaxed using GFN2-xTB (Bannwarth et al., 2019), and the Boltzmann-weightee properties of the generated and ground truth ensembles are compared. Specifically, using xTB (Bannwarth al.,
7

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2019), we compute properties such as energy (E), dipole moment (μ),, HOMO-LUMO gap (Ac), and the minimum energy (Em.n). Table 3 illustrates the median errors for ET-Flow and the baselines, highlighting our method's capability to produce chemically accurate ensembles. Notably, we achieve significant improvements over both TorsionDiff and MCF across all evaluated properties.
Table 3: Median averaged errors of ensemble properties between sampled and generated conformers
(E, Ae, Emin in kcal/mol, and j in debye).
E
Ac
Emin
OMEGA
0.68
0.66
0.68
0.69
GeoDiff
0.31
0.35
0.89
0.39
GeoMol
0.42
0.34
0.59
0.40
Torsional Diff.
0.22
0.35
0.54
0.13
MCF
0.68
0.28 0.05
0.63:10.05
0.04:10.00
ET-Fllww
0.18 0.01
0.18 118+000 0.01
0.35
0.02++0.00
4.5 Inference Steps Ablation
In Table 1, our sampling process with ET-Flow utilizes 50 inference steps. To evaluate the method's performance under constrained computational resources, we conducted an ablation study by progres- sively reducing the number of inference steps. Specifically, we sample for 5, 10 and 20 time-steps. The results on GEOM-DRUGS are presented in Table 4. We observed minimal performance degra- dation with decrease in the number of steps. Notably, ET-Flow demonstrates high efficiency, maintaining performance across all precision and recall metrics even with as few as inference steps. Interestingly, ET-Flow with 5 steps still achieves superior precision metrics compared to all existing methods. This underscores ET-Flow's ' ability to generate high-quality conformations while operating within limited computational budgets.
4.6 Sampling Efficiency
Doyi moon (11
Co-Rinere (1 Iore
Tyntooal Dihace
Time 11
Th bl
NS2 (1.34)
CTilton 12,390
MWB P newn (av. Bio
AMR On TIee
Strpe
stree
Tine 151
Trme S1
Figure 4: Sampling efficiency as a measure of the quality of Inference time with respect to the number of time steps GEOM-DRUGS.
We demonstrate the ability of ET-Flow to generate samples efficiently. We evaluate the inference time per molecule over varying number of time steps and report the average time across 1000 random samples from the test set of GEOM-DRUGE diffusion (Jing et al., 2022) in inference across all time steps. While ET-Flow may not achieve the fastest raw inference times (potentially due to MCF variants benefiting from optimized CUDA kernels for attention), it maintains competitive speeds while ensuring higher precision. We suspect that concurrent work on improving equivariant operations with optimized CUDA kernels (Leee et al., 2024) should lead to similar efficiency gains seen in transformer-bast based architectures.
Figure shows that ET-Flow outperforms Torsional
8

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Table 4: Ablation over number of inference steps on GEOM-DRUGH (Ó 0.75A). Performance of ET-Flow at steps is competent across all metrics while also retaining oftte-of-the-arr performance on precision metrics when compared with previous methods.
Recall
Precision
Coverage f
AMR +
Coverage
AMR
mean
median
mean
median
mean
median
mean
median
ET-Flow (5 Steps)
77.84
82.21
0.476
0.443
74.03
80.8
0.55
0.474
ET-Flow (10 Steps)
79.05
84.00
0.451
0.415
74.64
81.38
0.533
0.457
ET- FT-Flow (20 Steps)
79.29
84.04
0.449
0.413
74.89
81.32
0.531
0.454
ET-Flow (50 Steps)
79.53
84.57
0.452
0,419
74.38
81.04
0.541
0.470
ET-Flow effectively balances performance and speed. making it ideal for tasks that require high sample quality with efficient computation. With the ability to generate high-quality samples in fewer time steps, e.g., 5 time steps, as indicated in Table 4, ET-Flow is well-suited for scenarios demanding large number of samples, we encountered difficulties running MCF-L for 20 and 50 steps, so those results have not been included. summary, ET-Flow demonstrates efficient sampling, balancing precision and speed, making it highly effective for generating high-quality molecular samples while remaining competitive in inference time.
fewer steps lead to lower inference time per molecule. Additionally,
Conclusion
In this paper, present simple and scalable method ET-Flow, which utilizes equivariant trans- former with flow matching to achieve state-of-the-art performance on multiple molecular conformer generation benchmarks. By incorporating inductive biases, such as equivariance, and enhancing probability paths with a harmonic prior and RMSD alignment, we significantly improve the precision of the generated molecules, and consequently generate more physically plausible molecules. Impor- tantly, our approach maintains parameter and speed efficiency, making it not only effective but also accessible for practical tigh-throughhut applications.
Limitations And Future Works
While ET-Flow demonstrates competitive performance in molecular conformer generation, there areas where it can be enhanced. One such area is the recall metrics, which capture the diversity of generated conformations. Another area is the use of additional chirality correction step that is used to predict conformations with the desired chirality. Moreover, although our performance on the GEOM-XL dataset is comparable to MCF-S and TorsionDiff, there is still room for improvement.
We propose three future directions here. First, we observe during experiments that a well-designed sampling process incorporating stochasticity can enhance the quality and diversity of generated samples. An extension our current approach could involve using Stochastic Differential Equations (SDEs), which utilize both vector field and score in the integration process, potentially improving the diversity of samples. Second, we propose to scale the number of parameters ET-Flow, which has not only been shown to be useful across different domains of deep learning, but has also shown be useful molecular conformer generation for MCF (Wang et al., 2024). Third, to better handle the chirality problem, we aim to explore alternatives for incorporating S0(3) (equvvaavvrr into the model in the future.
Acknowledgements
The authors sincerely thank Cristian Gabellini, Jiarui Ding, and the NeurIPS reviewers for the insightful discussions and feedback. Resources used in completing this research were provided by Valence Labs. Furthermore, we acknowledge a grant for student supervision received by Mila Quebec's 's AI institute and financed by the Quebee ministry of Economy.

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A
Implementation Details
A, G
X Tij
Pi
7j
X2
3j
dj
xj
RBF
Embeciding Layer
attantion
max(H-Lo
Equivariant Attention Layer
Q
D
D
a
a
X N
LNO
LNO
Equivariant Attention Layer
(22)
Representation Module
(s>)
Equivariant Vector Output Layer
Equiveriant Attention Layer
Multi-Head
Attention
Vi
& x
sij Sij
Figure 5: (a) Overall Architecture of ET-Flow consisting of 2 components, (1) Representation Layer based on TorchMD-NET Thölke and De Fabritiis (2022) and (2) Equivariant Output Layer from (Schütt et al., 2018). (b) Equivariant Attention Layer with all the operations involved, (c) Multi-Head Attention block modified with the LayerNorm.
A.1 Architecture
The ET-Flow architecture (Figure 5) consists of major components, representation layer and output layer. the representation layer, we use modified version of embedding equivariant attention-hased update layers from the equivariant transformer architecture of TorchMD-NEE (Thölke and De Fabritiis, 2022). The output layer utilizes the gated equivariant blocks from (Schütt et al., 2018). We highlight our modifications over the original TorchMD-NET architecture with this color. These modifications enable stabilized training since we use a larger network than the one proposed in TorchMD-NET (Thölke and De Fabritiis, 2022) paper. Additionally, since our input structures are interpolations between structures sampled from prior actual conformations, it is important to ensure our network is numerically stable when the interpolations contain two atoms very close to each other.
The embedding layer maps each atom's physical and chemical properties into learned representation space, capturing both local atomic features and geometric neighborhood information. For the i-th atom in molecule with atoms, we compute an invariant embedding x; through following process:
Embedding Layer:
2, = embed (&))
(7)
hg = MIP((A.)
(8)
where 2; is atomic number and represents atomic attributes (detailed in Appendix A). The ML. projects atomic attributes into feature vector of dimension d,.
Next, we compute a neighborhood embedding v4; that captures local atomic environment:
embed (z,) byy)
T1:
Here, embed" (2,) provides a separate embedding for neighboring atomic numbers, dej is the distance between atoms i and j. and ly encodes edge features (either from radius-based graph molecular bonds). The interaction function g(daj, (e)) combines distance and edge information:
g(de;, 4y) = F [o(dg)) (et (de)).... .., (lde, eek REF
(duy), Lg]
(10)
RBF are K exponential radial basis functions following (Unke and Meuwly, 2019), and ((d,,) is smooth where eg cutoff function:
coos(
sd.; danel + 11) if day S deuatt otherwise
((d.))
(11)
14

---

Finally, we combine all features into the atom's 's embedding through a linear projection:
2;
W (mmbed"(a))) h,,t,
(12)
where t represents the time-step. and [. denotes concatenation. The resulting embedding E R serves as input to subsequent layers the network.
The multi- head dot-product attention operation uses atom features 2,, atom attributes ht, time-step and inter-atomic distances di to compute attention weights. The imput atom-level features 2; are mixed with the atom attributes hı and the time-step i using an MLP and then normalized using a LayerNorm (Ba al., 2016). To compute the attention matrix, the inter-atomic distances di are projected into two dimensional filters D and D as:
Attention Mechanism:
(BF() DK = =WOD RBF (diy) bOx)
D" =a (WOD RDF ( + )00))
(13)
The atom level features are then linearly projected along with a LayerNorm operation to derive the query Q and key K vectors. The value vector V is computed with only the linear projection of atom-level features. Applying LayerNorm on Q. K vectors (also referred to as QK-Norm) has proven to stabilize un-normalized values in the attention matrix (Dehghani et al., 2023; Esser et al., 2024) when scaling networks to large number of parameters. The Q and K vectors are then used along with the distance filter D for a dot dot-product operation over the feature dimension:
(oo Q LayerNorml (W xx.), K LayerNorm (W xx), V W F.
(14)
do K, K,D")
Σ I0.- KA DE
(15)
The attention matrix is derived by passing the above dot-product operation matrix through a non-linearity and weighting using cosine cutoff (das)) (similar the embedding layer) which ensures the attention weights are non-zero zero only when two atoms are within specified cutoff:
A = dio.Udot( (Q, K, )) ((dr))
(16)
Using the value vector and distance filter Dy. we derive equally sized filters by splitting along the feature dimension,
Sej: Siy: Siy split(V, DK)).
(17)
A linear projection is then applicd to combine the attention matrix and the vectors s to derive an atom level feature y. Wo [EN Aug- -33) The output of the attention operation are y, (an atom level fcature) and two scalar filters 5,y and oř (edge-level features).
Update Layer: The update layer computes interactions between atoms in the attention block and uses the outputs update the scalar feature and the vector feature ih. First, the scalar feature output yi from the attention mechanism split into three features (q), , q q²), out of which of and qf are used for the scalar feature update as,
Az, क! o -C17-U22)
(18)
where (U, U2ú;) is the inner product between linear projections of vector features T with matrices U1, U2.
The edge vector update consists of two components. First, we compute a vector wi, which for each atom is computed as weighted sum of vector features and a clamped-norm of the edge vectors over all neighbors:
N
+
(19)
7. P cl.)))
AT, + q. Uz
(20)
where Ui and U' are projection matrices over the feature dimension of the vector feature n.. In this layer, we clamp the minimum value of the norm (to e 0.01) to prevent numerically large values in cases where positions of two atoms are sampled too close from the prior.
SO(3) Update Layer: We also design an SO(3) equivariant architecture by adding an additional cross product term Equation 19 follows,
N
F F3
+ s/ (7xx
F. F3 max [i - 1
17, = Σ s/y U. +
(21)
max F - F,ll, c)
15

---

where sij" is derived by modifying the split operation Equation 17 in the attention layer where the value vector V and distance filter Dy is projected into equally sized filters instead of 3.
Output Layer: The oulput layer consists of Gated Equivariant Blocks from (Schitt al., 2018). Given scalar 24 and vector features i. the updates in each block is defined as,
Z updandd W. split(MLP((z,, U1ū4]))
(22)
updaatd = (U2i)) 18,,
(2.3
Here, Un and U2 are linear projection matrices that act along feature dimension. Our modification use LayerNorm in the MLP to improve training stability.
A.2 Input Featurization
Atomic features Node Features) computed using RDKit (Landrum al., 2013) features described in Table For computing edge features and edge index, we use combination of global (radius based edges) and local (molecular graph edges) similar to (Jing et al., 2022).
Name
Description
Range
chirality
Chirality Tag
[unspecified, tetrahedral CCW, other)
degree
Number of bonded neighbors
{x:0 0 2 10, 2}
charge
Formal charge of atom
te:-55x 5,
num_H
Total Number of Hydrogens Number of Radical Electrons
(r:05 (m:05ag1,ne)
number_rant caabbrrradica__eeee
hybrization
Hybrization type
(sp. sp'. .p'...... sp'..pp sp'd', other (True, False) (True, False]
aromatic
Whether aromatic ring
in_ring
Whether in ring
Table Atomic features included in ET-Flow.
A.3 Evaluation Metrics
Following the approaches of (Ganca et al., 2021; Xu ct al., 2022; Jing et al., 2022), we utilize Average Minimum RMSD (AMR) and Coverage (COV) to assess the performance of molecular conformer generation. Here, Cg denotes the of generated conformations, and C, denotes the set of reference conformations. For both AMR and COV, we calculate and report Recall (R) Precision (P). Recall measures extent to which generated conformers capture the ground-truth conformers, while Precision indicates the proportion of generated conformers that are accurate. The specific formulations for these metrics are detailed in the following equations:
AMR-R(C,, C+)
Σ min RMSD(R, R)
Cr
RECT REC,
COV-RIC,, C+)
(R C.JRMSD(RR. R 6, R C2)
AMR-P Cb)
Σ
min RMSD(R, R.) REC,
IC
REC,
COV-P(C,, C.
[[R C,/RMSD(R, R) < 8, R.H.H.
lower AMR score signifies improved accuracy, while higher COV score reflects greater diversity the generative model. Following (Jing et al., 2022), the threshold 8 is set to 0.5A for GEOM-QM9 and 0.75A for
A.4 Training Details and Hyperparameters
For GEOM-DRUGS, we train ET-Flow for fixed 250 epochs with batch size of 64 and 5000 training batches per epoch per GPU on 8 A 100 GPUs. For the learning rate, use the Adam Optimizer with a cosine annealing learning rate which goes from maximum of 10 to minimum 10-7 over 250 epachs with weight decay of 200 epochs with batch size 128, and use all of the training dataset per epoch on 4 A100 GPUs. We use the cosine annealing learning rate schedule with maximum of 8 10 to minimum of 10 over 100 epochs, post which the maximum is reduced by factor of 0.05, We select checkpoints based on the lowest validation emror.
10-10 10 For GEOM- RMM,, we train ET-Flow
16

---

Hyper-parametee
ET-Flow
num, mam_eaers channels
20
160
num num_heads Slaghbor_embeddiie
True
autoff_lower
0.0
cutoff cutoff_higher _iiheer
10.0
node_attr_dir
edge_attr_din
reduce_op
True
activation
SiLU
attn activatii
SiLUU
param
8.3M
Table 6: Hyperparameters for ET-Flow
B Training and Sampling Algorithm
The following algorithms go over the pscudo-code for the training and sampling procedure. For each molecule, we use up to 30 conformations with the highest boltzmann weights as provided by CREST (Pracht et al., 2024) similar to that of (Jing et al., 2022)
Algorithm 1: Training procedure
Input: molecules [Go..., Gv] each with true conformers [Ca.,,...Car ke. the harmonic prior Po. learning rate a, number of epochs Ne. initialized vector field dq Output: trained flow matching model ti for 4- to N₂ do
for G in [oo ..., GN] do Sample ~ (1 1] and C₁ E [CG.,,,.CG..al)
Sample prior Co (G);
Align Co + RMSDAlign((( Co. C1);
Sample C. = tC1 + (1 t)Co + 0²t(1 - -)),
2~
NOOII):
Construct vector field + x1 40 + 22((1-())
1-2t
Compute loss < [t, C.) url
Take gradient step f + 0 - VVo:
Algorithm Inference procedure
Input: molecular graph G, number conformers K, number of sampling steps N Output: predicted conformers [C1,...Ck]
for C. in [CI,..CCk] do
sample prior C ~ (G):
for n < N
Set <- Ni
Set Al N
Predict i = (t Ō);
Update Ć = Ć i
17

---

Algorithm 3: Stochastic Sampler
Input: molecular graph G, number conformers K, number of sampling steps N, stochasticity level churn, stochastic sampling range [iin maar)
Output: predicted conformers [Cl
for C in [Ck, do
sample prior C ~ Pu(G); for n
N do
Set / +- N
Set At N
Set Y
churn,
N
i [bmins mmaz] then Sample ~ N (0,1);
Al + 7(1 1);
8 marat Ař, 0);
Ó+ Aiv f2 Pe:
Predict à = nf. Ō):
Set At + At + At;
else
Predict é = (t, (t, C);
Update C = C GAt
Proofs
C.1 Designing SO(3) Equivariance
We show that satisfics rotation equivariance and reflection asymmetry. Let 0 and 82 be lincarly independent non-zero vectors | > 0, || > 0, and a scalar. We implement SO(3) equivariance by adding a vector with a cross product. We show that vector = J. + =tū x 02)), where ह x i2 denotes cross product of ij and iz satisfies anti-symmetry while maintaining rotation equivariance as follows,
modify the architecture in Section A.1 (Equation 18) to produce final vector output that
Rū; + s(RU,
R5y) R(T,) sR( 6
(24)
R(ú s 12))
(25)
3 + s 3.
- s( (g)
(26)
# --6 st x (2)
(27)
This concludes the proof for rotation equivariance and reflection anti-symmetry.
D Additional Results
D.I Design Choice Ablations
We conduct series of ablation studies to assess the influence of each component in the ET-Flow. Particularly, the experiments with (1) 0(3) equivariance without chirality correction, (2) Absence of Alignment, (3) Gaussian Prior base distribution. We demonstrate that improving probability paths and utilizing an expressive equivariant architecture
correct symmetries are key components for ET-FFow achieve state of the art performance. The ablations were ran with reduced settings (50 epochs; Ir = le 4; A 100 gpus). Results are shown in Table D.1.
D.2 Results on GEOM-XL
We now assess how well model trained on GEOM- DRUGS generalises to unseen molecules with large numbers of atoms, using the GEOM-XL XL dataset containing total of 102 molecules. This provides insights into the model's capacity tackle larger molecules and distr--distriuution tasks. Upon executing the checkpoint
18

---

Table 7: Ablation results GEOM-DRUGS.
Recall
Precision
Coverage 1
AMR
Coverage 1
AMR
mean
median
mean
median
mean
median
mean
median
ET-Flow
75.37
82.35
0.557
0.529
58.90
60.87
0.742
0.690
Flow (0(3))
72.74
79.21
0.576
0.556
54.84
54.11
0.794
0.739
ET-Flow (w/o Alignment)
68.67
74.71
0.622
0.611
47.09
44.25
0.870
0.832
ET-Flow (Gaussian Prior)
66.53
73.01
0.640
0.625
44.41
40.88
0,903
0.864
Table Generalization results on GEOM-XL.
AMR-P L
AMR-R R J
# mols
mean
median
median
GeoDiff
2.92
2.62
3.35
3.15
GeoMol
2.47
2.39
3.30
3.14
Tor. Diff.
2.05
1.86
2.94
2.78
MCF- S
2.22
1.97
3.17
2.81
102
MCF-
2.01
1.70
3.03
2.64
102
MCF-L
1.97
1.60
2.94
2.43
102
ET-Flow (ours)
2.31
1.93
3.31
2.84
102
Tor: Diff.
1.93
1.86
2.84
2.71
77
MCF S
2.02
1.87
2.9
2.69
77
MCF-
1.71
1.61
2.69
2.44
77
MCF -T
1.64
1.51
2.57
2.26
77
ET-Flow (ours)
2.00
1.80
2.96
2.63
75
provided by Torsional Diffusion, we encountered 27 failed cases for generation likely due to RDKit failures, similar to the observations in MCF albeit with slightly different exact numbers. In both experiments involving all 102 molecules and subset of 75 molccules, ET-Flow achieves performance comparable to Torsional Diffusion and MCF-S, but falls short of matching the performance MCF-B and MCF-L. It's worth noting that MCF-B MCF-L are significantly larger models, potentially affording them an advantage in generalization tasks. As part of our future work, we plan to scale up our model and conduct further tests to explore its performance in this regard.
D.3 Additional Out-of-Distrihution Results
Table 9: Additional OOD results. We use RS and SS to indicate Random Split and Scaffold Split respectively.
Recall
Precision
Coverage
AMR I
Coverage
AMR |
mean
median
mean
median
median
mean
median
ET-Flow (QM9 RS)
96.47
100.00
0.073
0.047
94.05
100.00
0.098
0.039
Fl-Flow (QM9 SS)
95.00
100.00
0.083
0.029
90.25
100.00
0.124
0.053
Fl-Flow (DRUGS + QM9)
86.68
100.00
0.218
0.160
68.69
75.30
0.369
0.317
ET-Flow (DRUGS RS)
79.53
84.57
0.452
0.419
74.38
81.04
0.541
0.470
ET-Flow (DRUGS SS)
76.06
80.65
0.644
0.545
67.83
74.19
0.511
0.473
To further evaluate the generalization performance of ET-Flow, we conduct two more out-of-distributiin experiments in addition to GEOM-XL. First, we test the model on scaffold-based splits of the GEOM-QM9 and GEOM-DRUGS dataset, which offers more challenging alternative the standard random split. We split the datasets based on Murcko scaffolds the molecules into an 80:10:10 ratio train, validation, and test sets. evaluate our method on 1000 randomly sampled molecules from the resulting test set. The second experiment involves training the model on -RDGS and assessing its performance on GEOM GEOM-QM9, a dataset with significantly smaller molecules. This experiment complements the generalization task to larger molecules in
19

---

GEOM-XL by assessing ability for ET-Flow generalize to smaller molecules. The results, presented in Table 9, indicate that the model's performance degrades only marginally the scaffold-based split. Furthermore, the model demonstrates robust performance on GEOM-QM9 even when trained on GEOM-DRUGS.
E
Visualizations
Figure shows randomly selected examples of sampled conformers from ET-Flow for GEOM-DRUGS. The left column the reference molecule from the ground truth, the remaining columns are samples generated with 50 sampling steps. Figure showcases the ability for ET-Flow to generate quality samples with fewer sampling steps.
Reference
good
Cor
Figure 6: Examples of conformers of ground truth and ET-Flow for GEOM-DRUGS.
20

---

ET-Flow-5
ET-Flow-10
ET-Flow-20
ET-Flow-50
Reference
Yo
Eoj
ad,
ad
Figure 7: Examples of conformers of ground truth and ET-Flow for different number of sampling steps.
21

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Reviews:
[{"review_id": "8eAQHyfCBd", "reviewer": "Reviewer_281V", "summary": "The paper describes an equivariant flow matching model for conformer generation.  The stated contributions are accurate - the model performance is state-of-the-art and largely due to good engineering.", "questions": "What is the out-of-distribution performance?  How well does the model generalize to different chemotypes (not just larger) and how does this compare to conventional and other generative models?\n\nIt would be interesting to see MCF trained using the same frame work as ETflow - e.g. how much of the performance is due to the choices in Table 6 versus the model architecture.\n\nCan you perform evaluations that demonstrate these conformers are better for downstream tasks than those generated through conventional approaches?", "limitations": "Some recent work has pointed to the fact that improved recall/precision on these GEOM benchmarks does not necessarily result in better performance on downstream tasks that use conformers.  At least some discussion of this limitation would be appreciated.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "The paper is well written and the evaluations follow other published work.  Informative ablation studies are performed.  The evaluation of ensemble property averages is particularly appreciated.\n\nI appreciate the authors highlight the modifications required for stable training.", "weaknesses": "Very little evaluation of out-of-distribution performance is considered (a small evaluation is in the appendix).  The ensemble property averages evaluation seems to be done on molecules drawn from the training data (if this is not the case, it needs to be more clear).  The paper would be stronger if generalization performance was more comprehensively evaluated.\n\nAlthough efficiency is one of the main claims, there's no evaluation of inference time.\n\nI think \"DRUGS\" is used as a short-hand for GEOM-DRUGS - this is confusing, use consistent nomenclature throughout.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer,\n\nWe first want to thank the reviewer for taking the time to review our work and ask thought-provoking questions. We hope that we are able to address said questions and concerns below.\n\n> **Very little evaluation of out-of-distribution performance is considered (a small evaluation is in the appendix).**\n\n> **The paper would be stronger if generalization performance was more comprehensively evaluated.**\n\n> **What is the out-of-distribution performance?**\n\nWe first want to acknowledge that we followed the evaluation protocols as done with prior work [1, 2, 3]. In the prior works, out-of-distribution (OOD) evaluation is done on GEOM-XL (Appendix C.1 table 7), which consists of molecules >100 atoms, using a model trained on GEOM-DRUGS (average 44 atoms per molecule). We demonstrate that our method performs slightly better than the MCF-S while having 5 million fewer parameters. However, we agree that more experiments that evaluate OOD performance is needed. Accordingly, we also evaluate our model on molecules significantly smaller than the training distribution via GEOM-QM9 (average 18 atoms per molecule) [4]. We provide these results in the table 1 of the global rebuttal statement. We are open to more suggestions and feedback regarding evaluation strategies.\n\n> **How well does the model generalize to different chemotypes (not just larger) and how does this compare to conventional and other generative models?**\n\nThank you for bringing this to our attention. The idea of exploring different chemotypes is an interesting suggestion, but it is unclear what chemotypes are being referenced. More details about this would be appreciated so we could discuss further. Moreover, we would like to make clear that our domain of application is in drug discovery and therefore we focus on drug like molecules from the GEOM dataset [4].\n\n> **The ensemble property averages evaluation seems to be done on molecules drawn from the training data (if this is not the case, it needs to be more clear).**\n\nWe would like to make clear that the ensemble averages experiment was done on the 100 randomly sampled molecules **from the test set**. We added a statement to section 4.4 to make that more clear.\n\n> **Although efficiency is one of the main claims, there's no evaluation of inference time.**\n\nThank you for bringing this up. We included a plot of inference time with respect to the number of steps in figure 1 in the attached pdf for global rebuttal. We also include a plot on performance (precision) with respect to inference time in figure 2 in the attached pdf for global rebuttal.\n\n> **I think \"DRUGS\" is used as a short-hand for GEOM-DRUGS - this is confusing, use consistent nomenclature throughout.**\n\nWe made the adjustment to the instances where we use “DRUGS” and just use “GEOM-DRUGS” for consistency. \n\n> **It would be interesting to see MCF trained using the same frame work as ETflow - e.g. how much of the performance is due to the choices in Table 6 versus the model architecture.**\n\nMCF follows a different framework than ETFlow. We would like to ask if the reviewer is implying to evaluate the architecture of MCF (Perceiver IO) with the flow matching framework? Any clarifications on this statement would be much appreciated, thank you. \n\n> **Some recent work has pointed to the fact that improved recall/precision on these GEOM benchmarks does not necessarily result in better performance on downstream tasks that use conformers. At least some discussion of this limitation would be appreciated.**\n\nWe would like to point out to the reviewer that the ensemble properties table in Section 4.4 - table 3 in the manuscript indicates that our method has the lowest errors for the chemical properties evaluated against GFN-xTB oracle, demonstrating our model produces more physically-correct molecules compared to other methods. We would be open to benchmarking our model on other metrics if the reviewer could refer us to said metrics and alternative downstream evaluations.\n\n[1] Ganea, O., Pattanaik, L., Coley, C., Barzilay, R., Jensen, K., Green, W. and Jaakkola, T., 2021. Geomol: Torsional geometric generation of molecular 3d conformer ensembles. Advances in Neural Information Processing Systems, 34, pp.13757-13769.\n\n[2] Jing, B., Corso, G., Chang, J., Barzilay, R. and Jaakkola, T., 2022. Torsional diffusion for molecular conformer generation. Advances in Neural Information Processing Systems, 35, pp.24240-24253.\n\n[3] Wang, Y., Elhag, A. A., Jaitly, N., Susskind, J. M., & Bautista, M. Á. Swallowing the Bitter Pill: Simplified Scalable Conformer Generation. In Forty-first International Conference on Machine Learning.\n\n[4] Axelrod, S. and Gomez-Bombarelli, R., 2022. GEOM, energy-annotated molecular conformations for property prediction and molecular generation. Scientific Data, 9(1), p.185.", "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": "wtbih4gB45", "reviewer": "Reviewer_AAmo", "summary": "The paper proposes the Equivariant Transformer Flow (ET-Flow) which predicts low-energy molecular conformations given the molecular graphs. Unlike existing methods that rely on large transformer-based models for conformed fields or complex internal geometry calculations, ET-Flow leverages flow matching with equivariance and harmonic prior which directly operates on all-atom coordinates with minimal assumptions. The extensive experimental results illustrate that ET-Flow achieves state-of-the-art performance in molecular conformer generation benchmarks with fewer parameters and faster inference times, outperforming or matching previous methods while maintaining high accuracy and efficiency.", "questions": "1. The proposed model utilizes an equivariant model architecture. Meanwhile, it is widely argued that equivariance is not a requirement for molecular machine learning (as suggested in the MCF paper). Without equivariance, the model is more easily generalized and scaled, and the symmetry could be implicitly learned from the data. I would like to ask the authors about the thoughts on this. In particular, I'm interested in the scaling performance (with respect to data or model size) of ET-Flow compared to MCF. \n\n1. Have the authors considered conducting data augmentation to improve the diversity of generated conformations and enhance recall metrics?", "limitations": "The authors have discussed the limitations of recall performance and additional chirality correction steps. No potential negative societal impact is involved.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The proposed approach represents a significant innovation by leveraging flow matching with equivariance and harmonic prior in order to simplify the conformer generation process while maintaining high accuracy. \n\n1. ET-Flow achieves SOTA model performance on molecular conformer generation benchmarks, outperforming or matching existing methods with much fewer parameters and faster inference times without sacrificing accuracy. The high accuracy and efficiency make it a promising candidate for practical applications. \n\n1. The paper provides comprehensive experimental results, comparing ET-Flow with several leading approaches across various datasets. The inclusion of various evaluation metrics strengthens the validity of the findings.", "weaknesses": "1. ET-Flow achieves SOTA or near-SOTA performance on the benchmarks. However, my concern is that TorchMD-Net is already a strong model with a well-established architecture that leverages equivariant transformers for molecular modeling. Given this, it can be challenging to attribute the performance improvements of ET-Flow to the flow matching approach. Ablation studies focusing on the flow matching component or results from simpler architectures might be useful. \n\n1. The need for a post hoc chirality correction step suggests that the model does not inherently handle stereochemistry (baseline models like MCF do not require such an explicit correction). Such a weakness may result in issues in practical applications. \n\n1. ET-Flow primarily combines TorchMD-Net with the flow matching technique. While both components are robust and effective, the novelty of the approach is limited as it largely builds on existing methodologies. This integration, although leading to performance improvements, does not significantly advance the state-of-the-art in terms of methodological innovation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer,\n\nWe first want to thank the reviewer for taking the time to review our work and ask thought-provoking questions. We hope that we are able to address said questions and concerns below.\n\n> **It can be challenging to attribute the performance improvements of ET-Flow to the flow matching approach. Ablation studies focusing on the flow matching component or results from simpler architectures might be useful.**\n\nWe first want to acknowledge that the performance of our model is not only attributed to the use of flow matching but is a result of a combination of different components such as choice of prior, rotational alignment, sampling method. We additionally included ablation studies demonstrating the effect of these design choices in table 2 in global rebuttal.\n\nWe agree that leveraging the expressive equivariant architecture combined with apt modification and engineering is indeed one of the key reasons for improved performance. However, the strengths of our approach, such as faster inference and flexible choice of prior are enabled by adopting the flow matching framework. Therefore, we view our work as addressing different challenges to create a harmonious integration of these components. \n\n> **The need for a post hoc chirality correction step. Such a weakness may result in issues in practical applications.**\n\nTorchMDNet is O(3) equivariant, thus necessitating a post-hoc chirality correction (CC) step to break the reflection symmetry. We conduct an additional experiment with SO(3) equivariance, alleviating the need for an additional correction step. This is achieved by modifying the vector output of TorchMDNet using the cross product in Equation(18). The empirical results are shown in table 3 for GEOM-DRUGs and table 4 for GEOM-QM9 in global rebuttal.\n\nFor GEOM-QM9, we show that the O(3) with CC and SO(3) TorchMDNET achieve roughly similar results, however in the case of GEOM-DRUGS the advantage of O(3) with CC is more pronounced. An important point to highlight is the negligible cost of the CC step as highlighted in figure 1 in the global rebuttal pdf. In practice, one may choose to use ET-Flow combined with CC or directly use the SO(3) version of the model.\n\nWhile we acknowledge the reviewer's concern, we respectfully disagree about the practical weaknesses. MCF uses data augmentation and more model parameters to learn physical symmetries. As shown in figure 1 of the attached PDF in the global rebuttal, our method’s inference time remains significantly faster, even with the CC step. Furthermore, ET-Flow’s inference time primarily depends on the number of parameters. \n\n> **The novelty of the approach is limited as it largely builds on existing methodologies. This integration, although leading to performance improvements, does not significantly advance the state-of-the-art in terms of methodological innovation.**\n\nWe argue that the correct integration and modification of existing methodologies to achieve significant improvements should not be overlooked. One of the strengths of our work lies in identifying the core problem with existing methods and devising a straightforward approach with precise engineering, which is often crucial for empirically important tasks, especially in the application of machine learning to chemistry. We also conduct design choice ablations as shown in Table 2 in the global rebuttal.\n\n> **It is widely argued that equivariance is not a requirement for molecular machine learning. Without equivariance, the model is more easily generalized and scaled, and the symmetry could be implicitly learned from the data. I'm interested in the scaling performance (with respect to data or model size) of ET-Flow compared to MCF.**\n\nThe MCF paper advocates for a more general approach to molecular generative tasks by not embedding geometric inductive biases into the model, which is a valid and valuable perspective. However, our stance is that in the application of AI to scientific domains, leveraging known inductive biases is crucial for effective performance (scale vs performance tradeoff). Explicitly incorporating inductive biases ensures that the model inherently respects physical symmetries, leading to more reliable and physically consistent predictions. This approach is particularly important in domains where data can be scarce or expensive to obtain, as the inductive biases help the model generalize better from limited information.\n\nIn our work, we used a dataset of comparable size to MCF but kept our model parameters(8.3M) significantly smaller. Although our computational limitations prevented us from scaling up to a similar number of parameters as used in MCF-L(242M), this is an avenue we plan to explore in the near future. Despite these limitations, our approach demonstrates the effective use of equivariant models in achieving competitive results, suggesting that incorporating inductive biases can lead to more efficient and scalable solutions in molecular machine learning.\n\n> **Have the authors considered conducting data augmentation to enhance diversity and recall metrics?**\n\nWe believe this suggestion may be coming from the context of its use in MCF and AlphaFold3. In their cases, data augmentation was primarily employed to compensate for the lack of inductive bias, enabling the models to be more expressive while learning equivariance in a soft manner. In contrast, our model inherently incorporates symmetries by design, eliminating the need for such techniques. Additionally, while our recall metrics are slightly lower than those of MCF-M and MCF-L, our precision metrics show a significant improvement over all versions of MCF. In practice, this improvement in precision is crucial because this produces more physically-correct molecules as evidenced in our ensemble properties table having the lowest errors amongst the existing methods.", "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": "Kd4w5nDKVd", "reviewer": "Reviewer_eBNM", "summary": "The paper proposes the Equivariant Transformer Flow (ET-Flow) to generate high-quality molecule conformations. The authors use rotational alignment, stochastic sampling, and chirality correction to improve the flow matching framework for this task. Additionally, the paper modifies the TorchMD-NET equivariant transformer architecture to parameterize the target vector field. Experiments on the GEOM dataset show that  ET-Flow can perform well on the conformation prediction task. Furthermore, conformations generated by ET-Flow can also be used to predict ensemble properties with great performance.", "questions": "See the sections above.", "limitations": "N/A", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Very clear writing.\n2. The method is simple and easy to understand.", "weaknesses": "1. Lack of experiments. The experiments on the GEOM dataset are not enough to show the empirical performance of the framework. Some further experiments on large-scale datasets with more data and larger system size are necessary. I suggest the authors do experiments on OC20/OC22(Open Catalyst 2020/2022) datasets to test the performance of the ET-Flow framework.\n2. The novelty of this work is not enough. I think the idea and method of this paper is very similar to [1], but in this work, the authors use a transformer-based model. Additionally, this paper proposes several tricks such as rotational alignment, stochastic sampling, and chirality correction. So the authors should clarify the novelty of this work.\n\n[1]. Klein, Leon, Andreas Krämer, and Frank Noé. \"Equivariant flow matching.\" Advances in Neural Information Processing Systems 36 (2024).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> **Lack of experiments. The experiments on the GEOM dataset are not enough to show the empirical performance of the framework. Some further experiments on large-scale datasets with more data and larger system size are necessary. I suggest the authors do experiments on OC20/OC22(Open Catalyst 2020/2022) datasets to test the performance of the ET-Flow framework.**\n\nThank you for your valuable feedback. Our primary focus in this work is on sampling molecular conformers, and therefore, the dataset and benchmark metrics we used align with the standards established by previous works that addressed the same objective from different perspectives[1,2,3]. While we appreciate the suggestion to test our framework on the OC20/OC22 datasets, this would be a nontrivial task due to the fundamentally different nature of the catalyst datasets. Consequently, existing methods for molecular conformer generation do not benchmark performance on OC20/22 and therefore is beyond the scope of our work. However, we agree that evaluating our framework on a diverse set of datasets is an important and promising direction for future research.\n\n> **The novelty of this work is not enough. I think the idea and method of this paper is very similar to [1], but in this work, the authors use a transformer-based model. Additionally, this paper proposes several tricks such as rotational alignment, stochastic sampling, and chirality correction. So the authors should clarify the novelty of this work.**\n\nWe argue that the correct integration and modification of existing methodologies to achieve significant improvements should not be overlooked. One of the strengths of our work lies in identifying the core problem with existing methods and devising a straightforward approach with precise engineering, which is often crucial for empirically important tasks, especially in the application of machine learning to chemistry. We include ablation studies in table 2 of the global rebuttal demonstrating the effect of each design choice on the result.\n\n[1] Ganea, O., Pattanaik, L., Coley, C., Barzilay, R., Jensen, K., Green, W. and Jaakkola, T., 2021. Geomol: Torsional geometric generation of molecular 3d conformer ensembles. Advances in Neural Information Processing Systems, 34, pp.13757-13769.\n\n[2] Jing, B., Corso, G., Chang, J., Barzilay, R. and Jaakkola, T., 2022. Torsional diffusion for molecular conformer generation. Advances in Neural Information Processing Systems, 35, pp.24240-24253.\n\n[3] Wang, Y., Elhag, A. A., Jaitly, N., Susskind, J. M., & Bautista, M. Á. Swallowing the Bitter Pill: Simplified Scalable Conformer Generation. In Forty-first International Conference on Machine Learning.", "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:
We perform these additional experiments to support our rebuttals.

### **Out-of-Distribution Evaluation on GEOM-QM9**
To improve upon out-of-distribution evaluation, we test our model trained on GEOM-DRUGS on GEOM-QM9 (significantly smaller molecules).

| Method | Recall Coverage (mean) | Recall Coverage (median) | Recall AMR (mean) | Recall AMR (median) | Precision Coverage (mean) | Precision Coverage (median) | Precision AMR (mean) | Precision AMR (median) |
|--------|------------------------|--------------------------|-------------------|---------------------|---------------------------|-----------------------------|-----------------------|------------------------|
| CGCF | 69.47 | 96.15 | 0.425 | 0.374 | 38.20 | 33.33 | 0.711 | 0.695 |
| GeoDiff | 76.50 | **100.00** | 0.297 | 0.229 | 50.00 | 33.50 | 1.524 | 0.510 |
| GeoMol | 91.50 | **100.00** | 0.225 | 0.193 | 87.60 | **100.00** | 0.270 | 0.241 |
| Torsional Diff. | 92.80 | **100.00** | 0.178 | 0.147 | 92.70 | **100.00** | 0.221 | 0.195 |
| MCF | **95.0** | **100.00** | 0.103 | 0.044 | **93.7** | **100.00** | 0.119 | 0.055 |
| ET-Flow | **94.99** | **100.00** | **0.083** | **0.035** | 91.00 | **100.00** | **0.116** | **0.047** |
| ET-Flow-OOD | 86.68 | **100.00** | 0.218 | 0.160 | 68.69 | 75.3 | 0.369 | 0.317 |

Table 1: Molecule conformer generation results on GEOM-QM9 (δ = 0.5Å). For our method, we sample conformations over 50 time-steps. Bold indicates best performance. ET-Flow-OOD is the model trained on GEOM-DRUGS and tested on GEOM-QM9.

### **Design Choice Ablation**

We conduct a series of ablation studies to assess the influence of each component in the ET-Flow. Particularly, we re-run the experiments with (1) $O(3)$ equivariance without chirality correction, (2) Absence of Alignment, (3) Gaussian Prior as a base distribution. We demonstrate that improving probability paths and utilizing an expressive equivariant architecture with correct symmetries are key components for ET-Flow to achieve state of the art performance. The ablations were ran with reduced settings ($50$ epochs; lr $=1e-4$; $4$ A100 gpus).

| Method | Recall Coverage (mean) | Recall Coverage (median) | Recall AMR (mean) | Recall AMR (median) | Precision Coverage (mean) | Precision Coverage (median) | Precision AMR (mean) | Precision AMR (median) |
|--------|------------------------|--------------------------|-------------------|---------------------|---------------------------|-----------------------------|-----------------------|------------------------|
| Our Method | 75.37 | 82.35 | 0.557 | 0.529 | 58.90 | 60.87 | 0.742 | 0.690 |
| Our Method (O(3)) | 72.74 | 79.21 | 0.576 | 0.556 | 54.84 | 54.11 | 0.794 | 0.739 |
| Our Method (w/o Alignment) | 68.67 | 74.71 | 0.622 | 0.611 | 47.09 | 44.25 | 0.870 | 0.832 |
| Our Method (Gaussian Prior) | 66.53 | 73.01 | 0.640 | 0.625 | 44.41 | 40.88 | 0.903 | 0.864 |

Table 2: Comparison of different variants of our method. Coverage (↑) is better when higher, AMR (↓) is better when lower.

### **Chirality correction and SO(3) Study**

We conduct ablation experiments evaluating ET-Flow with and without Chirality Correction (CC). Additionally, we also report performance with an SO(3) equivariant version of ET-Flow without Chirality correction.

| Method | Recall Coverage (mean) | Recall Coverage (median) | Recall AMR (mean) | Recall AMR (median) | Precision Coverage (mean) | Precision Coverage (median) | Precision AMR (mean) | Precision AMR (median) |
|--------|------------------------|--------------------------|-------------------|---------------------|---------------------------|-----------------------------|-----------------------|------------------------|
| GeoDiff | 42.10 | 37.80 | 0.835 | 0.809 | 24.90 | 14.50 | 1.136 | 1.090 |
| GeoMol | 44.60 | 41.40 | 0.875 | 0.834 | 43.00 | 36.40 | 0.928 | 0.841 |
| Torsional Diff. | 72.70 | 80.00 | 0.582 | 0.565 | 55.20 | 56.90 | 0.778 | 0.729 |
| MCF - S (13M) | 79.4 | 87.5 | 0.512 | 0.492 | 57.4 | 57.6 | 0.761 | 0.715 |
| MCF - B (62M) | 84.0 | 91.5 | 0.427 | 0.402 | 64.0 | 66.2 | 0.667 | 0.605 |
| MCF - L (242M) | **84.7** | **92.2** | **0.390** | **0.247** | 66.8 | 71.3 | 0.618 | 0.530 |
| ET-Flow (8.3M) O(3) | 78.6 | 83.33 | 0.479 | 0.455 | 67.16 | 72.15 | 0.637 | 0.563 |
| ET-Flow (8.3M) O(3) + CC | 79.53 | 84.57 | 0.452 | 0.419 | 74.38 | 81.04 | 0.541 | 0.470 |
| ET-Flow (9.1M) SO(3)    | 78.18 | 83.33 | 0.48 | 0.459 | 67.27 | 71.15 | 0.637 | 0.567 |

Table 3: Molecule conformer generation results on GEOM-DRUGS ($\delta$ = 0.75Å). For all ET-Flow methods, we sample conformations over 50 time-steps. Bold indicates best performance. 

| Method | Recall Coverage (mean) | Recall Coverage (median) | Recall AMR (mean) | Recall AMR (median) | Precision Coverage (mean) | Precision Coverage (median) | Precision AMR (mean) | Precision AMR (median) |
|--------|------------------------|--------------------------|-------------------|---------------------|---------------------------|-----------------------------|-----------------------|------------------------|
| CGCF | 69.47 | 96.15 | 0.425 | 0.374 | 38.20 | 33.33 | 0.711 | 0.695 |
| GeoDiff | 76.50 | **100.00** | 0.297 | 0.229 | 50.00 | 33.50 | 1.524 | 0.510 |
| GeoMol | 91.50 | **100.00** | 0.225 | 0.193 | 87.60 | **100.00** | 0.270 | 0.241 |
| Torsional Diff. | 92.80 | **100.00** | 0.178 | 0.147 | 92.70 | **100.00** | 0.221 | 0.195 |
| MCF | 95.0 | **100.00** | 0.103 | 0.044 | 93.7 | **100.00** | 0.119 | 0.055 |
| ET-Flow (8.3M) O(3) + CC | 94.99 | **100.00** | 0.083 | 0.035 | 91.00 | **100.00** | 0.116 | 0.047 |
| ET-Flow (9.1M) SO(3) | **95.98** | **100.00** | **0.076** | **0.030** | **94.05** | **100.00** | **0.098** | **0.039** |

Table 4: Molecule conformer generation results on GEOM-QM9 (δ = 0.5Å). For all ET-Flow methods, we sample conformations over 50 time-steps. Bold indicates best performance.

Meta-review: The paper presents a conformer generation approach using flow matching. In this task, given a graph a generative model generates 3D configurations of molecules which  coincide with energy minima (conformers). The approach is of limited technical novelty, but a nice display of how a simple, well-engineered model can achieve state of the art results. During discussion the authors engage well with the concerns raise by the reviewers, in particular, they add a number of important ablations.
Decision: Accept (poster)