Efficient Molecular Conformer Generation with SO(3)-Averaged Flow Matching and Reflow
Authors: Zhonglin Cao
Mario Geiger
Allan Dos Santos Costa
Danny Reidenbach
Karsten Kreis
Tomas Geffner
Franco Pellegrini
Guoqing Zhou
Emine Kucukbenli
Venue: ICML 2025 poster (2025)
Keywords: Flow-matching
few-shot generation
equivariance
small molecules
Primary area: deep_learning->generative_models_and_autoencoders
TL;DR: We present a novel flow-matching objective and combine it with reflow to accelerate training and inference of molecular conformer generation

Abstract: Fast and accurate generation of molecular conformers is desired for downstream computational chemistry and drug discovery tasks. Currently, training and sampling state-of-the-art diffusion or flow-based models for conformer generation require significant computational resources. In this work, we build upon flow-matching and propose two mechanisms for accelerating training and inference of generative models for 3D molecular conformer generation. For fast training, we introduce the SO(3)-*Averaged Flow* training objective, which leads to faster convergence to better generation quality compared to conditional optimal transport flow or Kabsch-aligned flow. We demonstrate that models trained using SO(3)-*Averaged Flow* can reach state-of-the-art conformer generation quality. For fast inference, we show that the reflow and distillation methods of flow-based models enable few-steps or even one-step molecular conformer generation with high quality. The training techniques proposed in this work show a path towards highly efficient molecular conformer generation with flow-based models.

Efficient Molecular Conformer Generation with S0(3)-Averaged Flow Matching and Reflow
Zhonglin Cao 1
Mario Geiger" I Allan dos Santos Costa 12 Danny Reidenbach Karsten Kreis
Tomas Geffner Franco Pellegrini
Guoqing Zhou' I Emine Kucukbenli
Abstract
termined by the conformers. In the domain of drug dis- covery, molecular conformer generation is a prerequisite for both structure-based (Trott & Olson, 2010) and ligand- based (Rush et al., 2005) compound virtual screening. For established computational chemistry molecular conformer generation tools, there is trade-off between generation speed and the quality diversity of generated conformers (Axelrod & GomezzBoobarrlli, 2022). For example, en- hanced molecular dynamics simulation (Grimme, 2019) can generate diverse conformers by sampling the conformation space rather exhaustively, but this approach slow due to the need of multiple energy function evaluations. RDKit (Landrum, 2016) and some rule-based tools (Hawkins et al., 2010) are faster but may miss many low-energy conformer, and the generation quality can deteriorate when molecule size increases. Thus, deep generative models are being sought as potential solutions to overcome such trade-off and bring fast, diverse, and high-quality molecular conformer generation.
Fast and accurate generation of molecular con- formers is desired for downstream computational chemistry and drug discovery tasks. Currently, training and sampling state-of-the-ar diffusson flow-based models for conformer generation re- quire significant computational resources, In this work, we build upon flow-matd cow-matching and propose two mechanisms for accelerating training and in- ference of generative models for molecular conformer generation. For fast training, we intro- duce the SO(3)-Averaged Flow training objective, which leads to faster convergence to better gen- eration quality compared to conditional optimal transport flow or Kabsch-aligned flow. We demon- strate that models trained using SO(3)-Averaged Flow can reach of-te-of-the-art conformer gener- ation quality. For fast inference, we show that the reflow and distillation methods of flow-based models enable few-steps even one-step molec- ular conformer generation with high quality. The training techniques proposed in this work show a path towards highly efficient molecular conformer generation with dow-based models.
Many earlier works are based on generative models (Simm & Hernández-Lobato, 2019; Zhu et al., 2022; Luo et al., 2021; Shi et al., 2021; Xu et al., 2022), given the stochastic nature of the molecular conformer generation task. However, established cheminformatics tools such as OMEGA (Hawkins et al., 2010) still have better generation quality and faster sampling speed compared to early deep- learning based methods. Torsional diffusion (Jing al., 2022) is the first diffusion model to achieve better genera- tion quality than cheminformaties model. By restricting the degree-of soreee-offfreede fr-fereeoo on the torsion angles, torsional diffusion can generate diverse conformers with a lightweight model and fewer number of reverse diffusion steps. More recent works, such as Molecular conformer field (MCF) (Wang et al., 2024) and ET-Flow (Hassan et al., 2024), perform diffusion flow-matching directly on the Cartesian coordi- nates of the atoms. With more scalable transformer architec- tures, they have achieved the state-of soute-of-the-aae the-art conformer gen- eration quality. However, iterative ODE or SDE solving with large transformer models to generate every conformer can still be computationally infeasible when the virtual screen- ing library contains billions of compounds (Bellmann et al., 2022).
1. Introduction
Molecular conformer generation is the task to predict the ensemble 3D conformations of molecules given their molecular graphs (Hawkins, 2017). Generating high quality molecular conformers that fit their natural 3D struc- tures is erucial task for computational chemistry because many physical and chemical properties (Guimarães et al., 2012; Schwab, 2010; Shim & MacKerell Jr, 2011) are de-
'Equal contribution 'NVIDIA MIT Center for Bits and Atoms WWrk was completed during internship with NVIDIA. Correspon- dence to: Zhonglin Cao <zhonglinc @eaiiia.viioinnnnn Mario Geiger <mgeiger@nvidii.comm nvidia.com>.
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Efficient Molecular Conformer Generation with SO(3)-Averaged Flow Matching and Reflow
(a)
Xo
(b)
Flow
overlaps
X1
R
X1 R2
X
X R4
Rs
Conditional OT
Kabsch + Flow
Before Reflow
After Reflow
SO(3) Averaged Flow
Figure 1. (v3))rereed Flow and Reflow (a) We illustrate a comparison between our approach Averaged Flow, conditional OT and Kabsch Flow. While conditional OT randomly assigns any rotation of the data, Kabsch Flow assigns the rotation of largest overlap. Our method instead computes the expected flow across all rotations. (b) Flow trajectory visualization before and after the reflow with 100 Euler steps. The flow trajectories are effectively straightened after reflow. Carbon, hydrogen, oxygen, and nitrogen are colored as black. gray, red, and blue, respectively.
In this work, we propose novel flow-matching training ap- proach to improve the efficiency of deep learning model training and sampling for molecular conformer genera- tion. To improve training efficiency, we design new flow- matching objective called SOv3)-Averaged Flow (Fig. la). As an objective, Averaged Flow avoids the need to rota- tionally align prior and data distributions by analytically computing the averaged probability flow path from the prior all the rotations of the data sample. Models trained with Averaged Flow are experimentally shown to converge faster to better performance. To improve the sampling efficiency, we adopt the reflow and distillation technique (Liu et al., 2022) to straighten the flow trajectories (Fig. lb). Straight- ened trajectories enable high quality molecular conformer generation with few-step or even one-step ODE solving, thus significantly reducing the computational cost.
2. Background and Related Works
2.1. Generative Models for Conformer Generation
The task of molecular conformer generation in its core is to sample from the intractable conformer distribution con- ditioned on the 2D molecular graph. Therefore, generative deep learning models are well-suited for this task, and many methods have been proposed. Deep learning model usually trained datasets containing molecular conform- ers generated by CREST (Pracht et al., 2020), which uses computationally expensive semi-enne cemi-empirical quantum chem- istry method (Bannwarth et al., 2019) under the hood. The earliest works in this field use variational autoencoder to gen- erate intrinsic inter-atomic distance (Simm Hernández- Lobato, 2019; Xu al., 2021). Shi al. (2021) proposes ccore-matching method that learns the gradient of intrinsic atom coordinates in molecular graph. Ganea et al. (2021) addresses molecular conformer generation by designing a message passing neural network to predict the local 3D structure and torsion angles. Xu et al. (2022) adopts a diffu- sion model and equivariant graph neural network to generate molecular conformers by iteratively denoising the Euclidean atom coordinates from sampled noise. Besides generating conformers from scratch, some works focus on optimizing molecular conformers to lower energy states (Lee et al., 2024a; Guan et al., 2021). Torsional diffusion (Jing et al., 2022) reduces the -f by refining the torsion freedom angles of RDKit-generated (Landrum, 2016) initial conform- ers with diffusion process on the hypertorus. Such design allows torsional diffusion to significantly reduce sampling steps. One drawback of torsional diffusion is that it relies on an RDKit- generated conformer as the starting point of diffusion, which adds computational overhead to the gener- ation process. The generation quality of RDKit, especially for atom coordinates in rings, can also impact the sample
Our main contribution can be summarized as follows: (i) We propose novel SOv3)-Averaged Flow training objective. Averaged Flow eliminates the need for rotational alignment between prior and data by training the model to learn the average probability path over all rotations of the data. This leads to faster convergence to better performance for molec- ular conformer generation, and can be extended to other sim- ilar tasks. Models trained using Averaged Flow achieve state-of-the-art conformer generation quality. (ii) We adopt reflow and distillation to reduce the number of ODE steps required for the model to generate high quality conform- ers. Such technique significantly improves the sampling efficiency of flow-matching models molecular conformer generation. (iii) Both Averaged Flow and resilv-distillation are model architecture-agnosiic,, meaning that our proposed methods be directly applicable both equivariant and non-equivariant neural network architectures.
2

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Efficient Molecular Conformer Generation with SOG3)-Averaged Flow Matching and Reflow
quality of torsional diffusion. Molecular conformer field (MCF) proposed by Wang et al. (2024) is recent work that leverages the scaling power of the transformer architec- ture (Jaegle et al., 2021) and diffusion model. MCF achieves state-of-the-ar performance in molecular conformer genera- tion by training models with tens to hundreds million of pa- rameters to denoise the atoms' Euclidean coordinates using DDPM paradigm (Ho et al., 2020). Equivariant Transformer Flow (ET-Flow) is concurrent work that trains an equiv- ariant flow-matching model to generate conformers from prior distribution. By combining harmonic prior (Jing et al., 2023), flow-matching, and Kabsch alignment that reduces transport cost, ET-Flow is reported to outperform MCF on several metrics with fewer ODE steps.
(1(()) - (2(2)))) (o(1))
(FMI1)
they prove that
(x(r)r))= o((21) o1(21) (x Me(xi)) + pb(x1))
(FM15)
It is noteworthy that we refer to the linear interpolant #; (x - (1 - ()xr between the noise and data distribution as conditional optimal transport (OT) following Lipman et al. (2023).
2.3. Techniques to Improve Sampling Efficiency
With the success of denoising diffusion probabilistic mod- els (Ho et al., 2020), much attention has been drawn im- prove the sampling speed of diffusion models. DDIM (Song et al., 2020) shows that the sampling steps can be signifi- cantly reduced by formulating the sampling process as ODE solving. Knowledge distillation techniques (Meng et al., 2023; Salimans & Ho, 2022; Song et al., 2023; Song Dhariwal, 2023) have also been proposed to reduce sam- pling steps and accelerate generation. Rectified flow (Liu et al., 2022; Liu, 2022) is method proposed to train the model to learn straight probability flow that bridges prior and data distribution. The reflow technique in rectified flow can straighten the flow trajectory and reduce the transport cost, allowing generation in very few steps with high qual- ity. After reflow, the model can be further distilled to im- prove one-step generation. The reflow and distillation tech- nique has been proven effective in enabling few-step or even single-step text-to-image (Esser et al., 2024; Liu et al., 2023b) and point cloud (Wu et al., 2023) generation.
Overall, the trade-off between conformer generation quality and speed is prevailing issue. Specifically, semi-empirical quantum chemistry can sample very high-quality conform- ers at high computational cost. Diffusion or fow-matching models can generate high-quality conformers but the itera- tive ODE/SDE solving process can be slow, making them less practical for large-scale virtual screening- Cheminfor- matics tools such as RDKit and OMEGA are very fast but generate conformers with limited diversity.
2.2. Flow-Matching
Averaged Flow is based on Flow Matching (Lipman al., 2023; Liu et al., 2023a; Albergo & Vanden-Eijnden 2023), which models probability density path (x(x) that grad- ually transforms an analytically tractable noise distribu- tion (t = 0) into a data distribution (t = 1), following a time variable E [0, 1]. Formally, the path (xx) corre- sponds to a flow 1hr that pushes samples from po to Pt via Pt = [w]] * Po. where * denotes the push-forward, In prac- tice, the flow is modeled via an ordinary differential equa- tion (ODE) dr, of ddt, defined through learnable vector field of (xx) with parameters f. Initialized from noise To ~ (xo). this ODE simulates the flow and transforms noise into approximate data distribution samples. The prob- ability density path (x(zz) and the (intractable) ground-truth vector field un (x4) are related via the continuity equation dpe(x:) (d) V (x)()))()))) To construct Pt. Lipman et al. (2023) introduce conditional probability pe (zr 21) and conditional vector field u. (arla)) both related to their unconditional counterparts as follow:
3. Method
3.1. SO(3)-Averaged Flow
The concept of Averaged Flow involves recognizing that the data distribution q may exhibit group symmetries, which can be explicitly integrated out. A symmetry group G of q consists of transformations g : +> g x that leave the distribution q unchanged, meaning q ((z) 4x)..)).
If we focus on Lie groups with a Haar measure (Zee, 2016; Nachbin & Bechtolsheim, 1965; Chirikjian & Kyatkin, 2000), we can express & as
g(xx - f
/ di
/i ((t) / dg (x)
Pc(x) f Pr (=(a)on)g(z1))d
(FM6)
(I)
uy (x) = (o(a)/r p((2)21)9(11) dr
where I represents the distribution over the group orbits, 2 is a representative point of the orbit, and the integral
(FM8)
Pr (x)
over G uses the Haar measure. By substituting this into cquation FM8, we obtain the vector field:
With the following simple choices of conditional probability and flow
Ue ux(x) = dā g(Ø) / dg ue(2lg 2) Pr(zlg f)
(2)
p((a)))) -
(FM10)
pc(x)
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Efficient Molecular Conformer Generation with SO(3)-Averaged Flow Matching and Reflow
Notice that pr (x( = f dă (()) dg pe 3) is the parti- tion function.
solution for
F log
dR (Rex((((FF()) SS(3)
(6)
Let's consider the case of conformer generation:
where F can be any x matrix. We can now leverage Eq. 6 to solve for log Zi(s,, at) Expand- ing the quadratic term in Eq. 4, 1 ||l-taR (rle"" R)- )r(a"
1. x is a N 3 matrix representing the 3D coordinates of N atoms.
(1-1)²
2. The group G is the rotation group SO(3). We will use R to denote the rotation matrix, which acts on a as
we obtain the
(1-1)²
x ++ .R..
following equation:
3. The goal to generate molecular conformers that correspond to at least local minima in the conforma- tional energy landscape. The orbits &tr in this case cor- respond to the different low-energy conformers of a given molecule and their permutations that leave the 2D molecular graph invariant. Therefore, the integral f då @()) in Eq.2, representing the entire conformer ensemble, can be written as Esex (()), where X' is the set of conformers and ((() is the weight associated with each conformer.
log Zz(2, a.)
log Σ (()) exp
log f
dRt 'r(la"
[1- "EJaR
Ssoss
closed-form solution using
17
828
where tr(an" Fan + Since we will take the 2(1-1)2
4. pe (ar) (m)) is Gaussian of the form:
derivative with respect to a to compute us (z), all terms that neither depend on a nor R (note that R R is the identity) will contribute multiplicative factors to the integral. Now, we can plug the analytically solved 0₂ log Zi(ze, a)) into Eq. 5 to directly learn the SO(3) -vvrraeee Flow u, (z,)) with:
(((()))))
x exp
Σ
(x (x1)2s
Di (x1))) Lx1)1o)
1)2 116
1 | 1x1 1
exp
LAvgFlow (0) = E [10 - we (x(e)))) with t E [0, 1].
2 (1 t)2
(8)
where M is IR NNN matrix. We will use the notation
We provide the Python implementation of this formula in Appendix C.I. While our Averaged Flow implementation is capable of handling multiple conformer states in the summa- tion in Eq 7, in practice, we approximate the expectation of the conformer ensemble by sampling one conformer in each training epoch. Following previous works (Jing et al., 2022), ((x) is taken as uniform distribution over all conformers. The computation time benchmark (Table C.2) shows that only a small overhead is introduced when using the Aver- aged Flow objective. We empirically find that the choice of interpolant (z²) during training should depend on the model architecture. If equivariant neural network is used as of linear interpolant (Liu et al., 2022; Lipman et al., 2023)
|| = EA).
We can rewrite the vector field (I),, averaged over all conformers and the SO(3) group, in this case as:
We (x)
Σ ((t)
dR
R
1
(1- 02
T 1
(x, 0)
&EX
5013)
t
(3)
and define Ze(x, a) as:
Z,(2, a) = Σ ((t) f S0(3)
1 13 (1-12
+er(a" "aaaa""r"" a
dRe
Tt = t Zy + (1 1--t) t) 1).2) 22
99)
REX
(4) where is an N. x 3 matrix that will be needed in the following steps.
can be used. However, if non-equivariant neural network is used as eg. training with Averaged Flow requires the interpolant to be solved with simulating the probability flow ODE:
Note ux(x) can be calculated as the derivative of log Zz a) with respect to a. evaluated at = Nx3,
F4 =0 + L. ut (x,)dr
(10)
(x)))) ((0,, log Ze (3re a))== x1)((1 1 (). (5)
where u+(+) is the ground-truth Averaged Flow function (Eq. 5). In practice, we simulate the ODE with fixed 20 Eu- ler steps to compute Dz, which we refer to as the integration interpolant.
The integral over R can be computed using the formula from Mohlin al. (2020), which provides closed-form
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Efficient Molecular Conformer Generation with SO(3)-Averaged Flow Matching and Reflow
3.2. Reflow and Distillation
with pairwise bias (DiT). Both model takes in the featurized molecular graph input. The featurization details can be found in Sec. B.2.
Diffusion and flow-based molecular conformer generation models typically require hundreds or even thousands of steps of numerical ODE or SDE solving during the sampling pro- cess. Such iterative processes add computational overhead and hinder the adoption of these model in industrial downstream applications, which require fast generation. One effective technique to reduce the sampling steps without significantly sacrificing generation quality is to straighten the trajectory. Inspired by the success of such technique in point-cloud generation (Wu et al., 2023) and text-to-image to-image generation (Esser et al., 2024; Liu et al., 2023b), we finetune our model vf. trained with Averaged Flow, using the reflow algorithm proposed in previous rectified flow works (Liu et al., 2022: Liu, 2022). Specifically, we first randomly sample atom coordinates z/ from standard Gaussian and generates the corresponding conformer æ² by simulating the learned flow with ODE. The coupling (z'g ) is then used in the rectified flow objective to finetune the model:
For the NequlP model (Fig.5), the features of cach atoms (Z) and bonds (E) are first embedded by the model into scalar features, along with coordinate of atoms (Ir) at given timestep t. These features are then mixed with the edge vector through 6 interaction blocks of the model. Finally, linear layer is used to make prediction of the vector field as type / = geometric features. Noteworthy modifica- tions to the original architecture include incorporating edge features into the graph convolution layer, adding residual connections, and equivariant layer normalization to stabilize training. Our modified NequlP model contains ~4.7 million parameters and its details can be found in Sec.A.1,
Algorithm Averaged Flow with Reflow+Distillation Train
Require: Molecule Dataset G [Co. Gp). each with con- formers X = [.....N
RRfflow (0) = EE [1] () (01 - = with & E [0,
Require: Learnable Velocity Field Network v² 1. Base SO(3) Averaged Flow Training
(11) Liu et al. (2022) proved that the coupling (1/) x1) yields equal or lower transport cost than (1), where 2o is sampled from the noise distribution and Φ from the data distribution. Therefore, applying the reflow algorithm to fine-tune model with Eq. 11 can effectively reduce the trans- port cost and straighten the trajectory.
L, w, G G~ (0,
on ~ x²
if of is equivariant then () xn (linear interpolant Eq.9)
2t +- L.1
else
2t + xu So ur (x )dr (integration interpolant Eq. 10)
end if
We empirically find that the transport trajectories bridg- ing Gaussian noise and molecular conformers demonstrates high curvature when is close (one example shown in Fig. 1b). Therefore, inspired by Lee al. (2024b), we sam- ple t from an exponential distribution with the probability density function as p(f) & Exp(At), where À = 1.2 by selection to focus the training more on / < 0.5. The distri- bution of t is visualized in Fig. 7. After reflow, the sampling speed can be further reduced by distilling the relationship of the coupling x1) into model uo to enable one-step step transport and eliminate the need of ODE solving. During the distillation stage, we fine-tune the reflowed model vo with the following loss function:
Ue (x,))) + Solve closed-form Eq. 5 for x. and t
Gradient Step 11of (z+[G) - us (21
2. Reflow
2/ ~ (0.
2 ~ (vf
Finetune model with coupled pair (#6 s1) through Eq. 11
3. Distillation
Train model with coupled pair (26, x1) through Eq. 12
Our DiT model (Fig. 6) is based on Diffusion Transform- ers (Peebles & Xie, 2023). To include the critical covalent bond information and add extra structural details, pairwise representation constructed using the pairwise distances between atoms and bond features, and is used to inject ad- ditional learnable bias in the attention mechanism. This design is inspired by AlphaFold3 (Abramson et al., 2024) and has proven highly effective in the protein generation task (Geffner et al., 2025). Our DiT model contains ~522 million parameters. We also trained a slightly larger vari- ant of it, DiT-L, which contains ~64 million parameters to match the size of a baseline MCF (Wang al., 2024). Details of our DiT models can be found in Aec.A.2. Both model are trained and fine-tuned using Averaged Flow + reflow + distillation following the Algorithm 1. Details of model training and sampling are included in Sec. B.3.
Disum (0) = B [10 1 (16. ( (x]]=
(12)
which is equivalent to Eq. 11 with f == O.
3.3. Flow-Matching Model Architecture
To demonstrate that the Averaged Flow training objective (Eq. 8) is architectur--anoosiic, we implemented and trained two different neural network for flow-mater motchin including (i) a -Euuivaqua graph neural networks (NequIP) which is modified based from Batzner et al. (2022), and (ii) a non-qquivariant yet highly scalable diffusion transformer
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Efficient Molecular Conformer Generation with S0(3)-Averaged Flow Matching and Reflow
(a)
NequiP- Recall
NequiP- Precision
(b)
DIT Recall
DIT - Precision
80
70
60
COV (%)
60
40
COV (%)
60
40
50
30
40
20
20
0.8
1.1
AMR (A)
1.0
AMR (A)
0.8
0.7
1.0
0.9
0.8
0.6
0.8
20
36
68 Epochs
52
84
100
20
36
52 68 Epochs
84
100
4
20
36
52 Epochs
68
84
100
20
36 52
68
84
100
Epochs
Cond. NeNequ
Kabschhequp
AvgFlowsequi
Cond. OToT
Kabscherr
AvgFlowot
Figure 2. Comparison between training objectives. Both (a) NequlP and (b) DiT model trained with Averaged Flow consistently
converge better performance on 300-molecule GEOM-Drugs subset. The other objectives we compared Avenaged Flow are: (i) Conditional OT and (ii) Kabsch alignment of noise with conformer before conditional OT (Hassan et al., 2024).
4. Experiments
quality with more simulation steps. Moreover, to fairly validate the effectiveness of the Averaged Flow objective, compare the performance of both the NequlP and DIT architectures (architecture details in Appendix A) trained with different objectives on the Drugs dataset. Similarly, we compare the performance of the NequlP architecture before and after eeflow-dittlllnn to show the necessity reflow for few-step generation.
Following previous works, we train and evaluate our model on the GEOM-QM9 and GEOM-Drugs datasets (Axelrod & Gomez-Bone Bimeer-Bmmbarlli, 2022). We follow the splitting strategy proposed by Ganea et al. (2021); Jing et al. (2022), and test containing 1000 molecules
model the same for both QM9 and Drugs datasets. Dataset and splitting de- tails are included in Sec. B. The major model evaluation metrics are the average minimum RMSD (AMR, the lower the better) and coverage (COV, the higher the better). Both AMR and coverage are reported for precision (AMR-P and COV-P) and recall (AMR-R and COV-R). The definition of metrics are specified in S.c.B.6. Intuitively, coverage measures the percentage of ground truth conformers that generated (recall) or the percentage of generated con- formers that are close enough to ground truth (precision), while AMR measures the average RMSD between each ground truth and its closest generated conformer (recall), or vice versa (precision). There are three types of baselines in this work: (a) methods with fast inference speed such cheminformatics tools (RDKit, OMEGA) and regression model GeoMol (Ganea al., 2021); (b) lightweight dif- fusion models with reduced degree-of-freedon (Torsional Diffusion); and (c) large transformer-based diffusion or flow-based model operating on Euclidean atomistic coor- dinates (MCF and ET-Flow).
4.1. Averaged Flow Leads to Faster Convergence to Better Performance
To showcase the advantage of the Averaged Flow over other training objectives, we evaluate the performance of mod- els trained on different objectives using a randomly sam- pled GEOM-Drugs test subset containing 300 molecules. The two other objectives compared conditional OT and Kabsch alignment. The Kabsch alignment objective is to rotationally align the sampled noise Zp with conformer before training with the conditional OT objective. Mod- els are evaluated every 8 epochs of training starting from to 100 epochs. Fig. 2 demonstrates that both models trained with Averaged Flow are consistently better than those trained with conditional OT and Kabsch alignment on all four metrics.
With only 68 epochs of training, AvgFlowNequi has COV- R higher and AMR-R lower than the same architecture trained with the other two objectives for 100 epochs (Fig.2a). The COV-P (49.3%) and AMR-P (0.831 of ArgFlowNequIP trained for 52 epochs are better than those of Cond. OT (COV-P 49.1% and AMR-P 0.832A) trained for 100 epochs. Also, AvgFlowNequIP outperforms Kabsch trained for 100 epochs on
worth mentioning that MCF has three variants, with number of parameters ranging from 13 242 millions. MCF uses DDPM sampler for full SDE simulation and DDIM sampler for few-step genera- tion. ET-Flow has two variants: ET-Flow and ET-Flow-SS. The former is shown to produce better few-step generation quality, while the latter shown to have better generation
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Efficient Molecular Conformer Generation with SOG3)-Averaged Flow Matching and Reflow
AMR-P Neuur 0.814A and NebUU 0.815A) and on COV-P (AugFlowNequir 50.9% and KabS = 50.5%) after 76 and 84 epochs, respec- tively.
the-art conformer generation performance on smaller scale molecules. Table. also shows that reflow and distillation can effectively maintain the conformer generation quality of www-based model with only or 2 steps of ODE solving,
The benefit of Averaged Flow is more significant with the non-equivariant DiT architecture (Fig.2b). AvgElowDIT trained for only 12 epochs has better performance on all met- rics than Kabschper trained for 100 epochs. Compared to Cond. OED trained for 100 epochs, ArgFlowDer achieves better performance precision metrics after 36 epochs of training and on recall metrics after 52 epochs of training. Overall, models trained with Averaged Flow converge in fewer epochs to better performance in molecular conformer generation.
4.3. GEOM-Drugs
We then train and benchmark our models on GEOM-Drugs, which is a larger dataset containing conformers of drug- like molecules. The top section of Table 2 shows compari- son between our models and all baselines with full simu- lation (no SDE/ODE step limit). ArgFlowNequIP demon- strates good performance on GEOM-Drigs by outperform- ing torsional diffusion on all metrics. Compared with MCF-S which has approximately 3 times more parameters, AvgFlowNequiP achieves better COV-P and AMR-P, indi- cating more conformers generated by AvgFlowNeqqiP are close to ground truth conformers. With a more scalable and expressive architecture, AvgElowDer achieves performance on par with both MCF and ET-Flow-SS in full simulation conformer generation. Specifically, it outperforms MCF in precision metrics and surpasses ET-Flow-SSS in recall met- rics. Moreover, when scaled up the sames number of pa- rameters as MCF-B (64M), AvgFlowDrT-L outperforms all MCF variants in precision metrics and MCF-B in AMR-R.
4.2. GEOM-QM9
Table 1. GEOM-QM9 Benchmark. Quality of generated con- former ensembles for GEOM-QM9 (8 = 0.5A) test set terms of Coverage (COV) and Average Minimum RMSD (AMR). Best scores are bold and second best underlined. Baseline values are taken from the corresponding papers.
COV-R (SI1
AMR-R (A))
(S)1
AMR-P (Ay)
Model
Mean
Med
Mean
Med
Mean
Med
Menn
Med
Full Simulation and Now-diffusionfton Bavelines
KDKE
$5.1
100
0.235
0.199
86.8
100
OMEGA
85.5
100
0.177
0.126
82.9
100
0.224
0.186
AvgFlowDIT-L also outperforms ET-Flow-SS in all metrics
Tor. Dift.
GeoMol
92.8
91.5
100
100
0.178
0.225
0.147
0.193
87.6
92.7
100
100
0.221
0.27
0.241
0.195
except only for AMR-P.
ET-Flow-SS. (8.3M)
95.0
100
0.083
0.055
91.0
100
0.027
MCF-B (64M)
0.105
92.7
0,119
In the lower two sections of Table 2, we demonstrate the two-step and one-step generation benchmarks of our mod- els, MCF, and ET-Flow. Nequuul (two-step) can outperform cheminformatics tools and GeoMol on all met- rics, with large margin specifically on the recall metrics. Avg outperforms all baselines in coverage metrics of two-step generation. Most notably, our Avg AogFlDWDDT-D significantly outperforms all baselines by a wide margin in one-step generation, thanks to the straightened trajec- tory (Fig. 8). We want to emphasize that it surpasses 20 steps of Tor. Diff. with one-shot generation, despite Tor. Diff. starting generation with RDKit-generater conformers. Furthermore, AugFlowpir-D (one-step) outperforms MCF-S (1000 steps full SDE simulation) across all precision metrics and exceeds the performance of all MCF and ET-Flow (two- step) models in the coverage metrics. Overall, the training strategy combining Averaged Flow with reflow and distilla- tion enables a scalable transformer-bases architecture like DiT to achieve exceptional one-shot conformer generation quality and diversity. More visualizations of generated con- formers and ODE trajectories before and after reflow and distillation are included in Fig. 8. The exceptional one-step generation quality of AugFlowprT.D pushes the limit of the quality-speed trade-off in molecular conformer generation, giving it the potential to be adopted for scale virtual screen use cases.
ArgFloosoqua (M)
96.4
100
0.084
0.012
92.8
100
0,132
0.0K4
AugFloweer (52M)
96.0
100
0.082
0.030
95.0
100
0
0,039
Two-step Generation
Arg (4.7M)
95.9
100
0.104
87.7
100
0.236
0.207
Ous-stop Geveration
AugFlowss D (4.7M)
95.1
100
0.220
0.195
848
100
0.304
0.283
On the CEOM-QM9 dataset, we compare our model with two widely used cheminformaties tools: RDKit and OMEGA ! along with GeoMol (Ganea et al., 2021). Tor- sional Diffusson (Jing et al., 2022), ET-Flow-SS (Has- san et al., 2024), and MCF (Wang et al., 2024). We de- note our NequlP model trained with Averaged Flow as AugFlowNequIP and the DiT model as AvgFlowDIT. The model finetuned with reflow and distillation are denoted as AvgFllowarhittettuee and Avg archttetttttttre--D. respec- tively. Table. I shows that AugFlowNequIr outperforms all other models in the COV-R metrics and nearly matches the AMR-R of ET-Flow-SS, indicating it is capable of gener- ating very diverse conformers on the GEOM -MM dataset. The AvgFlowDer model establishes a new stateoof-the-ar on the GEOM-QM9 dataset by outperforming all baselines in all metrics. More importantly, the ArgFlowNequi--R and AvgFlowNequlP.D .D achieve higher COV-R than other models with only 2-step and 1-step ODE sampling, respec- tively. AvgFlowNequtt..t also outperforms all cheminfor- matics tools and GeoMol in all metrics. The benchmark on GEOM- QM9 shows that our model can achieve of-
Results adopted from (Jing et al., 2022)
7

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Efficient Molecular Conformer Generation with SO(3)-Averaged Flow Matching and Reflow
Table GEOM-Drugs Benchmark. Quality of generated conformer ensembles for GEOM-DRUGS (6 0.75A) test in terms Coverage (COV) and Average Minimum RMSD (AMR). Best scores are bold and second best underlined. Baseline values are taken from the corresponding papers. *Due the adaptive step size, the number of steps AvgPlowNequIP an average value over test set molecules.
COV-R (%)
AMR-R (A) +
COV-P (%)
AMR-P (Å) 4
Method
Step
Mean
Med
Mean
Med
Mean
Med
Mean
Med
Full Simulation and Non-diffusionffow Baselines
RDKit
38.4
28.6
1.058
1.002
40.9
30.8
0.995
0.895
OMEGA
53.4
54.6
0.841
0.762
40.5
33.3
0.946
0.854
GeoMol
44.6
41.4
0.875
0.834
43.0
36.4
0.928
0,841
Tor. Diff.
20
72.7
80.0
0,582
0.565
55.2
56.9
0.778
0.729
ET-Flow-SS (8.3M)
50
79.6
84.6
0,439
0,406
75.2
81.7
0.517
0.442
MCF-S (13M)
1000
79.4
87.5
0,512
0.492
57.4
57.6
0.761
0.715
MCF-B (64M)
1000
84.0
91.5
0.427
0,402
64.0
66.2
0.667
0,605
MCF-L (242M)
1000
84.7
92.2
0.390
0.247
66.8
71.3
0.618
0,530
AugFlowNequtP (4.7M)
102*
76.8
83.6
0.523
0.511
60.6
63.5
0.706
0.670
ArgFlowDer (52M)
100
82.0
86.7
0,428
0.401
72.9
78.4
0.566
0.506
ArgFLOWDIT-L (64M)
100
82.0
87.3
0.409
0.381
75.7
81.9
0.516
0.456
Two-step Generation
MCF-B (64M)
2
46.7
42.4
0.790
0.791
21.5
13.2
1.155
1.160
MCF-L (242M)
2
54.2
54.4
0.752
0.746
25.7
18.8
1.119
1.115
ET-Flow (8.3M)
2
73.2
76.6
0.577
0.563
63.8
67.9
0.681
0.643
AvgFlowNequll-It (4.7M)
2
64.2
67.7
0.663
0.661
43.1
38.9
0.871
0.853
AugFlowDrT-R (52M)
2
75.7
81.8
0.545
0.533
57.2
59.0
0.748
0.705
One-step Generation
MCF-B (64M)
1
22.1
6.9
0.962
0.967
7.6
1.5
1.535
1.541
MCF-L (242M)
1
27.2
13.6
0.932
0.928
8.9
2.9
1.511
1.514
ET-Flow (8.3M)
27.6
8.8
0,996
1,006
25.7
5.8
0,939
0.929
AvgFloowoouutP-D (4.7M)
55.6
56.8
0.739
0.734
36.4
30.5
0.912
0.888
AvgFlOWDIT-D (52M)
76.8
82.8
0.548
0.541
61.0
64.0
0.720
0.675
4.4. When is Reflow Really Necessary?
Recall
Precision
80
From the benchmark results on GEOM-Drugs and GEOM- QM9, we understand that our models after reflow/distilla- tion can achieve better performance than cheminformatics methods on all metrics. However, is evident that the models' performance drops after reflow especially for the precision metrics. mottmng models generally have high generation quality with fewer steps compared to de- noising diffusion models (Lipman al., 2023), thanks to the ODE sampling process. In this section, are trying to answer the question: when is reflow really necessary to generate high-quality molecular conformers? We address this question with a case study of the NequlP model.
COV mean (%6)
60
40
40
AvgFiomrague
20
AvgFlowwezur
20
AugFiomaasuee
AvgFlowwesdn
AugFlomeegues
AvgFlowweup
0
0
100
10a
10a
10a
10a
10a
AvgFlommenue
AvgFJownwai
AMR mean (A)
3
AvgFiomneuus
AvgFJowweuns
AvgFiownauur
AvgFlowwaan-
2
Fig. shows the the performance of our NequlP model using Euler solver with number of steps Natep E 2,3,, 5, 10, 20, 50, 100). The performance of the mod- els is evaluated with the same four metrics on subset of the GEOM-Drugs test set containing 300 molecules. Over- all, NegFlow performs better when Nstop 2 10 than
100
10
10a
100
10a
10c
steps
steps
Figure 3. Effeet of the number of ODE steps on model's perfor- mance. Comparison between the performance of AvgFlowNequiP before and after reflow with different number of ODE steps.
AvgFlowxequIP.R. When Natep < 10, the performance of AvgFlowNeuuIr starts collapse and eventually reaches 0% coverage for both recall and precision when Natep
The performance gap becomes significant for all metrics when Nstop < AvgFlow R. on the other hand, has
8

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Efficient Molecular Conformer Generation with S0G3)-Averaged Flow Matching and Reflow
minimal loss in performance until Natep due the straightened flow trajectory. However, the one-step genera- tion quality of the model still suffers even after reflow. Dis- tillation can effectively reduce the RMSD of one-step gen- erated conformers and improve both the COV-R and COV-P. In summary, reflow is critical when generating molecular conformers with very few ODE steps (Nstep < 5).
rectness" of generated samples is invariant rotation (e.g. protein structure generation). The effectiveness of Averaged Flow objective is architecture-dgnosiicc, meaning that it can be applied to both equivariant and con-equivariant architec- tures, as demonstrated through experiments. We have also applied the reflow and distillation techniques to straighten the flow trajectory and enable few-step or even one-step molecular conformer generation. Our models reach state- of-the-art performance the GEOM-QM9 dataset. Our Arg/lowDrt model matches the performance of other strong baselines and our ArgFlowDsT.L. model achieves state-of- the-art precision metrics performance on the GEOM-Drugs dataset. By analyzing the effect of number of ODE steps on model generation quality, we find that reflow and distil- lation are necessary when few-step (NVstep 5) conformer generation is desired. Most importantly, our ArgFlowDsT--D model significantly outperforms both diffussion/llow-bide and cheminformatic baselines in one-step conformer gen- eration. Combining efficient implementation and one-step generation capability, our method sheds light on the path toward highly efficient yet accurate conformer generation using flow-based generative models. Overall, our method bridges the gap between diffusion/flow-basse models and practical molecular conformer generation application by pushing the boundary of quality-speed trade-off.
4.5. Sampling Time
To demonstrate the sampling efficiency of our model, we first compare the single function call wall-clock time of our model with several strong baselines. Fig. 4 shows that both ArgFlowNequIP and ArgFlowDer are significantly faster than other baselines models in terms of function call time. AugFlowDer with 64M parameters is comparable to MCF-S and slightly faster than ET-Flow. The major speed-up of the our model is due to the JAX implementa- tion and a smaller number of parameters (NequlP). Table 6 show that the average sampling time of Nogullr (2 steps) for each conformer in the GEOM-Drugs test set 2.68 microseconds, which is 21 to 50x faster than dif- ferent variants of MCF sampled with DDIM for 3 steps. It is also 48x faster than torsional diffusion sampled with 5 steps. Meanwhile, AvgFlowNeqqlP.. (2 steps) outperforms torsional diffusion and MCF-S (3 steps) by a large mar- gin with only a fraction of the sampling time. Moreover, AvgFlowDIT-D faster than the MCF variants (3 steps) with better generation performance in all metries, AvgFlowDeT also has better recall metrics performance than ET-Flow (2 steps) with ~3 faster sampling. With reflow/distillation finetuning that ensures high-quality generation with only 2 or even I ODE step, our models achieve extraordinary sampling efficiency.
Impact Statement
The method proposed in this work improves the training and sampling efficiency of flow-based molecular conformer generation models, advancing the application of deep gen- erative model in the field of computational chemistry and drug discovery. The Averaged Flow objective bears broader impact of improving flow-based models for other similar applications such as protein design. Direct consequences of our work include more accurate molecular property predic- tion and faster compound virtual screening, which lead to societal impacts like accelerated drug discovery and molec- ular design for environmental purpose.
Single step wall time (ms)
40
34.0
30
25.6
21.2
20
16.9
14.6
10
1.34
Tor. DIff.
ET-Flow
MCF-S
MCF-8 8
MCF-L
NequiP
DIT
DIT-L
Figure 4, Single function call wall-clock time comparison.
5. Conclusion
We have presented SO(3)-Averaged Flow new objective to accelerate the training of flow-matching models for molec- ular conformer generation. Averaged Flow leads to faster convergence and better performance compared with condi- tional OT and Kabsch alignment. The Averaged Flow objec- tive can also be extended other applications when the "cor-

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Efficient Molecular Conformer Generation with SO(3)-Averaged Flow Matching and Reflow
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Efficient Molecular Conformer Generation with SO(3)-Averaged Flow Matching and Reflow
Appendix
A. Details of Model Architectures
A.1. Modified NequlP Architecture
The equivariant model used in this work is a modified variant (Fig. 5) of the NequlP model (Batzner et al., 2022). The model takes inputs including the atomic features Z, coordinate of atoms zt, relative distance vector between atoms edge (bond) features E, and the flo--matching timestep The output model is vector field corresponding the probability flow at Compared to the original NequIP model, our variant has residue connection and equivariant layer normalization (Liao et al., 2023) after each interaction block, which more than 4 layers. Bond information in the 2D molecular graph is critical inductive bias for the molecular conformer generation task. To add bond information into the model, we featurize the edges in the molecular graph and concatenate the edge features E with the radial basis embedding of relative distance vector F. The concatenated message is then fed into the rotationally invariant radial function implemented as an multi-layer perceptron. To keep long-range information in the graph convolution during intermediate time-step I, we remove the envelop function from the radial basis and keep only the radial Bessel function.
found to be highly effective in stabilizing the training of model with
For both the BG0M-Druug and GEOM-QM9 dataset, we train model with 6 interaction blocks. The multiplicity is set to 96 and maximum order of irreps / is 2. The radial function MIP has 2 layers and hidden dimension of 256. Molecular graph are fully-connected with non-bond as an specified bond type. The relative distance vectors are scaled down by a soft cutoff' distance of 10Å and 20A for QM9 and Drugs dataset, respectively. we used 12 Bessel radial basis functions in the model. The model is implemented using e e3n- jax ax (Geiger Smidt, 2022; Geiger et al., 2022).
(a)
(b)
(c)
x
(G,, E)
0,1,2
(f. B)
1= 0,2,
Embedding
Norm
Interaction Block
Self-interactinn
(
Basis
N
Convolution
Concal.
Add & LayerNorm
(Z. x,).
MLP
Concatonation
Tensor
Output Block
Seif-interaction
Product
Figure Modified NequlP model architecture (a) Overview the modified NequlP architecture the flow vector field prediction. (b) Details of the interaction block, where atomic features are mixed and refined with relative distance vectors F and edge features E. (c) In the convolution block, learnable radial function MLP incorporate basis embedding of and edge features E. Tensor product used to combine the output of the MLP and the spherical harmonics y(()) projection of f.
A.2. DiT Architecture
The diffusion transformer (Peebles Xie, 2023) is a powerful yet scalable architecture for generative modeling. Inspired by recent successes of such architecture in protein structure prediction (Abramson et al., 2024) and generation(Geffner et al., 2025), we implemented the diffusion with pairwise attention bias (DiT) model and modified specifically for molecular conformer generation task. Fig. 6 shows schematic of the model and the details of the main trunk of the architecture. Besides the adaptive bias and scaling used in standard diffusion transformer, we also added: (i) auxiliary register tokens (Xiao et al., 2023; Darcet et al., 2023) and (ii) QK normalization (Dehghani et al., 2023) for the training stability. The inputs to the model are the same input features to the NequlP model. The atomic features are concatenated with the coordinate of atoms 2, and project to atomic representation of each atom. sinusoidal embedding of the timestep is used as the condition embedding. The pairwise distance || between atoms and the bond features E, which is embedded from bond
13

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Efficient Molecular Conformer Generation with SO(3)-Averaged Flow Matching and Reflow
types Sec.B.2, used construct pairwise representation, which is then injected bias into the attention. Gating mechanism applied over the output of attention block to control the update of the atomic representation. In this work, trained two variants of the DiT model: DiT (52M params.) and DiT-L (64M params.). The hyperparameters of the two variants are tabulated in Table. 3.
(a)
(b)
(Ilie ll, E)
(Z. X
t
Pair Rape
Cond
Sole
Sohmex
Acletox
SWISL
Adopthie Blased Mult-fead Allentoo
Adaptive Transiion
Condiona
Atomic
Unee
3g (C)
Figure DiT model architecture. (a) The overview of DIT. (b) The details adaptive multi-hest head attention with pairwise bias and adaptive transition block.
Table 3. Hyperparameters for the architecture of the DiT models.
Model
DiT
DUT-I
Architecture Component
initialization
random
random
atomic repr. dim
512
576
register tokens
cond. dim
512
196
sinusoidal enc dim
512
196
pair repr.
128
attention heads
8
12
transformer layers
8
10
trainable parameters
52M
64M
14

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Efficient Molecular Conformer Generation with SO(3)-Averaged Flow Matching and Reflow
B. Experiments Details
B.1. Datasets
The dataset we train and benchmark our model on are GEOM-Drugs and GEOM-QM9(Axclrod & Gomez-Bombarelli,
2022). We follow the exact splitting defined and used previous works (Ganea al., 2021; Jing et al., 2022; Wang et al.,
2024). The train/val/test set of GEOM-Drugs contains 243473/30433/100 molecules, respectively. The train/val/test set of
GEOM-QM9 contains (66566/1333/1000 molecules, respectively.
B.2. Molecular Graph Featurization
We followed the atomic featurization from GeoMol (Ganea et al., 2021). Details of the atomic featurization included in Table. 4. Graph Laplacian positional encoding vector (Dwivedi al., 2023) with size of 32 is concatenated with the atomic features for each atom in molccular graph to form the final atomic feature vector Z. The edge features is the one-hot encoding of the bond types: {No Bond, Single Bond, Double Bond, Triple Bond, Aromatic Bond).
Table Atomic features as input the model
Name
Description
Range
atom.type
Atom type
One-hot encoding of the atom type
degree
Number of bonded neighbors
{ar < x S 6, E Z
charge
Formal charge of atom Implicit valence of atom
}}
valence
te:0<z56.rE2
hybridization
Hybridization type
(sp. sp" sps. spåd, spdd other]
chirality
Chirality Tag
(unspecified, tetrahedral CW, tetrahedral CCW, other}
num_H
Total number of hydrogens Whether on aromatic ring
(x:0 8,8 18, Z}
aromatic
(Tre,, False)
num_rings
Number of rings the atom on
(=:0 S x S 3, E Z} (True, False)
ring.size.3- 3-8
Whether on ring size of 3-8
B.3. Training and Sampling Details
B.3.1. NEQUIP TRAINING AND SAMPLING
The NequIP model is trained with the Averaged Flow for 990 epochs on the GEOM-Drigs dataset and 1500 epochs on the GEOM-QM9 dataset using 2 NVIDIA A5880 GPUs. We used dynamic graph batching to maixmize the utilization of GPU memory and reduce JAX compilation time. The effective average batch size is 208 and 416 for Drugs and QM9 dataset, respectively. We used Adam optimizer with learning rate of 1e-2, 2, which decays to 5 after 600 epochs and to 1e-3 after 850 epochs. We selected the top-30 conformers for model training.
To sample coupled =1) for reflow and distillation, we generate 32 noise-sample pairs for each molecule in the Drugs and 64 for each molecule in the QM9 dataset. The reflow and distillation are done using 4 NVIDIA A100 GPUs and doubling the effective batch size of each dataset. During the reflow stage, the model is finetuned for 870 epochs Drugs and 1530 epochs on QM9. We used Adam optimizer with learning rate of 5e-3, which decays to 2.5e-3 after 450 epochs for Drugs (500 epochs for QM9), and to 5e-4 after 650 epochs for Drugs (900 epochs for QM9). During the distillation stage, the model is finetuned for 450 epochs on Drugs and 1200 epochs on QM9. We used Adam optimizer with learning rate of 2c-3, which decays to 1e-3 after 300 epochs for Drugs (500 epochs for QM9), and to 2e-4 after 450 epochs for Drugs (900 epochs for QM9). We used exponential moving average (EMA) with decay of 0.999 for all Averaged Flow, reflow, and distillation training.
To generate the benchmark results of AvgFlowNequIP (Table. 1 and Table. 2), we use the Tsitouras' 5/4 solver (Tsitouras, 2011) 1) implemented in the diffrax package with adaptive stepping. The relative tolerance and absolute tolerance are set to 1c-5 and 1c-6 when sampling for GEOM-Drugh respectively. The relative tolerance and absolute tolerance are both set to 1e-5 when sampling for GEOM-QM9. Euler solver is always used for AvgFlowNegule and AlgFloo When comparing the effect of ODE steps to models, Euler solver used.
15

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Efficient Molecular Conformer Generation with SO(3)-Averaged Flow Matching and Reflow
B.3.2. DIT TRAINING AND SAMPLING
We trained 2 variants of the DiT model, DiT (52M) and DiT-L (64M), using Averaged Flow. We used the cosine learning rate decay with warm-up steps for learning adjustment and AdamW optimizer (Loshchilov & Hutter, 2017). The details of hyperparameters in training finetuning of the 52M DiT model on the GEOM-Drugs dataset. 32 noise-sample pairs were generated for each molecule in the Drugs using 100 Euler steps to construct the reflow/distillation dataset. We used EMA with decay of 0.999 for all Averaged Flow, reflow, and distillation training. We used 100 Euler steps to generate conformers for all benchmarks of AvgFlowDrr
tabulated in Table. 5. Due to computational cost, only did reflow and distillation
Table 5. Hyperparameters for DiT models training.
Model
DIT (GEOM-Drugs)
DiT-L (GEOM-Drugs)
DIT (GEOM-QM9)
Training Details
train epochs
900
900
batch size per GPU
64
64
256
GPUs
8
8
2
GPU name
NVIDIA A100
NVIDIA A100
NVIDIA A5880
optimizer
AdamW
AdamW
AdamW
init learning rate
1e-6
1e-6
1e-6
peak learning
2e-4
2e-4
2e-4
warm-up steps
10k
cosine decay steps
IM
1M
IM
ending learning rate
1e-6
1e-6
1e-6
reflow epochs
260
reflow peak learning rate
le-4
distill epochs
130
distill peak learning rate
5e-5
B.4. Chirality Correction
A simple post-hoc chirality correction step is used during sampling. When the conformer of a molecule that contains at least 1 chiral center atom is generated, we use RDKit (Landrum, 2016) to perform the chirality correction withe following steps: (i) Given a Mol object with generated conformer, re-assign chiral tag to atoms. (ii) Obtain the the canonical SMILES (Weininger, 1988) of the Mol object and compare it to the original canonical SMILES, (ii)) If there is mismatch of the two canonical SMILES strings, mirror the generated conformer. The post-hoc chirality correct is simple, fast, and end-to-end solution to the chirality mismatch during generation.
B.5. Distribution of during Reflow
0.0175
0.0150
0.0125
Probability
0.0100
0.0075
0.0025
0.0000
0.0
0.2
0.4
0.6
0.8
1.0
Figure 7. The distribution of during reflow
The distribution of during reflow is sampled from p(t) oc Exp(At), where = -1.2 The distribution is visualized in Fig.
16

---

Efficient Molecular Conformer Generation with SO(3)-Averaged Flow Matching and Reflow
B.6. Evaluation Metries
We report the average minimum RMSD (AMR) between ground truth and generated structures, and Coverage for Recall and Precision. Coverage is defined the percentage of conformers with minimum enror under specified AMR threshold. Recall matches each ground truth structure its closest generated structure, and Precision measures the overall spatial accuracy of the each generated structure. Following Ganea et al. (2021); Jing et al. (2022), we generate two times the number of ground truth structures for each molecule. More formally, for K = 2L, let {C} heu1, and {Ck respectively be the sets of ground truth and generated structures:
COV-Precision ==
{ [L..K] en.. RMSDDC 0?) < <6}
K
(13)
AMR-Precision
Σ
min.e(1...)] RMSD(C,, Cf),
K kE[1..K]
where ò is the coverage threshold. is set 0.75A for the Drugs and 0.5A for the QM9 dataset. The recall metries are obtained by swapping ground truth (K) and generated conformers (L) in the above equations.
B.7. ODE Trajectory Visualization
In this section, we visualize the ODE trajectory of few selected test set molecules. For each molecule, the 100-steps ODE trajectory of the AvgFlowDrr model before reflow is shown orange curves, and the 1-step trajectory of the Arg model after reflow-distillaiion is shown as green curves.
B.8. More Comparison of Few-step Generation Performance
Here provide compare the inference time of models few-step generation while maintaining reasonable generation quality.
Table Sampling time and performance comparison between models few-step sampling. In this table, we record sampling time of baselines and our models with the minimum number of steps to generate reasonably good conformers. Mean are reported for all metrics. Bold results are best.
Method
Step
Time (ms) +
COV-R (%) f
AMR-R (A) 4
COV-P (%) f
AMR-P (A) 4
Tor. Diff.
5
128
58.4
0.691
36.4
0.973
ET-Flow
5
106
77.8
0.476
74.0
0.550
ET-Flow
2
42.8
73.2
0.577
63.8
0.681
MCF-S
3
57.3
56.9
0.725
30.8
1.014
MCF-B
3
102
66.5
0.665
39.9
0.951
MCF-I
3
134
71.6
0.636
45.3
0.686
2
2.68
64.2
0.663
43.1
0.871
AugFlowniT-D
1
14.6
76.8
0.548
61.0
0.720
17

---

Efficient Molecular Conformer Generation with SO(3)-Averaged Flow Matching and Reflow
(a)
beclcccccca-cchncecccc311)o2o
100-steps AvgFlow
1-step after distillation
(b)
100-steps AvgFlow
1-step after distillation
(c)
CHFCNNc22ccccc23cccccce11
100-steps AvgFlow
1-step after distillation
(d)
100-steps AvgFlow
1-step after distillation
Figure 8. Comparison between ODE trajectories. Visualization of selected generated conformers (SMILES attached) and the ODE trajectories. Orange trajectories are from Algwlor hefore reflow, and green trajectories are from AugFlowDIT after reflow-distillation Carbon atoms colored black, hydrogen gray, oxygen red, nitrogen blue, sulfur yellow, clorine green, and boron pink.
18

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Efficient Molecular Conformer Generation with SO(3)-Averaged Flow Matching and Reflow
C. Averaged Flow Details
C.1. Python Implementation
In this section, we provide the Python implementation of the Averaged Flow training objective as supplmentary information Sec.3.1.
Listing Averaged Flow
def avg_hacnonic (low(
Aex.Arrey. (]
Aes.Artey, [aun_ratees,
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edgest
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18, xly netrio, weights;
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t: jest Arrey. []]
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sigma_t t) signa0 sigmal
return dap.
eigna_tev2
ava_xi arg_taroct Mx, xl, by matric, woighes:
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def avo_target!
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matric: callabiell3ak Array, Aax.Arrayl, Arroyl, taugeta
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logeP(s
(target, Larget.7 alphal
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swauen jex.gred (log3) (log))
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anee tap (a, wherewybere, init aaw Snp-inf)
19

---

Efficient Molecular Conformer Generation with SO(3)-Averaged Flow Matching and Reflow
amax 5ax. stop.. gradieed
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be jas.lex.sstppp t clu, ano.sion (Tap, linaiq.dat vhj) soturn 5.3t mnl 1s1gm)
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def Loge [slc (ex,Array, a2: jax.Arcay, a3: Jan.Arsay) -> jax. assune al >u >u
secuen (ep.aearray jnp.anarr (82), jap.asarsay
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10.39094228, 0.1328592e-1, 9.2253196-2, -0.151505e-2, 0.916291e-2]
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return 1np, (abs x.im X, 91
20

---

Efficient Molecular Conformer Generation with SO(3)-Averaged Flow Matching and Reflow
x xix_iim, abs_x)
secuen
inp. Peg--pplyeii (b wiobs_x K_1in, 1.0) ** 2) Inp happccxp(aaa__
ppp.pallpcccoosseee..... MC..s X_lin / abs_x_i) snp.ogrt (abs_X_),
C.2. Speed Benchmark
We benchmarked the time used by our Python implementation to solve the Averaged Flow objective for batched graphs. Each graph set to have 50 nodes (the average number of atoms GEOM- molecules is 44). The benchmark done on single NVIDIA A5880 GPU.
Table Computation time of Averaged Flow batched graphs (50 nodes per graph). Unit is in Mbatch the number of graphs in batch and Neonformer is number of conformers used in Averaged Flow solving.
N conformer
Nbathh
10
100
1000
0.6
0.5
0.5
0.6
10
0.5
0.5
0.6
1.0
100
0.5
0.6
1.1
7.6
1000
0.3
0.9
7.3
73.5
21


Reviews:
[{"review_id": "0ZUBMfWJQ8", "reviewer": "Reviewer_aynM", "summary": "This paper presents SO(3)-Averaged Flow Matching and Reflow-based Distillation, a novel approach aimed at improving the computational efficiency of molecular conformer generation. By explicitly incorporating rotational symmetries into the flow-matching framework and refining the transport trajectories through Reflow and distillation, the authors significantly reduce the computational cost of both training and inference. The method achieves a substantial speedup compared to existing diffusion and flow-based models, demonstrating up to 20–50× faster sampling while maintaining competitive conformer quality. The approach is evaluated on standard molecular datasets (GEOM-QM9 and GEOM-Drugs), where it is shown to outperform prior methods in terms of sampling efficiency and convergence rate.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "*We want to thank the reviewer for the comprehensive review. Please see below for responses:*\n\n**Essential references**\n\nThe references suggested by the reviewer are indeed relevant to this paper. However, we want to point out that we have explicitly compared our model to GeoMol for both the QM9 and Drugs benchmarks. We have also discussed DiSCO as a related work in the field of molecular conformer optimization in Sec 2.1 (line 60-62). From our perspective, DiSCO is a great method to optimize molecular conformer, which is a fundamentally different application compared to generating conformer from scratch (noise). Therefore, we did not benchmark explicitly against DiSCO. As far as we know, Equiformer has not yet been used for conformer generation. It would be nice if the reviewer could point us to related literature. \n\n**Weaknesses**\n\n>Conformer generation in virtual screening\n\nWe want to respectfully correct the reviewer's statement about that the conformer generation being an offline task during virtual screening. During ultra-large virtual screening campaign (~$10^8$-$10^9$ compounds), conformer generation is indeed an *online* task, rendering the acceleration crucial. Our method, which achieves significant speedup, makes diffusion/flow model more practical for the use in virtual screening. \n\n>Degradation of performance after reflow\n\nWe want to argue that the goal of reflow is to improve the model's performance in few-steps sampling, which is critical for achieving speedup in generation. Comparing to the model before reflow, the model after reflow demonstrates significantly better performance in <5-step sampling. We also want to emphasize that the reflow technique is model architecture-agnostic and can be applied to other flow-based models to accelerate generation. That being said, we are planning on reducing possible performance degradation by excluding lower quality generated conformer ($X_1'$) from the reflow fine-tuning dataset. In that way, we expect to alleviate the problem of sampling error propagating to fine-tuning stage.\n\n**Responses to comments**\n\nWith the increasing demand of billion-level virtual screening, the  efficiency gain would be critically impactful to the application of flow-based conformer generation model. We would add more discussion about the impact in addition to line 42-43 of right column. Please see the previous answer for performance degradation of reflow and proposed future solution. We will perform a thorough grammar check for the future version of manuscript. We will release the model upon publication. \n\n**Answers to questions**\n\n1. Many molecules in the Drugs test set are with >100 ground truth conformers. Therefore the method's ability to generalize to flexible molecules is reflected well by the benchmark results in Table 2.\n\n1. Theoretically, *AvgFlow* as an training objective should not be affected by molecular size and bonds because those features are not used in the close-form solution.\n\n1. We want to clarify that the major contributions of this work, including *AvgFlow* and reflow/distillation techniques, are training schemes rather than new model architectures. We believe the proper way of showing the advantage of *AvgFlow* is by comparing same architecture trained with different flow-matching objective (as shown in Fig.1). Similarly, we show in Fig.3 the necessity of reflow/distillation. To strengthen this point, we have recently trained a diffusion transformer with pairwise biased attention with the *AvgFlow* which achieves on par performance MCF and ETFlow (please see details in the response to reviewer iSgP). This demonstrates that the improvements brought by *AvgFlow* are architecture-independent. \n\n1. We believe that RL-based fine-tuning can help models generate better conformers (more consistently at lower energy state) but may not improve sampling efficiency, which is the primary motivation of this work.\n\n1. Yes the model can be extended to larger-scale drug discovery pipeline as it achieves significant speedup for flow-based model in conformer generation. The generation quality of our 1-step distilled model has surpassed cheminformatic tools.", "claims_and_evidence": "The paper presents a novel approach for molecular conformer generation by integrating SO(3)-Averaged Flow Matching and Reflow-based Distillation. The primary claim is that by explicitly incorporating rotational symmetries into the flow-matching framework, the proposed method improves computational efficiency and convergence speed without compromising conformer quality. The authors provide empirical evidence demonstrating that their method is 20-50× faster in sampling compared to existing diffusion-based and flow-based methods while maintaining competitive performance on molecular benchmarks.\n\nA secondary claim is that Reflow and distillation significantly enhance inference efficiency, reducing the number of sampling steps required for high-quality conformer generation. This is supported by experimental results showing that Reflow enables high-quality sampling in as few as two ODE steps, while distillation further reduces this to a single-step generation process. The authors further claim that their model outperforms larger transformer-based models (such as MCF and ET-Flow) in terms of speed and parameter efficiency.", "methods_and_evaluation_criteria": "The proposed methods, SO(3)-Averaged Flow Matching and Reflow-based Distillation, are designed to improve the efficiency of molecular conformer generation by addressing rotational symmetries and reducing the number of function evaluations (NFE) required for high-quality sampling. SO(3)-Averaged Flow Matching eliminates the need for explicit rotational alignment, reducing computational overhead during training, while Reflow and distillation straighten transport trajectories, enabling one-step or few-step sampling.\n\nThe evaluation criteria are well-aligned with standard practices in molecular conformer generation. The model is assessed using Coverage (COV), which measures how well the generated conformers match the diversity of ground-truth conformers, and Average Minimum RMSD (AMR), which quantifies structural accuracy. Efficiency is evaluated through sampling speed (microseconds per molecule) and NFE, directly comparing the computational cost of the proposed approach to state-of-the-art methods.\n\nThe experiments are conducted on GEOM-QM9 and GEOM-Drugs, two widely used benchmarks in conformer generation research. GEOM-QM9 consists of small, well-characterized molecules, while GEOM-Drugs includes more complex molecular structures, providing a rigorous test of the method’s generalizability.", "theoretical_claims": "The paper does not present new theoretical results or formal proofs but instead focuses on methodological advancements and empirical validation. While SO(3)-Averaged Flow Matching is conceptually motivated as a variance-reduced training objective, and Reflow-based Distillation is inspired by rectified flow methods, these are implemented as practical improvements rather than rigorously derived theoretical contributions. The method's effectiveness is demonstrated through empirical results rather than formal guarantees.", "experimental_designs_or_analyses": "The authors evaluate their method on GEOM-QM9 and GEOM-Drugs, two widely used benchmarks, ensuring that the results are comparable to existing methods. The study includes quantitative metrics such as Coverage (COV) to measure the diversity of generated conformers, Average Minimum RMSD (AMR) to assess structural accuracy, and computational efficiency metrics (e.g., function evaluations per sample and total sampling time).\n\nA key strength of the experimental setup is the ablation study, which isolates the effects of SO(3)-Averaged Flow Matching, Reflow, and Distillation. The results demonstrate that each component contributes to improved efficiency while maintaining competitive conformer quality. Additionally, the study benchmarks against a range of baseline models, including flow-based (MCF, ET-Flow) and diffusion-based (Torsional Diffusion) approaches, providing a fair comparison.", "relation_to_broader_scientific_literature": "The paper builds upon prior work in flow-matching, diffusion-based molecular generation, and optimal transport methods. It extends conditional flow-matching by integrating rotational symmetry considerations, reducing computational overhead and improving efficiency. The introduction of SO(3)-Averaged Flow Matching eliminates explicit rotational alignment, while Reflow-based Distillation enhances sampling efficiency by reducing the number of required function evaluations.\n\nHowever, the paper does not explicitly compare its approach to recent developments in rectified flow, flow-straightening, and diffusion-bridge-based techniques for conformer generation. Notably, DiSCO (Diffusion Schrödinger Bridge for Molecular Conformer Optimization, AAAI 2024, https://ojs.aaai.org/index.php/AAAI/article/view/29238) proposes a Schrödinger bridge-based diffusion model for molecular conformer refinement, offering an alternative perspective on optimizing transport trajectories. Additionally, transformer-based and SE(3)-equivariant generative models (e.g., GeoMol, Equiformer) have demonstrated strong performance in similar tasks but are not addressed in this study.\n\nA more detailed discussion of these methods, along with empirical comparisons where feasible, would better contextualize the contributions of SO(3)-Averaged Flow Matching and Reflow-based Distillation within the broader landscape of molecular generative modeling.", "essential_references_not_discussed": "The paper situates itself within the broader context of flow-based molecular generative models but does not explicitly compare its approach to certain relevant recent works. For example, the DiSCO (Diffusion Schrödinger Bridge for Molecular Conformer Optimization) model, which employs Schrödinger bridges for molecular conformer refinement, shares conceptual similarities with the proposed approach in terms of leveraging probabilistic flow-based modeling. However, there is no direct experimental comparison with DiSCO or other related works in diffusion-based molecular generation. A direct comparison, particularly in terms of sampling efficiency, quality trade-offs, and robustness across different molecular types, would be valuable in further contextualizing the contributions of this work.\n\nhttps://ojs.aaai.org/index.php/AAAI/article/view/29238", "other_strengths_and_weaknesses": "The paper introduces a computationally efficient approach for molecular conformer generation, leveraging SO(3)-Averaged Flow Matching to eliminate rotational alignment and Reflow-based Distillation to enable fast sampling. It demonstrates a significant 20–50× speedup while maintaining competitive conformer quality, making it highly relevant for large-scale molecular screening. Comprehensive empirical validation and ablation studies further support its effectiveness.\n\nHowever, while the complexity is reduced, the extent of performance improvement beyond computational efficiency is unclear, and the study does not fully establish whether the method leads to better conformer accuracy or diversity compared to prior approaches. The necessity of extreme speedup in practical applications remains uncertain, as conformer generation is typically an offline task. Additionally, performance degradation with Reflow on larger datasets (GEOM-Drugs) raises concerns about robustness, suggesting potential trade-offs between speed and quality. The lack of direct comparisons with transformer-based generative models (e.g., GeoMol, Equiformer, DiSCO) further limits a full assessment of its advantages.\n\nA more detailed analysis of performance beyond efficiency, direct comparisons with a broader set of generative models, and a discussion of real-world impact would strengthen the study’s contribution. \n\nI will decide whether to maintain the final score based on the authors' response to these issues.", "other_comments_or_suggestions": "The paper would benefit from a clearer discussion on the practical impact of efficiency gains, particularly in real-world molecular modeling workflows.\n\nPerformance degradation with Reflow on GEOM-Drugs should be further analyzed, with potential strategies to mitigate quality loss. \n\nReleasing the complete official code would improve reproducibility and adoption by the research community.\n\nThorough grammar check could improve clarity (e.g. Line 312 (\"to rotationally aligning\") → \"to rotationally align\")", "questions_for_authors": "How well does the method generalize to highly flexible molecules with multiple low-energy conformers?\n\nDoes SO(3)-Averaged Flow perform robustly across different molecular sizes and bond constraints?\n\nHow does the approach compare with transformer-based generative models for molecular structures?\n\nWould further fine-tuning with reinforcement learning improve molecular generation accuracy?\n\nCan this method be extended to larger-scale drug discovery pipelines without significant modifications?", "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": "b9S6S7k20H", "reviewer": "Reviewer_Adi6", "summary": "The paper introduces a new method for molecular conformer generation task called Averaged Flow.  Averaged Flow is an SO(3) Flow Matching method that addresses rotational symmetry in 3D molecular structures by integrating overall SO(3) group transformations during training. The authors combined their approach with rectified flow to reduce the number of sampling steps. The method has been evaluated on two common benchmarks in molecular conformer generations: GEOM-QM9 and GEOM-Drugs datasets.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "*We thank the reviewer for acknowledging the novelty of integrating reflow and distillation method for accelerating the sampling of conformer generation model. Please see our response below to other questions and comments:*\n\n**Weakness**\n\n1. We want to emphasize that the major motivation of this paper is to improve the efficiency of diffusion/flow-based models to the level of cheminformatics tools for conformer generation through algorithmic innovations. Given the need of ultra-large scale (~$10^8$-$10^9$ compounds) virtual screening, diffusion/flow conformer generation model would become practical with improved sampling efficiency. Bearing the motivation in mind, we chose to implement a compact 4.7M-parameter equivariant GNN model that achieves significant sampling speedup while maintaining good generation quality. Comparing to larger transformer-based baselines, the model's speedup in few-step sampling compensates for reduced generation quality: the 2-step $\\mathrm{AvgFlow_Reflow}$ model outperforms 3-step MCF-B in precision metrics. It also achieves ~58% of the 5-step ETFlow with ~40x speedup in sampling. We believe that the current 4.7M model has fulfilled this motivation.\n\n   That being said, we agree with the reviewer that achieving the SOTA in generation quality is valuable. We have implemented a more scalable diffusion transformer (DiT) with pairwise biased attention, similar to AlphaFold3 [1] and Prote&iacute;na [2], for conformer generation. Due to the limited time, we trained a 52M-parameter DiT model with *AvgFlow* objective for only ~124k steps and benchmarked it on the Drugs test set (see results [here](https://anonymous.4open.science/r/confgen_icml25_rebuttal-8406/benchmark.png)). We have also compared the performance of the DiT model trained with *AvgFlow*, Kabsch, and Conditional OT (same experiment as Sec 4.1 of manuscript, see results [here](https://anonymous.4open.science/r/confgen_icml25_rebuttal-8406/dit_obj_comp.png)). The benchmark results of DiT demonstrates that:\n\n    - *AvgFlow* can be a better training objective than Conditional OT and Kabsch even with **non-equivariant model architecture**.\n\n    - A transformer-based architecture with more parameters trained with *AvgFlow* can achieve performance on-par with other SOTA models, even with a relatively limited number of training steps. \n\n   The performance of the DiT model is expected to improve further as it is trained for more steps. It can also be scaled up to similar size to MCF-L for better performance if resources allow. We are currently working on reflow fine-tuning the DiT model and we will share results once completed. \n\n1. We agree with the reviewer that an OOD evaluation on the GEOM-XL dataset can examine the model's generalizability to large molecules. The benchmark results of the 4.7M $\\mathrm{AvgFlow}$ model on GEOM-XL is attached [here](https://anonymous.4open.science/r/confgen_icml25_rebuttal-8406/geom_xl.png). In general, our model has slightly higher mean AMR but on-par median AMR for both recall and precision compared with SOTA model MCF and ETFlow. We also want to mention that the DiT model we recently trained with *AvgFlow* objective is expected to generalize better to larger molecules thanks to its scalable architecture. We will benchmark its performance on GEOM-XL as well. \n\n**Answer to the question**\n\nWe understand the confusion of the reviewer about the derivation in Sec 3.1. To clarify:\n- For the conformer generation problem, we define orbits $\\hat{x}$ as **low-energy conformers** of a given molecule. Therefore the integral $ \\int d\\hat{x}\\ \\hat{q}(\\hat{x})$ in Eq. 2, representing the entire conformer ensemble, can be written as $\\sum_{\\hat x \\in \\mathcal{X}} \\hat q(\\hat x)$, where $\\mathcal{X}$ is the set of conformers and $\\hat q(\\hat x)$ is the weight associated with each conformer (also elaborated by line 144-152, right column). \n- Your understanding is correct that for a given conformer $\\hat{x}$, the rotated molecule $g \\cdot \\hat{x}$ has the same energy state.\n- The *AvgFlow* method proposed is capable of integrating over all conformers (orbits) **and** the SO(3) group. In practice, we only integrate over the SO(3) group during training and sample one conformer in each epoch to approximate the expectation of the conformer ensemble. (line 168-170, right column). Hence, the flow-matching objective proposed in this work is SO(3)-*Averaged Flow*. \n\nReferences:\n\n[1] Abramson et al. \"Accurate structure prediction of biomolecular interactions with AlphaFold 3.\" *Nature* (2024)\n\n[2] Geffner et al. \"Proteina: Scaling Flow-based Protein Structure Generative Models.\" *ICLR* (2025).", "claims_and_evidence": "I did not find enough evidence for the claim: \"Averaged Flow leads to faster convergence to better performance for molecular conformer generation, and can be extended to other similar tasks.\"\n\nThe method did not outperform the state of the art, and also the paper about conformer generation.", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "No", "experimental_designs_or_analyses": "Yes", "relation_to_broader_scientific_literature": "The method is built on flow matching and rectified flow generative models but integrates the SO(3) symmetries in the training process.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "**Strengths**:\n\n- Integrating reflow and distillation methods to improve the sampling time in conformer generation is novel. \n- Ablation studies show the effectiveness of the proposed method, with less sampling time compared to the current SOTA.\n\n**Weaknesses**:\n\nDespite the improvement in sampling time, the paper still has several weaknesses. \n\n- Performance gap: The performance of the proposed method is far from previous approaches in most of the evaluation metrics. \n\n- Need for OOD evaluation: The paper lacks evaluation in out-of-distribution settings. For example, both MCF and ET-Flow have been evaluated on larger molecules dataset (GEOM-XL).", "other_comments_or_suggestions": "No", "questions_for_authors": "I did not understand how integrating over the group orbits gives all the possible conformers. Conformers of a molecule consist of different arrangements of atoms in 3D space, not just the result of direct rotation of the whole molecule. Also, if we have a molecule x, for example, and a rotated molecule g.x, should they not have different energy states?", "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": "NWFvOBmF9B", "reviewer": "Reviewer_iSgP", "summary": "This paper focuses on improving the training and inference efficiency of 3D molecular conformer generation while matching the performance of strong baselines. \nTo improve training efficiency, it introduces a new training objective, called SO(3)-Averaged Flow, which can avoid the need for rotational alignment between prior and data distribution by training the model to learn the average probability path over all rotations of the data.\nThen, it further introduces the reflow and distillation technique for fast inference, which can achieve high-quality molecular conformer generation with few-step or even one-step ODE solving.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "*We would like to thank the reviewer for reviewing and acknowledging the novelty of the SO(3)-Averaged Flow. Please see the response below:*\n\n**Theoretical Claims**\n\nThe major motivation behind the development of *AvgFlow* is to eliminate the need for data augmentation through rotation by training the model to learn the flow from interpolant $x_t$ to ground truth $x_1$ averaged over the SO(3) group. Sec 3.1 is the major contribution, as it mathematically derives a closed-form solution for the SO(3)-Averaged Flow so that training can be efficient without much of computational overhead. We have also attached the Python implementation of solving the *AvgFlow* objective in Sec A.3 to accompany the mathematical derivation. We would be happy to take further questions from the reviewer about specific steps in Sec. 3.1.\n\n**Weakness**\n\n1. We want to emphasize that the major motivation of this paper is to improve the efficiency of diffusion/flow-based models to the level of cheminformatics tools for conformer generation through algorithmic innovations. Given the need of ultra-large scale (~$10^8$-$10^9$ compounds) virtual screening, diffusion/flow conformer generation model would become practical with improved sampling efficiency. Bearing the motivation in mind, we chose to implement a compact 4.7M-parameter equivariant GNN model that achieves significant sampling speedup while maintaining good generation quality. Comparing to larger transformer-based baselines, the model's speedup in few-step sampling compensates for reduced generation quality: the 2-step $\\mathrm{AvgFlow_Reflow}$ model outperforms 3-step MCF-B in precision metrics. It also achieves ~58% of the 5-step ETFlow with ~40x speedup in sampling. We believe that the current 4.7M model has fulfilled this motivation.\n\n   That being said, we agree with the reviewer that achieving the SOTA in generation quality is valuable. We have implemented a more scalable diffusion transformer (DiT) with pairwise biased attention, similar to AlphaFold3 [1] and Prote&iacute;na [2], for conformer generation. Due to the limited time, we trained a 52M-parameter DiT model with *AvgFlow* objective for only ~124k steps and benchmarked it on the Drugs test set (see results [here](https://anonymous.4open.science/r/confgen_icml25_rebuttal-8406/benchmark.png)). We have also compared the performance of the DiT model trained with *AvgFlow*, Kabsch, and Conditional OT (same experiment as Sec 4.1 of manuscript, see results [here](https://anonymous.4open.science/r/confgen_icml25_rebuttal-8406/dit_obj_comp.png)). The benchmark results of DiT demonstrates that:\n\n    - *AvgFlow* can be a better training objective than Conditional OT and Kabsch even with **non-equivariant model architecture**.\n\n    - A transformer-based architecture with more parameters trained with *AvgFlow* can achieve performance on-par with other SOTA models, even with a relatively limited number of training steps. \n\n   The performance of the DiT model is expected to improve further as it is trained for more steps. It can also be scaled up to similar size to MCF-L for better performance if resources allow. We are currently working on reflow fine-tuning the DiT model and we will share results once completed. \n\n1. Clarification about Table 3:\n   - We agree with the reviewer that baselines should be compared with same number of sampling steps when possible. We took the benchmark values from corresponding papers because their checkpoints were not released before this manuscript was submitted. We have now benchmarked the 2-step and 1-step generation results of ETFlow and MCF-B/L (see updated Table 3 here). Our reflow model outperforms MCF for 2-step generation across all metrics. More importantly, our distill model outperforms both ETFlow and MCF by a large margin for 1-step generation, especially in the recall metrics. Despite the high 2-step generation quality of ETFlow, our reflow and distill models are still 16x and 32x faster in sampling, respectively. \n\n   - We want to respectfully disagree with the reviewer's suggestion that the effect of implementation should be excluded when comparing sampling speed. We would argue that faster implementation should be counted as a technical contribution as it validates a potential direction for optimization and acceleration. \n\n**Answers to questions**\n1. Please see above answers regarding experiments with larger model. \n\n1. Yes, we are working on reflow/distill fine-tuning the DiT model (non-equivariant transformer architecture) mentioned above. These experiments can be time-consuming as they require generating ($X_0'$, $X_1'$) couples. We will report results once completed.\n\nReferences:\n\n[1] Abramson et al. \"Accurate structure prediction of biomolecular interactions with AlphaFold 3.\" *Nature* (2024)\n\n[2] Geffner et al. \"Proteina: Scaling Flow-based Protein Structure Generative Models.\" *ICLR* (2025).", "claims_and_evidence": "One significant limitation of this paper is that many claims are experimental and empirical while lacking clear and convincing theoretical evidence. \nFor example, section 3.1 lacks further details on the derivation of the relevant formulas. \nIt's difficult for me to understand all these formulas without additional information and references, and I want to know whether they are supported by theoretical evidence.", "methods_and_evaluation_criteria": "As a novel concept, I think the proposed SO(3)-Averaged Flow makes sense for this problem.", "theoretical_claims": "The theoretical claims of this paper are mainly concentrated in section 3.1. However, it's difficult for me to check their correctness without further details.", "experimental_designs_or_analyses": "I think the experimental designs and analyses are sound for this problem.", "relation_to_broader_scientific_literature": "Since the key contributions of the paper are built upon flow-matching, which aims to improve the training and sampling efficiency of flow-based models, it's possible to bear an impact on other similar applications base on flow-matching, such as protein design.", "essential_references_not_discussed": "None.", "other_strengths_and_weaknesses": "Weaknesses:\n\n1.  As shown in Table 1 and Table 2, the performance of the proposed method is significantly weaker than strong baselines. In this case, the meaning of efficiency improvement is limited. \n\n2. The sampling efficiency comparison in Table 3 is unfair. \n    * Why not implement these baselines with the same steps? Reviewers need to know whether these baselines can still perform better with the same steps. \n    * The fair comparison should be entirely based on the methods themselves, and the influence of their specific implementation (e.g., JAX implementation) needs to be removed.", "other_comments_or_suggestions": "None.", "questions_for_authors": "1. Why not increase the model size to match those strong baselines? After all, it will be more convincing if the proposed method can achieve comparable or even better performance.\n\n2. Can you provide the relevant experimental support for the model architecture independence of the reflow and distillation algorithm?", "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:

Meta-review: The reviewers recognize that this paper introduces a novel and computationally efficient approach—SO(3)-Averaged Flow Matching combined with Reflow-based Distillation—for molecular conformer generation, significantly improving sampling efficiency by up to 20–50× compared to existing methods. They particularly appreciate the extensive empirical evaluations, including careful ablation studies on standard benchmarks (GEOM-QM9 and GEOM-Drugs), clearly demonstrating the value of each methodological component. Additionally, the scalability demonstrated by applying the proposed methods across different model architectures underscores its potential broad impact.
Decision: Accept (poster)