Accelerating 3D Molecule Generative Models with Trajectory Diagnosis
Authors: Zhilong Zhang
Yuxuan Song
Yichun Wang
Jingjing Gong
Hanlin Wu
Dongzhan Zhou
Hao Zhou
Wei-Ying Ma
Venue: NeurIPS 2025 poster ()
Keywords: 3D Molecule Generation
Fast Generation
Drug Design
Primary area: machine_learning_for_sciences
TL;DR: A novel method for fast 3D molecule generation

Abstract: Geometric molecule generative models have found expanding applications across various scientific domains, but their generation inefficiency has become a critical bottleneck. Through a systematic investigation of the generative trajectory, we discover a unique challenge for molecule geometric graph generation: generative models require determining the permutation order of atoms in the molecule before refining its atomic feature values. Based on this insight, we decompose the generation process into permutation phase and adjustment phase, and propose a geometric-informed prior and consistency parameter objective to accelerate each phase. Extensive experiments demonstrate that our approach achieves competitive performance with approximately 10 sampling steps, 7.5 × faster than previous state-of-the-art models and approximately 100 × faster than diffusion-based models, offering a significant step towards scalable molecular generation.

Accelerating 3D Molecule Generative Models with Trajectory Diagnosis
Zhilong Zhang 1.2-
Yuxuan Songl 1.3+ Yichun Wang** JingJing Gong'
Hanlin Wu
Dongzhan Zhou²
Hao Zhou & Wei-Ying Ma
Institute of AT Industry Research (AIR), Tsinghua University Qiuzhen College, Tsinghua University
3 Department of Department of Computer Science and Technology, Tsinghua University 4 Department of Industrial Engineering, Tsinghua University
5 Shanghai Artificial Intelligence Laboratory
Abstract
Geometric molecule generative models have found expanding applications across various scientific domains, but their generation inefficiency has become a critical bottleneck. Through a systematic investigation of the generative trajectory, we discover unique challenge for molecule geometric graph generation: generative models require determining the permutation order of atoms in the molecule be- fore refining its atomic feature values. Based on this insight, we decompose the generation process into permutation phase and adjustment phase, and propose a geometric-informed prior and consistency parameter objective to accelerate each phase. Extensive experiments demonstrate that our approach achieves competitive performance with approximately 10 sampling steps, 7.5 faster than previous state-of-the-an art models and approximately 100 x faster than diffusion- based mod- els, offering a significant step towards scalable molecular generation. Code is available at https://githuu.com/ wow.aaaoooooooeeeetTTtAAAAAAAAA
1 Introduction
Geometric generative models have achieved notable progress various important scientific tasks, including protein folding [1] de novo drug design [3,, and crystal generation [4]. Previous progress are largely driven by the adoption of advanced generative models, particularly diffusion models, which could refine cloud of atoms into precise molecular structures through iterative sampling. A notable example is AlphaFold33 which employs a diffusion- network to predict the joint structures of molecular complexes (I]
However, the iterative sampling process of these advanced geometric generative models can lead to inefficiencies in practice. For instance, the AlphaFold3-level structure prediction system requires around 200 diffusion steps, which significantly contributes to the heavy inference pipeline and usually 5 sampling procedures conducted with different seeds for single strueture prediction task to obtain comparable performance [5]. As application scenarios expand, generation inefficiency has become critical bottleneck, especially given the growing demand for scalable inference to produce large-scale synthetic data [6] and to integrate generative models into real-time scientific workflows in drug discovery 7 and material design
[4) Therefore, enhancing the efficiency of geometric generative models has emerged as a key research direction.
Recent efforts have adapted techniques from general domains such as image generation improve sampling efficiency in the molecular generation setting. However, even the most efficient methods
"qqual Contribution. Correspondence Hao Zhouthhouhao@@aitttisnuuuuuu
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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still require approximately a hundred or more steps for example, 90 steps in GOAT [ and 200 steps in EquiFM 9 In contrast, state-of -he--art image generation techniques only take as few 1-2 steps to achieve desirable results 10 11] This discrepancy raises two key questions: 1) What are the fundamental bottlenecks limiting the efficiency of 3D molecular generative models? 2) Can we achieve substantial improvement in efficiency, potentially reducing the required sampling steps by an order of magnitude?
The challenge of efficient molecule generation is fundamentally distinct from that of image generation due to the structural nature of the data. While images have fixed spatial ordering (e.g..ixeel positions remain constant), 3D molecular generative models that generate the structure holistically requires determining the permutation order of atoms before refining their atomic feature values. This permutation step is unique to 3D geometric generation and have not been adequately addressed by directly applying acceleration techniques designed for image domains.
In this paper, we first conduct a systematic and theoretical investigation of the generative trajectory to analyze this permutation component. Building on the insights from this analysis, we propose decomposing the geometric generation into two phases: the reordering of permutations and the adjustment of atomic features, and propose MOLTD (Molecule Trajectory Diagnosis) with novel acceleration methods for both phases. For the permutation phase, we introduce a geometric-informed prior the sampling process with a corresponding accuracy scheduler which could efficiently reduce the procedure of reordering; For the adjustment phase, we demonstrate that previous acceleration approaches from general domains, such as consistency training [0O, can be significantly beneficial for efficient generation when specifically adapted for geometric generation in the adjustment phase. To sum up, our paper makes the following contributions:
To analyze the intrinsic properties and underlying challenges in efficient geometric gener- ation, we propose a quantitative framework for analyzing the generative trajectory. This approach allows us to identify key considerations for developing improved methods while highlighting the fundamental differences between 3D molecular generation and general domains.
We further introduce effective modifications targeting different decomposed phases of the generation process. For the permutation phase, a geometric-informed prior is adopted to accelerate the generation of stable structures. For the adjustment phase, we propose consistency parameter objective to improve the accuracy of atomic feature adjustment.
We demonstrate the effectiveness of the proposed methods on two molecule datasets: QM9 [12] and GEOM-DRUG [13). For the first time, our approach enables the generation of large molecules using approximately 10 steps while reaching new state-of-the-arr performance on both datasets: on QM9, MOLTD achieves molecule stability of 16%, and on GEOM- DRUG it achieves Atom stability of 86.88% This represents significant improvement in generation efficiency: 7.5 x faster than flow matching models, 8.3 x faster than Bayesian Flow Network models, and nearly 100 ffater than cisttusion-bassee models.
We demonstrate the broad applicability of MOLTD by generalizing to different tasks and generative backbones. In the task of structure-based drug design, MOLTI achieves state-of- the-art performance using only 25 NFEs - a 40x speedup over diffusion- models. Ablation studies further confirm that our approach significantly accelerates both diffusion models and Bayesian Flow Networks.
2 Related Work
2.1 Efficient Generation in General Domains
Advancements have been made to enhance the sampling efficiency of generative models. For diffusion models, DDIM [4] introduced an adaptable diffusion process for faster inference, while DiffFlow [1] optimized the diffusion trajectory. Another line of works focuses on improving ODE solvers to reduce discretization error, for instance, DPM-Solver (16) introduced a high-order, training-free solver that reduces sampling steps to just 12. For flow matching models, recent efforts aim to improve the coupling hetween prior and target distributions. Coneurrent works [17 [8] approximate optimal
transport coupling to create more straight trajectory, while Rectified Flow [19 employs iterative
2

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Phase
Phase II
Geometrie-informedl Prior
Sample
Numbor of permi permutations
Align
Azom
Stable Point
El|0(- (0,,4) of (0
Intermediate time step
Consisteney Parameter
Figure 1: Illustration of MOLTD. To accelerate the generation in the first phase, we introduce a geometric-informed prior the sampling process based the aligned molecules in the dataset. For the second phase, we propose consistency parameter objective to accelerate the adjustment of atomic features.
distillation to straighten it. Most recently, Consistency Models [10 20
propose consistency training,
which enables one-step generation without sacrificing quality. While these advancements primarily focus on general domains like image generation, they have inspired improvements in geometric generative models.
2.2 Advancements in 3D Molecule Generative Models and Sampling Techniques
but there is growing interest in 3D molecule generation. Autoregressive methods like G-Schnet and GSphereNet iteratively connecting fragments to build molecules, but they require complex action design. Another approach models molecules as atomic density grids, generating densities over voxelized 3D
Extensive prior work has focused on generating molecules as 2D graphs 21 22 23
24 25
space [26].
Recent advances leverage diffusion models and flow matching for molecular generation 27 9
antibody design (28). and protein design (29] However, these methods typically require hundreds of sampling steps. To accelerate the generation, EquiFM (9) introduced optimal transport objectives with adaptive ODE solvers, GOAT [8] applied optimal transport in a joint latent space, and GeoLDM leveraged a latent space to reduce dimensionality. However, even most efficient methods still require 90-200 steps, whereas state-of-the-an thrtr--rtttttttt image generation achieves high quality in just 1-2
30
steps [10 1
3 Background
3.1 Notations and Definition
To differentiate between geometric representations and atomic property features, we represent 3D molecules using the tuple g = (x,, f). Here, x (x' (x',... x...,,V)) matrix, and (h! NN E R Nxd represents the node feature matrix, which includes attributes such as atomic types and charges. Here = {x E Nx3 N - x² = 0} is the Zero Center-of- Mass (Zero CoM) space, which means the average of the N elements should be O. For Bayesian Flow Networks, we use 02 to denote the parameter encoding the information of the molecule at time We use g to denote the data distribution and Pa as learned distribution.
..'....,,") E X' denotes the atomic coordinate
To analyze the generative trajectory, we introduce additional notations: E N N is the permutation of elements, and E N is rotation matrix in 3D space, V(æ;; r) is an open ball centered at with radius

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3.2 General Formulation of Generation Process
Most advanced geometric generative models employ iterative sampling processes to progressively transform noisy latent variables into valid molecule. This approach is exemplified by diffusion models (DMs), How-matching models (FMs), and Bayesian Flow Networks (BFNs) [2 611 and their
sampling process can be unified as follows:
gt a:9:-1 + (g
NE, E (1,2....,
(1)
where 9: is the noisy latent at time step t, T denotes the total number of sampling steps, C₁ is a standard Gaussain random vector, + represents the neural network, and az; Be, Ys are parameters to be instantiated according to specific generative model and sampling technique.
For diffusion-based models and Now-matching--asse based models 32 process aims to discretizing and solve the differential equation:
33
34
the iterative generation
dg: dt
Ug t).
which transports the prior distribution p(go) to the target distribution p(gr). Typically, the network in Equation (1 is trained to approximate the velocity term Vy by estimating conditioned version of the ground truth velocity. However, since discretization inevitably introduces errors into the sampling process, these models generally require large number of sampling steps to effectively approximate the learned differential equations and generate high-quality samples [35 16.
In contrast, Bayesian Flow Networks (BFNs) generate data without requiring discretization by performing Bayesian updates based on observed noisy random variables [36]. This approach provides unified framework for handling diverse data modalities, including continuous, discretized, and discrete data. Each modality is addressed with a specifically adapted Bayesian update rule, as outlined in Equation ( The network + is trained to reconstruct the ground truth molecules. In previous work [1I, effectively approximating the data distribution with a BFNs-based model required hundreds of sampling steps. We include more detailed introduction of BFNs in Appendix B
4 Method
In this section, we systematically and theoretically investigate the generative trajectory of geomet- ric generative models and introduce decomposition framework for 3D molecule generation in Section 4.1 Building on insights gained from this analysis, we introduce the geometric-informeed prior in Section 4.2 and the consistency parameter objective in Section 4.33 to accelerate the first and second phase, respectively. Proofs for the propositions presented this section are provided in the Appendix(]
4.1 Decomposition of the Generative Trajectory
We focus on advanced geometric generative models, including the diffusion- approach EDM (27] the flow- fow-matching. based approach EquiFM (9. and the Bayesian flow network- based approach GeoBFN [3.. These methods generate trajectory noisy molecules g 9T through iterative sampling described in Section 3.2] We quantify the change of molecular geometry along this trajectory to investigate the characteristics of the generation process.
To evaluate the changes in molecular geometry, we compare the intermediate molecules {g.) ) i=1 generated along the trajectory, against the final molecule gr obtained at the end of the trajectory. We define two metrics to evaluate the geometric changes: (g: measures structural changes at the molecular level, while D Drype (g:) captures changes in atom types at the atomic level:
Definition 4.1. (Metrics for the change of geometry)
The structural difference between g, and gr is defined as
|| l|o
D Dstuccure (g.)
(3)
N
where
m R argmin (R...) 27 ||2
(4)
R.R

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1.0
Number of Permutatons
15.0
0.9
Permutation
Non-permutation
12.5
0.8
10.0
point
0.7
Phase
Phase
0.6
7.5
0.5
x
0.4
0.3
EOuFFM
Boundary
0.0
0.22
0.2
Phaie
0.4
1.0
steps
The structural difference measured by Streeture. The Stable point Etabll represent the dividing point in time between the first generation phase and the second phase. Middle: The change of atom types measured by Dyye Right: Illustration of the two phases. In the trajectory of (az) = + + + (#): =ū)). the [=i,"3 lies in the second phase and is aligned with (ah at), while ( 22) lies in the first phase where permutation reordering is required.
Figure 2: Left:
Here 2; is the molecular structure, || performed by π/.
lu the number of permutation (i.e., transpositions)
And the difference of atom types is defined as
Deypo ΣE= 5(hg h2y
(5)
N
where d the Dirac delta function.
We randomly sample 1000 generative trajectories and report the mean of these metrics. More imple- mentation details about the investigation and additional experiments can be found Appendix C
The results of geometric changes along the generated trajectory are presented in the left and middle of Figure 2 We could observe the trajectory of generated molecules exhibits two-phase pattern across all analyzed generative models, and these two phases are delineated by the stable point: Definition 4.2. The stable point time step in the generative trajectory that distinguishes two phases:
fabbl argminngsaammm structure (g:), s.t. structure S
(6)
where e is a small positive number, e.g., 1e 3.
For Laable a; requires permutations to align with the final molecules, phase reordering of permutations. This process highlights a unique challenge in molecular generation, distinguishing from other generative tasks such image generation. In general image generation, intermediate samples inherently follow fixed permutation order consistent with the final output, i.c., T (xx;) where represents the identity map. In contrast, the geometric generative models must first resolve the correct permutation mapping for intermediate molecular structures.
the
Fort taable the molecular structure stabilizes, and the generative models refine atomic features including both atom coordinates and atom types to finalize the molecular geometry. In this phase, the molecule can be treated general data, as (ιx, I, allowing the application of advanced techniques from general domains to accelerate generation.
In fact, the existence of the two phases is theoretically justified follows: Theorem 4.3. Let a (abjrr), i (1.2,.... .N.. The radius is chosen as = mingz || 2ill where j E (1,2, ... .N} and + Then, argmin 7 ((x)) zolla INXN.
where n is an N x N permutation matrix, and INNN the identity matrix.
Theorem 4.3 establishes that if the two structures, .r. and af lie within the open ball of the corre- sponding target, as shown on the right side of Figure then no further permutation required. This explains why permutation for alignment is not needed in the second phase.
5

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For the first phase, because of the significant difference between the prior and the target, permutation inevitable the beginning of generation. The following theorem provides the conditions under which permutation occurs. Since all permutations can be decomposed into products of pairwise permutations, in the following theorem, we focus on pairwise permutations between two atoms:
Theorem 4.4. Assume zh & (= i E (1,2....,N) mutation between the i-th and j-th rows is required if and only if ary lies outside one sheet of the two-sheeted hyperboloid defined by the foci 2 and xiy. passing through z²z
For j E (1, N \ {i), a per-
We now apply the proposed tools to analyze the inefficiencies in advanced geometric generative models. Based on the results in the left and middle of Figure 2 we identify the permutation phase as a key bottleneck in the generation efficiency of current models. Both diffusion- based EDM and flow matching- baased based EquiFM require hundreds of sampling steps. and a large proportion of steps are spent in finding the stable structure in the first phase. In contrast, GeoBFN benefits from a smoother trajectory and the capability of unified modeling of different modalities, enabling it to reach stable point earlier than previous methods, and thus significantly reduce the necessary sampling steps. However, GeoBFN still experiences large number of permutations at the start of the first phase.
Building on this analysis, we gained valuable insights into accelerating each geometric generative model. In this study, we focus on optimizing GeoBFN, as it effectively disentangles the two phases of the generation process and exhibits the highest generation efficiency among the models analyzed. Although these methods are tailored for BFNs, we demonstrate that they can be readily generalized to other generative models, such as diffusion models, paving the way for efficient acceleration in large-scale molecular generation systems.
To address the challenges of each phase, we propose two techniques specifically designed for the first and second phases, respectively. These techniques are discussed in detail in the following sections. With the proposed methods, approach achieves earlier stable point, more accurate atom-type predictions, and largely reduces the redundant steps, as demonstrated in Figure 2
4.2 Accelerating the Bermutation-Roodderii Phase with Geometric-Informet Prior
In this section, we focus on accelerating the first phase of generation by leveraging the intrinsic structural information present in the molecular dataset. In drug discovery, is well recognized that molecules can be decomposed into stable and conserved substructures, such as scaffold-arm decom- position 3 and fragment-basse decomposition [38]. Inspired by this fact, we propose utilizing the "representative structures" within the molecule dataset that capture the general characteristics of most molecular structures, as an effective starting point for efficient generation of stable structure. To this end, we extract representative structural information from the dataset, summarize it into geometric-informed prior, and seamlessly incorporate it into the sampling process.
The extraction of representative structural information consists of two main steps:
In order to extract fine fine-grained geometric information, we stratify the molecules in the dataset based on their number nodes N, since molecules with different number of nodes
show distinct structural patterns [27] 39 support this claim.
We provide visualization in the Appendix G
We perform Equivariant Optimal Transport (EOT) [ to align the geometric structures within the stratified molecules. By eliminating variations due to rotation and translation while preserving the relative spatial configuration of atoms, tive structural features by simply taking the mean of all atomic features over the aligned molecules. The resulting representative structure is denoted as g-
could extract representa-
detailed description of the above process can be found in Algorithm
The framework of BFN provides an efficient approach to utilize g in the sampling process, without the need to train the model from scratch. First, g still lies in the sample space, utilize the Bayesian update distribution to map it to the parameter space:
0,
E ò(0,- h h ap))
Ps (v,
where 0, is the geometric-informed prior, Ps and are sender distribution and Bayesian update function, basic components of BFNs that are defined in Appendix B ap is the accuracy schedule,

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which determine the signal-to-noise ratio in the prior. We provide an ablation study on the choose of Op in Appendix F Finally, we perform standard Bayesian update in BFNs with 0, as the prior. Notably, the sampling process of MOLTD starts with pp. random vector embedding intrinsic structural information, in contrast to the fixed, uninformative prior used in the original BFNs.
Based on construction, the prior already specifies orientation in 3D, and thus Pa that starts from the prior 0g enjoys the rotational-equivarinnn equivariant property:
Proposition 4.5. The density induced by the geometric-informer prior rotationaleeuuivariant:
pp(g)op) = P4(Rg Rg Rtp),
(8)
where any orthogonal matrix.
4.3 Accelerating the Adjustment Phase with Consistency Parameter Objective
In the adjustment phase, the permutation orders of the generated molecules are fixed, allowing us to adapt state-of- the-art acceleration techniques from domains such as image generation.
While consistency training [TO] has shown remarkable success in accelerating generation tasks, primarily designed for continuous data like images. To address the challenges posed by the multi- modal nature of 3D molecules, we introduce consistency parameter objective-a novel objective function that enforces consistency in the parameter space of BFNs. Our key insight is that the continuous parameter space of BFNs provides a smooth and structured representation of multi-moda molecular information, which in turn enhances the stability and effectivvness of consistency training in this domain.
The proposed consistency parameter objective is formulated as follows (we highlight the differences with standard consistency training using color blue):
(g)
Σ
(,11((0r,,14)) + (0t 4
where Φ- is neural network with stop-gradient, laable is the stable point, ensuring that the consistency parameter objective is only computed in the second phase. And {4: J denotes discretized time steps based on pre-defined curriculum 20) More implementation details can be found in Appendix
Similar to standard consistency training. the goal of the consistency parameter objective is to accurately predict ground-trute molecule-but from the parameter space rather than the sample space. This objective is naturally aligned with the original training goal of BFNs, where the network & is optimized to perform accurate Bayesian updates by predicting the ground-truth molecule g:
Proposition 4.6. the original formulation BFNs, the objective is upper bounded by the estimation error of ground truth molecules:
E plg) DKr(p. (0l S E
d(g, ((,,))
(10)
((g):rr(0lg.
where d(.,.) is measure of differences that depends on the data modality.
This insight enabled us to jointly optimize consistency parameter objective as well as the original BFN objective to stabilize the training dynamics.
5 Experiments
In this section, we justify the advantages of MOLTD with comprehensive experiments. We first introduce experimental setup in Section 5.1 Then report and analyze the evaluation results in Section We also provide further ablation studies in Section 5.3 to investigate the effect several model designs. The ablation sampling steps is shown in the Appendix
5.1 Experiment Setup
Task and Datasets. Following the setting of prior works 24
41, we focus on molecular modeling and efficient generation, which measure the efficiency of the models to generate chemically
7

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Table 1: Results on QM9 and DRUG datasets, including NFE, atom stability, molecule stability, validity, and quality. Metries are calculated with 10000 samples generated from each model, we run the evaluation for 3 times and report the derivation. Compared with previous methods, MoLTD enables around 10x speed up while generate stable and valid molecules.
x (V&U%N). A higher number indicates better generation
QM9
DRUG
Metries
NFE
Atom (%)
Mol Sta (%)
Valid (%)
V&U&N (%)
Atom Sta (%)
Valid (%)
Data
99.0
95.2
97.7
86.5
99.9
ENF
85.0
4.9
40.2
G-Schnet
95.7
68.1
85.5
GDM-AUG
1000
97.6
71.6
90.4
66.8
77.1
91.8
EDM
1000
98.7
82.0
91.9
59.6
81.3
92.6
EDM-Bridge
1000
98.8
84.6
92.0
82.4
92.8
GcoLDM
1000
98.9
89.4
93.8
53.9
84.4
99.3
EquiFM
200
98.9
88.3
94.7
53.7
84.1
98.9
GeoBEN
100
87.2
GOAT
90
99.2
92.9
72.3
MOLTD
12
99.40 0.1
92.53
96.04
0.4
67.03
0.6
95.33
*Note that, for DRUG dataset, molecule stability and uniqueness metric are omitted since they are nearly 0% and 100% respectively for all methods, expeet that MoLTD achieves molecule stability with 6.37 %.
valid and structurally diverse molecules, and their capacity to learn molecular distribution. We evaluate benchmarks over two widely adopted datasets, including QM9 [12] and the GEOM-DRUG
QM9 is a standard dataset that contains 130k 3D molecules with a maximum of 29 atoms, while GEOM-DRUG is a more challenging dataset containing around 450K molecules, each with an
42
average 44 atoms and up to 181 atoms. And the data configurations directly follow previous works
43
30 31.
Evaluation Metries The evaluation configuration follows the prior works 22
After generating 10000 molecular geometries, the bond types are first predicted (single, double, triple, or none) based pair-wise atomic distance and atom types (27). To qualify the sampling efficiency, we report the number of function evaluations (NFE) utilized in sampling for each generative model. With the obtained molecular graph, we evaluate the quality by calculating both atom stability and molecule stability metrics. The validity (based on RDKit) is also reported, which is the percentage of valid molecules among all generated compounds. To comprehensively evaluate the capability of de novo molecule design 44]. we report validity x xunqqueness novelty (V%U%N), to quantify the percentage of valid, unique, and novel molecules among the generated samples. On QM9, we additionally evaluate how well the model learns the molecular distribution. We report the total variation distance of atom types, and the Wasserstein distance of the bond angles and bond lengths between generated molecules and test set. We also report the strain energy to qualify the overall structure of the generated structure.
41 30
Baselines We compare MOLTD to several competitive baseline models. G-Schnet (24] and Equiv- ariant Normalizing Flows (ENF) 400 previous equivariant generative models for molecules, built on autoregressive and flow- models respectively. Equivariant Graph Diffusion Models (EDM) with its oouiviuiva variant (GDM) [27] are based on diffusion models for molecule generation. EDM-Bridge [41) further boosts the performance of EDM with well-designed informative prior bridges. Furthermore. MOLTD is compared with recent advancements for efficient molecular generation. EquiFM 99] and GOAT 45 are flow-matching models with equivariant optimal transport objective. GeoBEN is the first work that leverages Bayesian flow networks for 3D molecular generation.
5.2 Main Results
The results of efficient molecular generation are presented in Table As shown, MOLTD establishes a new ofate-of-the-art in generating high-quality molecules with only 12 sampling steps on both QM9 and GEOM-DRUG datasets. Notably. MOLTD achieves 7.5 x faster generation than flow matching models, 8.3 x faster than Bayesian Flow Network models, and nearly 100 faster than diffusion-basee models, while maintaining superior generation quality. The actual runtime comparison Table shows that MOLTD's efficiency gain mirrors the improvements observed in the NFEs. Moreover, the results highlight MOLTD's ability to generate diverse and novel molecular

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Table 2: Ablation results on QM9 and DRUG datasets. P stands for geometric-informed Prior and C stands for Consistency parameter objective. Metrics include atom stability, molecule stability, and validity, A higher number indicates better generation quality, In all experiments, NFE is set to 12.
QM9
DRUG
# Components
Atom Sta (%)
Mol Sta (%)
Valid (%)
V&U&N (%)
Atom Sta (%)
Mol Sta (%)
Valid (%)
Data
99.0
95.2
97.7
86.5
99.9
and C
20.8
2.8
65.4
5.1
3.6
0.0
w/o P
87.54
48.51
64.10
54.06
69.57
5.42
94.20
w/o C
98.97
90.01
96.10
66.93
77.48
3.02
93.01
MOLTD
99.40 = 0.1
92.53 0.2
96.04 = 0.4
67.03 + 0.6
86.88
6.37
95,33
geometries, indicating that it is not over over-fitted specific subset of the training data. This strong generalization capability underscores the potential of the MoLTD for broad applications in molecular design and drug discovery.
Results on molecular distribution learning, presented in Table demonstrate that MOLTD achieves competitive superior performance across all evaluated metrics. These results highlight MOLTD'S capability for accurate distribution learning and fast sampling convergence.
5.3 Ablation Studies
Table Additional results on QM9, including NFE, atom type total variation, Wasserstein distance of bond length and bond angles, and strain energy. lower number indicates better generation quality.
First, we evaluate the effectiveness of the two proposed acceleration techniques. Results in Tableerdemonstraaee that, while each technique independently enhances the quality of few-step generation, nei- ther achieves state-of-the-art performance in isolation. This underscores the impor- tance of designing tailored methods for each phase of the generative process and integrating them to achieve optimal results.
Metrics
NFE
Atom TV
Length W1
Angles WI
S-Energy
Data
0.0
19.4
1000
25.8
GeoLDM
1000
0.9
1.4
9.2
26.0
EquiFM GeoBFN
200
1.5
0.6
9.2
25.1
100
1.3
1.4
25.3
MoLTD
12
1.2
1.0
8.1
24.9
*Results in the table are obtained by our own experiments.
Furthermore, we demonstrate the general applicability of proposed techniques two scenarios:
First, adapt both techniques EDM, a diffusion-based molecular generative model. Specifically, we inject rep- resentative structural information the final step of the forward process to construct an informed prior, and ini- tialize sampling from this prior. Since diffusion models lack an explicit parameter space, we apply consistency distillation on pre-trained EDM model to stabilize train- ing, following [10). As shown in Figure
Quality vs. NFE
MI-TD
112,
EDMinecoeed
Molocatle stability (94)
(30.
3 the adapted techniques significantly accelerate EDM just 30 NFEs while maintaining state-of-the-art stability. However, the acceleration remains limited-MoLTD achieves better quality with only 12 NFFs, owing to the advantage of the explicit parameter space BFNs for modeling molecular geometry.
Sou HM 2200 36.3)
11000,
Gevari (100, 87.7)
Eqarte
COM
ESNi expeaved
EDM
MoTD
[1003, 82.0
log INFEi
Secondly, we demonstrate the cross-data generalization ability of the Geometric-informer Prior. We constructed the geometric-prior using training data from GEOM- DRUG, and plugged it into the generative model trained on QM9-an a dataset with distinet structural patterns (smaller molecules, different atom types). Using 12 NFEs we gen- erated 10000 molecules and evaluated on QM9: Atom stability 99.2%. Molecule stability 90.22%, Validity 94.58%. As the results show, using prior
Figure 3: The acceleration effect of pro- posed method on BFN-based model and diffusion-based models. Note that the NFE is presented in log scale.

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Table 4: Comparison of structure- based drug design models. For Vina-based metrics, lower value indicate better performance. For others, larger value indicate better performance. MolTD achieves superior performance with significantly fewer NFEs.
Model
NFE
QED
Vina Score
Vina Min
Mean
Median
Mean
Median
TargetDiff
1000
0.48
0.58
-5.47
-6.30
-6.64
-6.83
Decomp-R R
1000
0.51
0.66
-5.19
-5.27
-6.03
-6.00
MoICRAFT
100
0.50
0.69
-6.59
-7.04
-7.27
-7.26
MoICRAFT*
25
0.51
0.65
-5.95
-6.70
-6.73
-6.89
MolTD
25
0.54
0.72
-6.60
-6.91
-7.36
-7.24
*Results the table ure oblained by our own experiments.
constructed from external dataset achieves the same accelerating effect as original MOLTD, demon- strating that the prior captures fundamental geometric structures shared across datasets.
Thirdly, we applied MOLTD structure-based drug design, integrating with MolCRAFT 3 state-of-the-arr method in this domain. The geometric-informed prior is constructed using Algorithm from the set of target ligands within the training set. Given that MoICRAFT also utilizes BFNs as its generative backbone. parameter space. We adhered to previously established evaluation settings to report our results, which is detailed in the Appendix B As the results show in Table 4 our method demonstrates comparable or superior performance with significantly fewer NFEs. This outcome underscores its considerable practical potential.
were able scamlessly apply consistency parameter objective to its
6 Conclusion
In this paper, we introduce MoLTD, a novel approach to accelerating 3D molecule generative models by addressing geometric generation challenges. Through theoretical and empirical analysis, we identify two-phase generative pattem--permnneee permutation reordering and atomic feature adjustment-and and propose two key techniques for accelerating each phase: geometric-infonne ordering and consistency parameter objective for accelerated adjustment. Extensive experiments demonstrate that MOLTD significantly improve sampling speed, achieving a speed-up of approxi- mately 8x compared to previous advancements, while maintaining state-of-the-art generation quality. Beyond improving efficiency, MoLTD offers new insights into geometric generative models, with applications in drug discovery, material design, and beyond.
prior for faster re-
Acknowledgments
The authors thank Keyue Qiu, Yanru Qu for the helpful discussions and proofreading of the paper, as well as the anonymous reviewers for reviewing the draft.
This work is supported by the Natural Science Foundation of China (Grant No. 62376133) and sponsored by Beijing Nova Program (20240484682) and the Wuxi Research Institute of Applied Technologies, Tsinghua University (2024200112))
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13

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A
Algorithms
For a better understanding of the whole procedure in construction the geometric-informed prior, we involve the detailed algorithms in Algorithm
Algorithm 1 Construction of Geometric-Informed Prior
Input: set of M molecules [goJne the same size, and accuracy level Ap.
Output: The ceometric-informee prior Op.
Initialize:
Set the reference molecule Jve 91, and initialize the aligned molecules list Mol [grd]] for i = to do
Compute the optimal permutation and rotation using the optimal transport (EOT) objective:
T R =
argmin|| (Rg.) 9rerll2
s,R
Append the aligned molecule to the list: Mol.append (R* (R)g:))
end for
Extract Information: Compute the average of the aligned molecules:
Mean(Mol)
Project to Parameter Space:
0p
E pe g.ap)
5(0ap
h(y,0,
B Basic Introduction of Bayesian Flow Networks
In this section, we introduce the key components of Bayesian Flow Networks (BFNs) (36 from the perspective of Bayesian inference. As standard Bayesian inference, a prior distribution P1 (0g) is specified for each data modality. For example, for continuous data, assume Gaussian prior where 0, represents the mean and variance, whereas for discrete data, we use a categorical distribution where On corresponds the probability of each category.
The sender distribution serves as the likelihood function in Bayesian inference, generating noisy observations y that iteratively update the prior parameters. These noisy signals are obtained by perturbing according to the following distribution:
4 ... Un (g)
II ps (v
(11)
where Ps. referred to as the sender distribution, is typically modeled as Gaussian. The parameters a; control the noise level, ensuring that the sequence (y1,... ,yn) exhibits an increasing signal-to-noise ratio.
These noisy signals are then used iteratively update the posterior distribution by modifying the parameter 6:
pr(g y, 0,, a:) = pr(gg 01+1)= - pr(g h1y, a)),
(12)
where the deterministic function h, known the Bayesian update function, governs the update rule:
0, - h10,011,1
1., as).
(13)
14

---

The explicit form of depends on the data modality and the choice of ps, as derived in [36]. With Equation (TT) and Equation (33). the distribution of the parameter can be formulated
qu (0, 0 0.-1. g. 04)) (a(2.00.1....)
5(0 ; h 0 0,)1,
(14)
As the sequence (y).... yn) provides increasing information about the posterior distribution is progressively refined, yielding a more accurate approximation the target distribution.
However, during generation, the target molecule g is unknown, meaning the true noisy observations required for Bayesian updates are inaccessible. To address this, BFNs adopt variational approach by training a neural network ( 1) to predict the noisy signal. Specifically, +(0) trained to approximate the ground truth sample via the ouiput distribution:
D II d=1
9 ~ po
Po(g (4) ((0)(a))
(15)
where Po is referred the outpul distribution, and the output of (0) lies in its parameter space. To generate predicted noisy signals ÿ. we marginalize over the predicted g:
ps (u (0))))) P
(16)
where PR. known as the receiver distribution, incorporates both the output distrihution and the sender distribution approximate the noisy signal.
BFNs is trained to approximate the distribution (y), ... yn). in order to acquire accurate signal for Bayesian updates. The variational lower bound is optimized:
log (g)) 2 By log Po y1,..., yn (p yn)
( Un g)
(17)
DrL. (gllp
(v). (y....., Vn))
E. y .r Marp [log (g yi..... yn.)
where (g yı. y/..... Vn) po (g)0 4). From this perspective, BFNs can be viewed latent variable model (y) ... yn) in the parameter space:
Pa (O.. ... 0n) II pu (0 0;)1,
(18)
E PR 0.)1.a.,0)
5(0: h( 0,1,,
could also be view as a latent variable model in the sample space:
Po (y1...., ynn = po II Pol ())=)1))
70
Pol Po(yi 0--1)
E Pola 1:4)
C Details of Investigation of Generative Trajectory
C.1 Implementation Details
In this section, we provide more details on the investigation of the generative trajectory. In the investigation, we utilize official checkpoints from the open-source repositories of the geometric
15

---

HOULD
15.0
GoolEN
Numbar of Poruustinn
Accurss st0m 53p2
Studii pour, Eqgure
10.0
StAdie pour. Centen
poin: MOOI
0.6
7.5-
0.4
5.0
25
0.21
0.24
EDM
Intermediate time
Intermediste time
Figure 4: Investigation on trajectory of predicted molecules. Left: D Satuuctree along the trajectory. Right: yypes along the trajectory. MOLTD achieve the earliest stable point among the baselines with accurate atom type prediction.
generative models, including EDM [27]. EquiFM [9]. and GeoBFN B1.. Each model requires different sampling steps achieve comparable performance, and thus we 1000 steps for EDM, 200 sampling for EquiFM with Euler discretization 46]. and 100 steps for GeoBFN. We randomly sample 1000 trajectories for each model and calculate the mean of Suuuuu and Drypes each time step.
C.2 Investigation on the Generative Trajectory of Predicted Molecules
Furthermore, we analysis the generative trajectory of predicted molecules. GeoBFN directly generate of sequence of predicted molecules in the sampling process, while EDM and EquiFM could predict the molecules by denoising the noisy sample. For EDM, the molecules time step are predicted by:
9o-
gt
Tt
(z(z))
(19)
at
at
where a₁ and o, are the noise schedules EDM, (z, 1) the predicted noise. For EquiFM, the noisy molecules generated by 9t (1 t)go fe, and the estimated velocity vo(ge, is trained to approximate -o + E. As a result, the predicted molecules at time step can be calculated by:
9: ltp
(20)
We conduct the same analysis as described the main body of the paper, with the results presented in Figure As shown, the trajectories of the predicted molecules exhibit similar pattern to those of the noisy samples analyzed earlier. These trajectories display a distinct two-phase structure, with MoLTD achieving the earliest stable point among the baselines while maintaining accurate atom type predictions.
C.3 Analyzing the Distinct Generative Pattern of Different Generative Models
EDM typically requires around 1000 sampling steps to generate valid molecules, with the majority of these steps spent in the first phase to achieve structural stability, shown in Figure This inefficiency can be attributed to the stochastic nature diffusion models, which necessitates numerous denoising steps to stabilize the structure.
EquiFM achieves minimal structural difference the start of generation by employing the EOT objective for training. However, our analysis reveals that EquiFM also spends large proportion of sampling steps in the first phase. This inefficiency can be attributed to the inaccurate prediction of atom types, as shown in the middle of Figure 2 which complicates the convergence to stable structures.
GeoBEN benefits from smoother, less noisy trajectory compared to EDM, which leads to faster convergence rate in the first phase. Additionally, its capability to handle multi-modal features in molecules enables accurate atom-type predictions. Consequently, GeoBFN reaches an earlier stable point compared with previous methods. However, GeoBFN experiences a large number of permutations at the start of the first phase, as its iterative sampling process begins with fixed, uninformative prior. Furthermore, with relatively long time span during the second phase, GeoBEN fails to apply acceleration techniques to further improve efficiency.
16

---

D Formal Proof of Theorems and Propositions
D.1 Proof of Theorem
Proof. Without loss of generality, assume that i and j = Consider a permutation operator no that swaps i and zī.
wo (i 1
such that To no(z)) = 2 and (=(2)) = z/. From the definition of r, we know that ø| & V( r) and of & V (irtir) r). Thus,
[xx((1) - zblla + || - zill2 || - zbll2 + ||= =1ll2 2r > || - zbllz + || x z0ll2
This shows that after the permutation, | zoll increases. Therefore,
argmin || [x(]) zoll2 = INNN.
D.2 Proof of Theorem 4.4
Proof. Without loss of generality, assume and j 2. Given the known condition, w2
r)) First, consider the 2D case, where z3 lies on the plane C₁ formed by 20, z and zz
The condition for permutation is
|| zblla + ||ا3 ril2 2212 < 11-2 z1ll2 + 11
x l12.
implies that the sum of the distances to the targets is smaller than the sum of the distances after the permutation. This equivalent
| zblle - ||ا - - < || - =bll2 - -110 =2112.
Further, this is equivalent to zi and 2/ lying outside one sheet of the hyperbola with foci at æj and 26. passing through 2/. Thus, have shown that the original statement holds for the 2D case.
When 22 is not on the plane C, we can rotate C1 around the line through zá and z as the axis, so plane C2. In this case, the distances from 222 and 2 a and x0
that ur2, 2:2, 20. and x lie remain unchanged, still satisfying the above equation. The hyperbola rotates along its imaginary axis, producing two-sheeted hyperholoid. Therefore, the original statement holds in 3D well.
D.3 Proof of Propasition 4.5
Proof. Based on the introduction of BFNs in Appendix B we could formulate the density function of generated molecules as:
(z( po p0(x 6p, 0 0
6 p (on (0
da)d0
ptn
pt 1:pt
-f ps(x (x 0 (ptn) II pu (0p+i (pti-lias) do ptiipn (Markov property) (21)
Note that (a 0 p+nn Po (Rx RØ + po (R(x) R(O (100t))) Φ) due to the property
of EGNN. Furthermore, based on the same argument used in Theorem 3.1 in [3I], we could
prove that po (0p+i 0pti-1: ap+i) satisfies the equivariant condition that pu (0p+i Opti-l:a;)
17

---

PU (R0p+: RØ, 1:0p+/11:000 1ppt-): Thus, we could prove pr(x)0p)) is rotational-equivariann
pp())(0p)= / Polar Op, 0 p+11"" 0 Optnlpo(08 p+1."
p+n O 0a)d0 p+1.pn
/ polar pptn II PC (0ptz 0pti-1; do"
(22)
p+ l:p+n
-f (o(R
RØ p
ptn
PU (R0p+) p+s Rd 1200-111:00 05) do
p+ppt 1:p+n
(23)
/ (Ræ
R
Pu (RO p+s RØ R0p-1-1;a:) [de p+1spt
(24)
ptn.
Po (Ræ) RRO,)
(25)
Here we used the change of variable formula and the fact that [det(R) 1, as R is an rotation matrix.
For completeness, we include the derivation of Py (0, 0;-1;0;) - Pu (RO, R0,-;;a) from
31: Recall that po (0: (.-1:a5) po(y1(0-1:a.)
E
5 (0. h (0,-1.y,, then have:
Pr (RO, RO R0:-1;0;)- po RO; 1801
8 (RO: - 11 (RO (R0
= - po (y: R0.-1;a)) 8 (RO; h v., dy:
(26)
Then we apply subeiitio and replace the variable y, with new variable v/, i.e. y: = Ry', into the Eq. 26
/ po (y: R0;-1;;0) (RO, - h (RO;- (R0. dy:
/ po (Ry' R0:-1:a;) 5 (RO, h (R0 Ry/, ax)) dRy;
/ po (Ry' R0 8 [RO, (RO, Ry,, ax))
(27)
The rotation matrix R is a SO(3) matrix, thus the det R)] = And for the continuous coordinate variable, the update function h for continuous data [36] is also equivariant:
h (RO,- Ry. o.) R0:-1P:-11 + Ry.a;
Rh (0:-1,99;(s))
(28)
Pi
Putting these conditions back to the Eq. 27 we have that
(RO: RR0-1;30) / po (y: R0,-;;a)) (RO, - (RO, as)) dy:
f po (Ry/ RR; 8 (RO: Rhh (0,-1, 8/. ax)) [d)
= f po (v( 0:-1:0s) (0, - h (0, y/ 0z)) dyf
pe (0; 00-1;00)
(29)
D.4 Proof of Proposition 4.6
Proof. We only need to show the conditioned inequality:
bxxlps(slg.t)llpxxyl... t)) S d(g, (0,
(30)
18

---

Table 5: Ablation results of different sampling steps QM9 and DRUG datasets. Metrics include NFE, atom stability, molecule stability, validity, and significance. higher number indicates a better generation quality.
QM9
DRUG
Metrics
Atom Sta (%))
Mol (%)
Valid (%)
V&U&N (%)
Atom (%)
(9)
Valid (%)
Data
99.0
95.2
97.7
86.5
99.9
98.6
87.2
93.0
78.9
93.1
GooBFN
98.8
88.4
93.4
62.1
81.4
93.5
1000
99.1
90.9
95.3
61.7
85.6
92.08
98.88
86.90
93.24
66.82
80.16
2.50
92.28
10
99.18
89.71
94.68
67.57
83.17
3,55
93.81
MOLTD
99.29
91.20
95.02
67.33
85,40
4.98
94.77
99.40
92.56
96.04
67.03
86.88
95.33
13
99.52
93.49
96.49
64.77
87.95
8.13
96.06
For continuous modality, ps(ylg, 6) Nyyl a,T) and pr((y ((yl t), az f),a;)) Since the KI. divergence between two Gaussains with the same variance is the square norm of their means, we have:
Del(ps(1) () (y)lppyyyl...... ()) &
(),
(31)
For discrete data (atom types) discretised data (number of charges), Ps and pn are mixture of Gaussain distributions, based on the formulation of BFNs [3).1]] data and the proof can be easily generalized to discretised data. For discrete data,
31]. Here focus on the discrete
Ps (KyaayKKe, 1), a,K))
(32)
where Eg is the one-hot encode of g and is the total number of classes. And
k
Σ
Φ O' (0.0W(ylas - 11),
(33)
where the कF is the ith component of + which determines the weights for each gaussian, and et is the one-hot encode of class We use p as a simplification of Vylas(Ke 1), KTK. Now have:
(x( dx Σ (ip'(r)
DRI(ps(vlg ()l
ps((x) log
(34)
ps(x) log(ps(:))drr (s())) log
p'p'(x))da
(35)
1=1
< ps((1) log(ps()))s
Σ Φb
ps((x) log(p' log(r'xx))de
(Jensen's Inequalily)
(36)
K
- /ps(O) log
(2(x)
dx
(37)
p¹
- C. E e e
(38)
where (1) we again used the fact that the KL divergence between two Gaussains with the same variance is the square norm of their means, C, is constant depend t. As result, the KL divergence is bounded by the probability of generating the wrong class Ei ob.
E Implementation Details
implement the Bayesian Flow Network using EGNNS 47 within the PyTorch framework 448 The latent invariant feature dimension & is set to I for QM9 and 2 for DRUG, significantly reducing
19

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the atomic feature dimensionality. Following the implementations of GeoBFN [3I, we only take atom charges as atomic features. For training the parameter network we configure EGNNs with 9 layers and 256 hidden features for QM9, and 6 layers with 256 hidden features for DRUG, both trained with a batch size of 64. The model employs SiLU activations and trained until convergence. Across all experiments, we adopt the Adam optimizer [4 with a fixed learning rate of 10-1 as the default training configuration. The training process requires approximately 2000 epochs for QM9 and 20 epochs for DRUG using RTX 3090.
We provide more details on the implementation of proposed techniques. Regarding the geometric- informed prior, we randomly sample 10 molecules for each molecular size to construct the prior. We use the accuracy level at 0.85 in all evaluation, which achieves the highest uniqueness of 97% and molecule stability of 90.0%. Regarding the consistency parameter objective, we use exponentially increasing curriculum. QM9, the discretization number of the time span N starts with 100 and is doubled every 80 training epochs. In GEOM-DRUG, N starts with 100 and is doubled every 40000 training iterations. We enforce consistency of the predicted mean for atoms coordinates and charges, and the predicted probability for atom types. As the proposed consistency parameter objective could significantly accelerate the adjustment of atomic features, we perform early stop at the second phase of sampling. Empirically, we found that the early-stop point for both QM9 and --DM- datasets could be chosen within (0.6, 0.8] to achieve balance between sample quality and efficiency. To further improve the sample quality of few-step generation, we employed the noise reduced sampling method propose in
0.94
0.96
0.9
Molecule Stability
0.90
0.90
Uniqueness
0.88
0.7
0.86
0.6
0.84
0.5
Accuracy level of the Prior
Figure 5: Ablation study on the accuracy level of geometric-infee inform prior.
For structure-based drug design, follow previously established evaluation protocol.
We use the CrossDocked dataset for training and testing, which originally contains 22.5 million protein-ligand pairs. and after the RMSD-based filtering and 30% sequence identity split by Luo et al (50), results in 100,000 training pairs and 100 test proteins. For each test protein, we sample 100 molecules for evaluation.
For baselines, we consider TargetDiff 2 DecompDiff and Molcraft which are state-of-the the-ttherrtta art methods in the fields.
Evaluation metrics include Vina Score, direct score of generated pose, Vina Min, which scores the optimized pose after local minimization of energy. Drug-likelines (QED) and synthetic accessibility (SA) ligand conformation. Sampling efficiency is evaluated by NFEs.
F
Ablation Study
In this section, we present additional experimental results on QM9 and GEOM-DRUG to evaluate the impact of the number sampling steps. As shown Table increasing the number of sampling steps enhances molecular stability and validity. Notably, with only 13 NFEs, MoLTD achieves
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record-breaking atom stability of 87.95% and a molecule stability of 8.13% on GEOM-DRUG, Furthermore, we did not observe significant decline the diversity or novelty of the generated molecules as the number of sampling steps increased.
Furthermore, investigate the impact of varying accuracy levels in the geometric-informed prior, which determine the starting point of the generation. We focus on molecule stability and the uniqueness of generated molecules in the QM9 dataset, presented in Figure 3 Since the prior derived from ground truth molecules within the dataset, higher accuracy improves molecular stability, However, the uniqueness metric follows a concave trend with respect to accuracy level. At lower accuracy levels, the prior introduces representative structural information and degree of randomness into the initial stages of generation, improving both stability and diversity compared the fixed prior commonly used in GeoBFN [3T] Conversely, at higher accuracy levels, the prior may lead MOLTD to collapse onto subset of molecules from which was constructed. As shown in Figure [ an accuracy level within 8 ~ 18 strikes an optimal balance, achieving uniqueness above 90% and stability around 90%.
In Table 6, we evaluate the efficiency gain in terms of actual runtime. We calculate the actual runtime required generate 1000 samples on the QM9 dataset, using a single RTX 3090 GPU with batch size of 64:
Table 6: Comparison of models based on NFEs and Time.
Model
NFEs
Time (seconds)
MOLTD
12
4.86
GcoBFN
100
34.52
EquiFM
200
180
EDM
1000
760
As the results demonstrate, MOLTD achieves sampling speed over 100x faster than diffusion- based models. Furthermore, the practical speed-up even greater than what is reflected by the reduction in NFEs alone, as MolTD benefit from more efficient implementation.
G Visualization
We provide T-SNE visualizations of the representations produced by Frad [52 on QM9 in Figure From the figure, it is evident that molecules with different sizes often have distinct modes in structures, which is reflected in their geometric representations learned by modern geometric encoders.
H Further discussion on the geometric-informed prior
In this section, we provide discussion and analysis on the effect of geometric-informed prior.
We analysis the effect of distribution of molecule size. The geometric-informed priors are tailored to the number of atoms (N) in molecule, and the number of molecules available for each size varies within different molecule datasets. And thus, with enlarged sample size, two factors contribute to improved generation quality:
A structural prior derived from more extensive data better captures the full spectrum of geometric features.
A Better Prior:
A Better Generative Model: More training data yields a more robust generative model.
However, we highlight that our method could boost the generation efficiency, even with only 10 samples to construct prior. We randomly sample 10 molecules for each to create prior and generated 10000 molecules on QM9. As shown in Table 7 10-sample prior provide similar acceleration effect as prior constructed form all training data. This result demonstrates the substantial practical value of the geometric-informed prior,
data- scarce settings.
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t-SNE Visualization
75
25
-20
15
-25
10
-50
-75
5
-100
-75
-50
-25
25
50
75
100
Figure 6: T-SNE visualizations of the representations produced by Frad. It evident that molecules with different sizes often have distinct modes in structures, which is reflected in their geometric representations learned by modern geometric encoders.
Table 7: MOLTD using only samples in the training set construct geometric-informed prior
Method
NFE
Atom Stability
Mol Stability
Validity
GcoBEN
100
98.6
87.2
93.0
EquiFM
200
98.9
88.3
94.7
MolTD
12
99.4
92.53
96.04
MolTD (Prior using 10 samples)
12
99.1
90.1
95.3
Limitation and Impact Statement
Although demonstrate the effectiveness of our algorithm across datasets at different scale with comprehensive ablation studies, our focus is limited to the class of generative models based on iterative denoising processes. We do not consider auto-regressive VAE-based models in this work. However, incorporating components from these architectures could potentially enhance the flexibility and performance of our approach, and leave this promising direction for future research.
In this work, we propose MOLTD, a pioneering framework that achieves effective molecule generation in approximately 10 sampling steps, 7.5x faster than previous state-of-the-ar methods. By reducing the sampling steps by an order of magnitude, our method paves a new way for scalable molecule generation, which may significantly enhance practical feasibility for real-world molecular design and drug discovery applications.
Furthermore, our systematic analysis of generative trajectories reveals fundamental limitations in conventional geometric diffusion paradigms, which drive our formulation of phase-decoupled generation framework. We believe the insights gained from our exploration will inspire more researchers and bring more powerful methods this field.
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Reviews:
[{"review_id": "qY1n5n0UOb", "reviewer": "Reviewer_Yv3A", "summary": "In this paper, the authors introduce MOLTD, a novel approach to accelerating 3D molecule generative models by addressing geometric generation challenges. Through theoretical and empirical analysis, they identify a two-phase generative pattern—permutation reordering and atomic feature adjustment—and propose two key techniques for accelerating each phase: a geometric-informed prior for faster re-ordering and a consistency parameter objective for accelerated adjustment.", "questions": "1. Can the findings and the generative framework proposed in this paper be extended to protein structure generation or Structure-based Drug Design?\n2. The Permutation stage and Refinement stage is similar to the graph structure generation and graph feature generation in Graph Generation Domain. Could you please provide some insights and open discussions on this interesting connection?\n3. The model architecture and loss could be SE(3) and Permutation-invariant, how about incorporating Permutation invariant loss that aligns molecule structure first?\n4. The geometric prior is connected with fragment-based molecular generation, how MOLTD perform against these fragment-based molecular generation methods? will the structure prior degrade the diversity of generated samples? As the diversity is important for exploring in the vast molecular space.", "limitations": "Yes, the authors have discussed the limitations in the Appendix.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strength\n\n1. The idea of Trajectory Diagnosis is interesting, as it also reflects, to some extent, how the molecular generative model understands the process of molecular generation. Moreover, the authors propose targeted improvements for both stages of the generation process.\n2. The authors propose a quantitative framework for analyzing the generative trajectory. This approach allows researchers to identify key considerations for developing improved methods while highlighting the fundamental differences between 3D molecular generation and general domains.\n3. The speed-up of molecular generation is impressive\n\n### Weakness\n\n1. Including more comparisons with relevant acceleration methods from the graph generation field would make the paper more comprehensive, especially considering that molecular generation is naturally more closely related to graph generation tasks.\n2. The term ‘Permutation’ is not that straight forward. (Graph) geometry or structure stage maybe more clear, as actually the ‘permutation’ comes with the structure alignment. And in the view of graph structure generation, maybe more methods in protein structure generation and graph structure generation could be discussed.", "was_revised": true, "final_justification": "This paper provides a two stage acceleration pipeline for various generative models in SBDD. I think this paper is above the bar in the specific domain but fails to provide more insights in ML sides. But the components and insights the authors propose are not new in generative models and even in another related application domains such as SBDD, protein and Graph Generation.  In general, I recommend acceptance for this paper.", "rebuttal": "## W1: Comparison with acceleration methods from graph generation\nWe thank the reviewer for this insightful suggestion regarding comparisons with graph generation acceleration methods.\n\nWhile related, a key distinction makes direct comparison challenging: 2D graph generation primarily models **discrete states** (e.g., the adjacency matrix and node types), whereas 3D molecule generation must model **continuous states** (i.e., the 3D atomic coordinates), which adds a significant layer of complexity. This fundamental difference in data modalities means that acceleration techniques are often not directly transferable.\n\nNonetheless, we agree that a comparison is valuable. We evaluated our method against two state-of-the-art 2D graph generation models known for their efficiency:\n- GraphBFN [1] : Using probabilistic interpolation for acceleration of graph generation\n- DruM [2]: Utilizes a mixture of diffusion processes to model graph topology.\n\n| Model | NFE  | Atom Stability on QM9 | Molecule Stability on QM9 | Atom Stability on DRUG | Molecule Stability on DRUG |\n|-|-|-|-|-|-|\n| GraphBFN | 100  | 99.4 | 94.7 | -  | -  |\n| DruM     | 1000 | 98.8   | 87.3  | 83.0  | 0.51  |\n| **MolTD**    | 12   | 99.4   | 92.5    | 86.9   | 6.37  |\n\nThe results show that while 2D methods perform well on stability metrics—a natural result of focusing on the simpler, discrete task of graph generation—our method (MolTD) achieves comparable stability with a drastically lower NFE (12 vs. 100-1000), highlighting its high efficiency.\n\n## W2: Terminology choice of 'Permutation' stage\nWe thank the reviewer of this useful comment! We agree that the 'permutation stage' could be ambiguous in the context of our full pipeline, and 'geometry alignment stage' is more consice. We will update it in the next version.\n\n## Q1: Application on Structure-based Drug Design\nThanks for the insightful comments! We have applied our method to structure-based drug design, integrating it with MolCRAFT [3], a state-of-the-art method in this domain. \n\nThe geometric-informed prior is constructed using Algorithm 1 from the set of target ligands within the training set. Given that MolCRAFT also utilizes BFNs as its generative backbone, we were able to seamlessly apply our consistency parameter objective to its parameter space. We adhered to MolCRAFT's established evaluation settings to report our results:\n| Model      | NFE  | QED  | SA   | Vina Score (mean) | Vina Score (median) | Vina Min (mean) | Vina Min (median) |\n|-|-|-|-|-|-|-|-|\n| TargetDiff | 1000 | 0.48 | 0.58 | -5.47 | -6.30  | -6.64 | -6.83 |\n| Decomp-R   | 1000 | 0.51 | 0.66 | -5.19 | -5.27 | -6.03 | -6.00 |\n| MolCRAFT   | 100  | 0.50 | 0.69 | -6.59 | **-7.04** | -7.27 | -7.26 |\n| MolCRAFT  | 25   | 0.51 | 0.65 | -5.95 | -6.70 | -6.73 | -6.89 |\n| **MolTD**      | **25**   | **0.54** | **0.72** | **-6.60** | -6.91 | **-7.36** | **-7.24** |\n\nFor Vina-based metrics, lower value indicate better performance. For others, larger value indicate better performance. Our method demonstrates comparable or superior performance with significantly fewer NFEs. This outcome underscores its considerable practical value.\n\n## Q2 & W3: Relation to protein structure generation\nThanks your for the insightful question! AlphaFold3, the state-of-the-art model for protein structure prediction, utilized the diffusion models to generate spatial coordinates of proteins. Our ablation study in Figure 3 demonstrates that our method can be **successfully applied to diffusion-based models**, yielding significant acceleration. Thus, we believe the protein structure generation could be similarly decomposed as permutation phase and adjustment phase, and applying our method with proper modification **will boost the generation efficiency**. Furthermore, we anticipate the permutation phase would be less computationally intensive for proteins, as the fixed amino acid sequence already provides strong conditional information that constrains the possible ordering of nodes.\n\nWe are actively exploring this promising direction as part of our ongoing research.\n\n## Q3: Discussion on the relation between permutation stage and refinement stage to graph structure generation and graph feature generation\n\nWe thank the reviewer for this insightful question, which highlights an interesting analogy between our method and graph generation:\n- Our **Permutation Phase** is analogous to generating the **2D graph topology** (e.g., an adjacency matrix), as it establishes the fundamental geometrical structure.\n- Our **Refinement Phase** is then similar to the **2D-to-3D \"lifting\"** step, where 2D topology is mapped to 3D geometric space.\n\nHowever, we wish to clarify a critical distinction: In 3D molecular generation models, we do not explicitly model edges or an adjacency matrix. Instead, the final molecular topology is **implicitly defined** by the geometric configuration of the generated atoms. While in graph generation, the topology is directly generated as the adjacency matrix. This distinction may suggest why specialized methods, such as those we introduce for the permutation and adjustment phases, may be more beneficial for advancing 3D molecule generation compared with general graph generation.\n\n## Q4: Using permutation invariant loss\nWe thank the reviewer for this insightful suggestion. The suggested approach, which involves aligning molecular structures before applying a permutation-invariant objective, is a valid strategy similar to methods like **EquiFM**.\n\nIn our preliminary experiments, we explored this exact direction. We used Equivariant Optimal Transport (EOT) to align molecular structures first, with the goal of simplifying the subsequent generation task. However, we found this approach to be **less effective than our current method**. \n\nThe primary issue was that pre-aligning the spatial features made it significantly harder for the model to predict the **discrete atom types**. As a result, the generation process became less efficient, nullifying the potential acceleration benefits. This performance trade-off is empirically demonstrated in Figure 2 of our manuscript, represented by EquiFM.\n\nConsequently, we opted for our current architecture, which we found addresses the challenges of both discrete and continuous generation more effectively, leading to a more robust and efficient model.\n\n## Q5. Comparison with fragment-based molecular generation\nThis is a great question! Adapting fragment-based methods to 3D generation is a **challenge due to the combinatorial complexity** of assembling fragments in 3D space. Consequently, few models have demonstrated high performance in this area.\n\nFor our comparison, we selected two representative state-of-the-art models:\n- Symphony: An auto-regressive model with E(3)-equivariance that iteratively adds molecular fragments using a spherical harmonics framework.[4]\n- HierDiff: A hierarchical diffusion model that generates a coarse fragment-based structure before refining it to a full atomic representation.[5]\n\nWe recognize that these models were developed for distinct tasks and were originally evaluated with different metrics. Symphony is primirly evaluated for validity and statistical difference between training data and generated molecules, while HierDiff is foused on drug-likeness. To ensure a fair and direct comparison, we report the **Validity and Uniqueness**, which are standard metrics consistently used across all compared methods. \n\n|     Model     | Type            | NFE  | Validity % on QM9 | Validity & Uniqueness on QM9 | Validity % on DRUG | Validity & Uniqueness on DRUG |\n|-|-|-|--|-|-|-|\n| Symphony | Auto-regressive | -| 68.1  | 66.5  | - | -  |\n| HierDiff | Diffusion  | 1000 | 87.8  | 85.9  | 94.0 | 94.0   |\n| **MolTD**    | BFN   | 12  | 96.04| 93.2   | 95.33 | 94.5 |\n\nAs the results show, our methods achieve superior generation quality and efficiency. \n\n## Q6: Impact of geometric-informed prior on diversity\nWe thank the reviewer for raising this important point regarding sample diversity. \n\nYou are correct that a structural prior can introduce a trade-off between sample quality and diversity. Our method is designed to make this trade-off **fully controllable**.\n\nAs shown in our ablation study in Figure 5, the influence of the prior can be precisely modulated. By applying a prior with lower accuracy level (or equivalently, inject the prior at the first few steps in generation), the model retains significant freedom to explore diverse chemical space while still benefiting from prior information.\n\nCrucially, our primary results demonstrate that this control does not hinder the model's ability to generate a diverse set of molecules. As reported in Table 1, our method achieves Validity, Uniqueness, and Novelty scores (67.03 combined) that are comparable to state-of-the-art models. Furthemore, we compare the **Validity & Uniqueness** on QM9 in the following:\n\n| Model       | NFE  | V&U  |\n|-|-|-|\n| EDM    | 1000 | 90.7 |\n| GeoBFN | 100  | 91.5 |\n| EquiFM | 200  | 93.5 |\n| GOAT   | 90   | 91.9 |\n| **MolTD**  | 12   | 93.1 |\n\nThis confirms that our approach successfully explores the vast molecular space to generate valid and diverse structures.\n\n## Reference \n[1] Song, Yuxuan, et al. \"Smooth Probabilistic Interpolation Benefits Generative Modeling for Discrete Graphs.\" ICML 2025\n\n[2] Jo, Jaehyeong, Dongki Kim, and Sung Ju Hwang. \"Graph generation with destination-predicting diffusion mixture.\" (2023).\n\n[3] Qu, Yanru, et al. \"MolCRAFT: Structure-Based Drug Design in Continuous Parameter Space.\" ICML 2024\n\n[4] Daigavane, Ameya, et al. \"Symphony: Symmetry-Equivariant Point-Centered Spherical Harmonics for 3D Molecule Generation.\" ICLR 2023\n\n[5] Qiang, Bo, et al. \"Coarse-to-fine: a hierarchical diffusion model for molecule generation in 3d.\" ICML 2023"}, {"review_id": "Kc7Wc4h94C", "reviewer": "Reviewer_dJLY", "summary": "Proposes a 3D molecule generation model that can generate molecules in a less number of sampling steps compared to existing flow-matching and diffusion-based models. The generation process split into permutation phase and adjustment phase. A geometric-informed prior is introduced to reduce the number of permutations. The adjustment phase is made efficient using consistency training.", "questions": "- Does a geometric-informed prior calculated for one dataset differ significantly from the prior for same-sized molecules obtained from a different dataset? If so, how would you transform the geometric-informed prior to another dataset?\n- Typo in line 44: analysis -> analyze", "limitations": "yes", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "### Strengths\n- This is a well-written paper with a clear flow. The method is well-motivated and introduced in an easy to understand manner.\n- Achieves better performance over existing 3D molecule generation methods such as EquiFM and GOAT in a fewer number of sampling steps.\n### Weaknesses\nThe weaknesses I noticed in this paper are very minor and do not affect the contributions.\n- The relative speed of MolTD is expressed as a proportion of sampling steps (i.e., 100 x faster than diffusion-based models). Do these speedups reflect in wall clock time?\n- The geometric-informed prior is separately calculated for each N number of atoms. How does the distribution of molecule size affect the quality of the generated molecules?", "was_revised": true, "final_justification": "I confirm my final review score.", "rebuttal": "## W1: Wall clock time comparison\nWe appreciate the opportunity to clarify the computational efficiency of our model. Apart from the fewer sampling steps, the **efficiency advantages in wall-clock time are more significant**. The following table presents the wall-clock times required to generate 1000 samples on the QM9 dataset using various generative models, all performed on a single RTX 3090 GPU with a batch size of 64:\n| Model  | Number of Function Evaluations (NFEs) | Wall-Clock Time (seconds) |\n|--------|---------------------------------------|---------------------------|\n| **MolTD**  |                                    12 |                      4.86 |\n| GeoBFN |                                   100 |                     34.52 |\n| EquiFM |                                   200 |                       180 |\n| EDM    |                                  1000 |                       760 |\n\nAs the results demonstrate, MolTD achieves a sampling speed over **100x** faster than diffusion-based models like EDM and **20x** faster than the flow-matching model EquiFM. Furthermore, the practical speed-up is even greater than what is reflected by the reduction in NFEs alone, as MolTD benefit from a more efficient implementation.\n\n## W2: How does the distribution of molecule size affect the quality of the generated molecules?\nThanks for the insightful question! Our approach leverages geometric-informed priors tailored to the number of atoms (N) in a molecule, since molecules with different numbers of atoms show **distinct structural patterns** (Figure 6 in appendix). This approach allows us to extract fine-grained geometric information within each cluster of molecular structures.\n\nOur method was evaluated on the public QM9 and DRUG datasets, following the established settings of previous work [1,2]. The number of molecules available for each size varies within these datasets. Two factors contribute to improved generation quality:\n- **A Better Prior**: A structural prior derived from more extensive data better captures the full spectrum of geometric features.\n- **A Better Generative Model**: More training data yields a more robust generative model.\nTo empirically validate this, we conducted an experiment on the QM9 dataset. For molecules with 12 and 19 atoms respectively, we generated 1000 samples and evaluated the results:\n\n| Number of atoms            | 12   | 19    |\n|----------------------------|------|-------|\n| Number of molecules in QM9 | 807  | 13364 |\n| Atom stability             | 98.8 | 99.6  |\n\nAs the results illustrate, molecule sizes corresponding to a higher distribution of training data exhibit demonstrably better generation quality compared to those with lower distribution. \n\nHowever, we highlight that our method could boost the generation efficiency, even with **only 10 samples** to construct the prior. We randomly sample 10 molecules for each N to create prior and generated 10000 molecules on QM9:\n\n|                                | NFE | Atom Stability | Mol Stability | Validity |\n|--------------------------------|-----|----------------|---------------|----------|\n| GeoBFN                         | 100 | 98.6           | 87.2          | 93.0     |\n| EquiFM                         | 200 | 98.9           | 88.3          | 94.7     |\n| **MolTD**                          | 12  | 99.4           | 92.53         | 96.04    |\n| **MolTD (Prior constructed from 10 samples)** | 12  | 99.1           | 90.1          | 95.3     |\n\nThis result demonstrates the substantial practical value of the geometric-informed prior, even in data-scarce settings.\n\n## Q1: How to transform prior to another dataset, the difference of prior constructed from different dataset and its impact on generation quality\nWe thank the reviewer for this insightful question. Our method utilizes distinct structural priors for each molecule size, derived from the DRUG dataset, and applies them to the QM9-trained model during inference. The effectiveness of this cross-dataset prior depends on the molecule size (N).\n\n**For small N**, the space of valid molecular geometries is relatively constrained. In this scenario, priors derived from different datasets (QM9 vs. DRUG) are quite similar, and we observe only a marginal performance difference when applying the DRUG prior.\n\n**For large N**, however, the geometric configuration space becomes vast and complex. The DRUG dataset provides significantly more data for these larger sizes, leading to a much richer and more informative prior. A prior constructed from this much larger and more diverse set captures a broader spectrum of valid geometric features, leading to a notable improvement in generation quality. \n\nTo empirically validate this, we conducted an experiment for larger molecules (N = 25 to 29), generating 1000 samples for each size and comparing the results when using priors derived from QM9 versus those from DRUG.\n\n| Number of Atoms                         | 29    | 28    | 27    | 26    | 25    |\n|-----------------------------------------|-------|-------|-------|-------|-------|\n| Number of molecules in QM9              | 25    | 0     | 266   | 48    | 1506  |\n| Atom stability of MolTD                 | 96.4  | 98.5  | 98.8  | 99.0  | 99.4  |\n| Number of molecules DRUG                | 56697 | 43809 | 31647 | 25672 | 17419 |\n| Atom stability of MolTD with **DRUG prior** | 98.3  | 98.8  | 99.2  | 99.2  | 99.4  |\n\nThese results demonstrate general applicability of geometric-prior, which could utilize external dataset to improve generation quality.\n\n## Q2: Typo in line 44: analysis -> analyze\nThanks for your careful assessment and for catching this typo! We will correct it in the revised manuscript.\n## Reference:\n[1] Hoogeboom, Emiel, et al. \"Equivariant diffusion for molecule generation in 3d.\" ICML 2022\n\n[2] Xu, Minkai, et al. \"Geometric latent diffusion models for 3d molecule generation.\" ICML 2023"}, {"review_id": "Y0K1NLPz60", "reviewer": "Reviewer_s7Zq", "summary": "This paper addresses the problem of 3D molecular generation. The authors empirically demonstrate that diffusion- and flow-based generative models tend to establish atom permutations early in the sampling trajectories. To analyze this phenomenon, they decompose the generation process into two distinct phases: (1) permutation reordering, which finds a coarse-grained 3D configuration, and (2) atom feature refinement. To accelerate the first phase, the authors propose a geometry-informed prior that initializes the generation process with a representative molecular structure, thereby facilitating early stabilization of atom permutations. Additionally, they introduce a novel training objective aimed at improving the consistency of model parameters throughout the generative trajectory. The method effectively accelerates both the permutation stabilization and atom type assignment phases.", "questions": "### 1. Table 1.:\n\n1. **Key Metrics**: Could the authors clarify which metrics in Table 1 are considered most indicative of model performance and why? From a drug design perspective, the V\\&U\\&N metric appears to be the most relevant. However, it is notably low for MolTD on QM9 and is omitted entirely for DRUG. Could the authors elaborate on the rationale behind this omission?\n2. **NFE Definition**: What specific function evaluations are counted under NFE (Number of Function Evaluations)? Is this definition consistent across all compared models?\n3. **Runtime Reporting**: The paper reports NFE but does not include actual runtime measurements. Given that BFN-based models may be computationally expensive, could the authors provide wall-clock runtimes or comment on how well NFE reflects real-world performance?\n\n### 2. Figure 2.:\n\n1. **GOAT Exclusion**: Could the authors explain why the GOAT model is excluded from Figure 2?\n2. **Figure Clarity**: The third subfigure in Figure 2 is somewhat difficult to interpret. Specifically:\n\n   * What is the boundary of $x^1_2$?\n   * What do the circles around $x^1_0$ and $x^1_2$ represent?\n   * What does the \"Phase I\" region denote?\n   * What is the distinction between \"Permutation\" and \"Phase I\", especially since Phase I seems to be when permutation is established?\n   * A more detailed explanation in the figure caption or main text, and possibly a refinement of the visualization, would be appreciated.\n\n### 3. Geometric-Informed Prior:\n\n1. **Construction of Representative Structure**: Could the authors clarify how the representative structure is constructed? Is it derived by aligning a set of molecules and averaging their features, including spatial coordinates?\n2. **Number of Representatives**: How many representative structures are used in total? Is there one representation structure per stratified subset of molecules?\n3. **Inference Procedure**: How is the representative structure selected during inference? The notation suggests $\\theta_p$ is defined for a single representative - how is it determined?\n\n### 4. Consistency Parameter:\n\n1. **Training Phase Separation**: In the introduction, the authors state that \"consistency training can be significantly beneficial when specifically adapted for geometric generation in the adjustment phase.\" This implies that decoupling the permutation and adjustment phases is crucial. Could the authors provide experimental evidence for this claim? For example, what happens if $t_{\\text{stable}}$ is removed from Equation (9) and the consistency objective is optimized across the entire trajectory?\n2. **Terminology**: Could the authors clarify what is meant by “naturally aligned” in line 251?\n3. **Proposition 4.6**: How does Proposition 4.6 support the joint training of the BFN and consistency objectives? A more explicit connection would help the reader understand the motivation.\n\n### 5. Training:\n\n1. **Stability Determination**: How does the method handle cases where the structure does not stabilize?\n\n### 6. Ablation:\n\n1. **Component Impact**: It would be helpful to include results for MolTD without the permutation and consistency components in Figure 2a and 2b. This would clarify the contribution of each component to the respective metrics.", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n1. The analysis in the section Decomposition of the Generative Trajectory is well-designed and provides a compelling justification for the proposed approach.\n2. The method effectively accelerates both the permutation stabilization and atom type assignment phases, leading to improvements in the Number of Function Evaluations (NFE) metric.\n\n**Weaknesses:**\n1. Evaluation Scope:\n    * One of the most practically relevant scenarios for molecular generation is conditional generation, where molecules are optimized for specific properties. Without demonstrating successful conditional generation, the practical significance of the work remains limited.\n    * A key evaluation metric, Validity, Uniqueness, and Novelty (V&U&N), is missing from the DRUG benchmark table.\n    * Actual runtime measurements are not reported. Although NFE is informative, real-world runtime is crucial, especially when comparing different generative frameworks.\n2. Clarity and Terminology:\n    * Several parts of the manuscript are confusing or imprecise:\n        * Line 149: Referring to transpositions or swaps as \"permutations\" may be misleading; a clarification is needed.\n        * Line 24: The phrase “comparable” lacks a clear reference—comparable to what?\n        * Line 39: The claim that \"3D molecular generation requires determining the permutation order of atoms\" is overly general. It applies to some model families but not all; for example, fragment-based generation methods followed by conformer prediction do not require this step.\n        * Line 40: Permutation ordering is not unique to 3D molecular generation; point cloud generation models may exhibit similar characteristics.\n        * Lines 53–57: This section describing the contributions should be more concrete. The current formulation is overly vague and lacks specific technical insights.\n        * Line 104: The permutation should be defined over $\\{0, 1\\}$, not real numbers.\n    * Several design choices and methodological details would benefit from further explanation (see the Questions section for specific points).\n\n**Overall Assessment**\n\nI find the paper to be interesting and promising. The proposed analysis and methods address a meaningful challenge (speed) in generative modeling for molecules. However, for the work to reach its full potential, it would benefit from a more comprehensive evaluation, particularly in the context of conditional generation, as well as clearer presentation. If these concerns are addressed in a revision, I will be inclined to increase my score.", "was_revised": true, "final_justification": "Following the authors' clarifications and the inclusion of additional benchmarks, I have raised my initial score and now recommend the paper for acceptance", "rebuttal": "Thank the thorough review and insightful suggestions! Your feedback will help us significantly improve our work. We address your concerns as follows:\n## W1: Conditional generation\nWe apply our method to **structure-based drug design**, integrating it with MolCRAFT [1], a state-of-the-art method in this domain. The prior is constructed using Algorithm 1 from the set of target ligands within the training set. Given that MolCRAFT also utilizes BFNs as its generative backbone, we were able to seamlessly apply our consistency parameter objective to its parameter space. We adhered to MolCRAFT's established evaluation settings to report our results:\n|Model | NFE| QED | SA | Vina Score (mean) | Vina Score (median) | Vina Min (mean) | Vina Min (median) |\n|-|-|-|-|-|-|-|-|\n| TargetDiff | 1000 | 0.48 | 0.58 | -5.47 | -6.30 | -6.64 | -6.83 |\n| Decomp-R |1000 | 0.51 | 0.66 | -5.19 | -5.27 | -6.03 | -6.00 |\n| MolCRAFT | 100 | 0.50 | 0.69 | -6.59 | **-7.04** | -7.27 | -7.26 |\n| MolCRAFT | 25 | 0.51 | 0.65 | -5.95 | -6.70 | -6.73 | -6.89 |\n| **MolTD** | **25** | **0.54** | **0.72** | **-6.60** | -6.91| **-7.36** | **-7.24** |\n\nFor Vina-based metrics, lower value indicate better performance. For others, larger value indicate better performance. Our method demonstrates comparable or superior performance with significantly fewer NFEs. This outcome underscores its considerable practical value.\n## W2: V&U&N metric on DRUG dataset\nOn DRUG, both our model and all compared works **consistently achieve near-perfect scores** for uniqueness and novelty. Our model achieves 99.7% Uniqueness and 99.9% Novelty. Given this saturation, we followed the established practice of prior works [2,3] by primarily reporting the validity metric.\n\nFurthermore, we are one of the few works to report **molecular stability** on the DRUG dataset, as detailed in Table 3. Including this metric provide a more comprehensive evaluation of generation quality.\n## W3: Actual runtime\nThe following table presents the wall-clock times required to generate 1000 samples on the QM9 dataset, using a single RTX 3090 GPU with a batch size of 64:\n| Model  | NFEs | Time (seconds) |\n|-|-|-|\n| **MolTD**  | 12 |4.86 |\n| GeoBFN |100 | 34.52 |\n| EquiFM | 200 | 180 |\n| EDM    | 1000 |760 |\n\nAs the results demonstrate, **MolTD** achieves a sampling speed over **100x** faster than diffusion-based models. Furthermore, the practical speed-up is even greater than what is reflected by the reduction in NFEs alone, as MolTD benefit from a more efficient implementation.\n\n## W4: Clarity and Terminology\nWe have revised the paper based on your valuable comments, a provide a point-to-point response in the following:\n1. Line 149: We change **permutations** to **transpositions** to improve clarity.\n2. Line 24: Compared to official results reported by AlphaFold3\n3. Line 39: We change the phrase to '..., 3D molecular generative models **that generate the structure holistically**, requires determining the permutation order ...', which is more rigorous.\n4. Line 40: We change the phrase to 'This two-phase segmentation is unique to **3D geometric generation** and...'\n5. Line 53–57: We have revised the paragraph to more precisely articulate our contribution and provide high-level insights.\n6. Line 104: We have corrected it in the revised manuscripts.\n## Q1: Which metrics in Table 1 is more indicative?\nBoth Validity and Stability are crucial for assessing a model's capacity to generate chemically sound molecules. However, **Stability is arguably a more robust indicator** of model performance than Validity, primarily because the **Validity is susceptible to being artificially inflated**.\n\nWe borrow the discussion from [2] : 'Experimentally, it is observed that validity could artificially be increased by reducing the number of bonds. For example, predicting only single bonds was enough to obtain close to 100% of valid molecules on GEOM-DRUGS. On the contrary, the stability metrics directly model hydrogens and cannot be tricked as easily.' \n\nFurthermore, the **Novelty** metric, which quantifies the percentage of generated molecules absent from the training set, also requires careful interpretation. While high novelty is generally desired, an **extremely high score can paradoxically suggest an underfitting** to the training data distribution.\n\nGiven these considerations, we believe that the **V&U&N metric is less indicative** of a model's true generation performance. It should be viewed as **a signal that ideally falls within a reasonable range**. Our method achieves a V&U&N of 67.03%, aligning well with the 50-70% range reported by previous works in Table .\n\nWe report two additional metrics to evaluate quality and diversity: Validity & Uniqueness (V&U) and Atom Stability & Validity & Uniqueness (S&V&U):\n|  Model | NFE   | V&U  | S&V&U |\n|-|-|-|-|\n| EDM    | 1000 | 90.7 | 89.5  |\n| GeoBFN | 100 |  91.5 | 90.2  |\n| EquiFM | 200 | 93.5 | 92.4  |\n| GOAT   | 90 | 91.9 | 91.1  |\n| **MolTD**  | 12 | 93.1 | 92.5  |\n\nAs the result shows, our method achieves significant speed-up while maintaining superior generation quality.\n## Q2: Definition of NFEs\nOne NFE denotes one forward pass of the neural network, which is equivalent to one sampling step in the denoising process. This definition is consistent across all compared models.\n## Q3 Exclusion of GOAT in Figure 2\nWe agree that GOAT is a key state-of-the-art model, and we have aimed to compare against it thoroughly.\n\nAs shown in **Table 1**, we conducted a comprehensive comparison with GOAT on all standard metrics, where our method demonstrates a significant speed-up while maintaining comparable generation quality.\n\nThe analysis in **Figure 2**, however, requires analyzing intermediate results, and thus requires model's implementation. Unfortunately, at the time of our experiments, the official implementation of GOAT is still under actively updating. Attempting our own reproduction, especially within a limited timeline, would risk introducing implementation biases.\n\nWe are keen to include GOAT in Figure 2 based official implementation or a faithful reproduction.\n## Q4 More explanation on the third subfigure in Figure 2\nWe have provided a detailed explanation on the subfigure in the revised manuscript, and a point-to-point response in the following:\n- The boundary of $x_2^1$ refers to one sheet of a two-sheeted hyperboloid with foci at $x_0^i$ and $x_0^j$, passing through the $x_2^1$. This surface defines the decision boundary for **whether a permutation is needed** between the i-th and j-th rows, supported by Proposition 4.3 and 4.4. \n- These circles around x01 and x21 represent the neighborhood $V(x_0^i; r) $, where a specifical radius $r$ is chosen to ensure that these neighborhood are non-overlapping. Within $V(x_0^i; r)$, the optimal matching is the identity permutation.\n- Phase I denotes the permutation phase, where permutation is needed. Phase II corresponds to the adjustment phase, where no further permutation is needed during Phase II.\n## Q5: Geometric-Informed Prior\n**1. Construction:** for size of molecules (number of atoms), prior is created by:\n- Alignment: Using Equivariant Optimal Transport to align all geometric structures within the subset.\n- Averaging: After alignment, we compute the mean of all atomic features (including spatial coordinates) across all molecules in the subset. \n\n**2. Number of Representatives:** We construct one representative structure for each molecule size contained in molecules. \n\n**3. Inference:** Following the previous frameworks [2,3], we first decide the number of atoms based on the distribution of training data, which is done by sampling from the histogram of molecule size. Once the number of atoms is fixed, we retrieve the corresponding prior and plug it in Equation (7) to create $\\theta_p$.\n\n## Q6: Consistency Parameter\n**1. Training Phase Separation:**\nConsistency training can suffer from high variance during training, because the consistency objective enforces self-consistent outputs across all time steps, creating complex optimization goals [4] .\n\nBy introducing $t_{stable}$, we apply the consistency objective only during the adjustment phase—once a stable atomic structure has been largely determined. This simplifies the learning task and substantially **reduces training variance**.\n\nWe trained our model on QM9 with and without the $t_{stable}$ threshold from Equation (9), repeated each experiment 5 times and reported the mean and standard deviation.\n| Method | Molecule Stability % |\n|-|-|\n| With $t_{stable}$ (Our Method) | 92.5 ± 1.8 |\n| Without $t_{stable}$ (Full Trajectory) | 82.7 ± 10.3   |\n\nThe results demonstrate the effectiveness of acceleration and variance-reduction of our method.\n\n**2. Terminology and Proposition 4.6**:\nProposition 4.6 establishes that the original BFN objective intrinsically trains the network to predict ground truth molecules from intermediate time steps, which is conceptually identical to that of consistency models. Consequently, both the original BFN and the consistency parameter objective are utilized to stabilize the training process, without introducing conflicting optimization goals.\n\n**3. How does the method handle cases where the structure does not stabilize?**\nFor the permutation phase, the network is trained using only the BFN loss, meaning the acceleration observed in this phase primarily stems from the geometric-informed prior.\n\n## Q8: Include ablation result in Figure 2\nThank for the constructive suggestion. We include visualization in Figure 2 in the revised manuscript\n\n## Reference\n[1] Qu, Yanru, et al. \"MolCRAFT: Structure-Based Drug Design in Continuous Parameter Space.\" ICML 2024\n\n[2] Hoogeboom, Emiel, et al. \"Equivariant diffusion for molecule generation in 3d.\"  ICML 2022\n\n[3] Xu, Minkai, et al. \"Geometric latent diffusion models for 3d molecule generation.\" ICML 2023\n\n[4] Song, Yang, and Prafulla Dhariwal. \"Improved Techniques for Training Consistency Models.\" ICLR 2024"}, {"review_id": "Qw13Ci92y8", "reviewer": "Reviewer_4XgW", "summary": "The paper proposes a novel approach to accelerate the generative process for 3D molecular data. Unlike in the Euclidean data domain, such as images, the authors observe that the generation of 3D molecular structures involves two distinct phases. In the first phase—permutation ordering—the model focuses on aligning the current sample with the correct atom ordering of the final structure. Once this is resolved, the second phase involves refining the atomic features. To accelerate the process, the authors introduce a geometry-informed prior to speed up the first phase and apply consistency learning to improve efficiency in the second phase. As a result, the proposed model achieves competitive performance with a significantly reduced number of function evaluations (NFEs).", "questions": "- Could the authors elaborate on how the novelty metric is computed? Is it based on SMILES string comparison with the training set?\n\n- In the context of 3D molecular data, multiple symmetries exist. Could the authors discuss how rotational symmetry evolves throughout the generative process, in addition to permutation symmetry?", "limitations": "yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "**Strengths**\n\nI find the discovery of two distinct phases in 3D molecular generative modeling both novel and interesting. As for the proposed solutions, using a geometry-informed prior instead of the standard BFN prior is a reasonable approach to accelerate molecular structure generation. Moreover, applying consistency training in the parameter space of BFNs also strikes me as a novel and promising idea.\n\n**Weaknesses**\n\n- The work builds on prior research [1,2] to identify the problem and design its solutions. From the perspectives of modeling, benchmarking, and architecture, the contribution appears incremental and not particularly substantial within the broader context of 3D molecular generation.\n- The method requires storing a separate geometry-informed prior for each group of molecules with the same number of atoms. This limits its generalization to out-of-distribution cases involving molecules with atom counts not seen during training, and may reduce scalability to datasets with a wide range of molecular sizes.\n- Some statements in the paper appear to be unclear (see Questions).\n\n**References**\n\n[1] Equivariant Flow Matching with Hybrid Probability Transport for 3D Molecule Generation. In NeurIPS23.\n\n[2] Unified Generative Modeling of 3D Molecules via Bayesian Flow Networks. In ICLR24.", "was_revised": true, "final_justification": "The paper is very well written. Their method appears novel in the field of 3D molecular generation.  In the rebuttal, the authors also addressed all my questions. I believe this is a very good paper, and I recommend accepting it.", "rebuttal": "## Weaknesses 1：Relationships to prior works\nWe thank the reviewer for their careful assessment of our contribution. While our framework follows prior research [1, 2], our core contribution lies in identifying and addressing a **fundamental, previously unappreciated characteristic** of 3D molecular generation that has significant implications for efficiency. \n\nDriven by this novel insight, we introduce two distinct and highly effective techniques. Our empirical results demonstrate that this dual approach delivers a **magnitude-order** improvement in sampling speed over the state-of-the-art. Furthermore, our techniques are complementary and can be integrated into various generative frameworks, highlighting their broad utility.\n\nBeyond the demonstrated speedup in 3D molecule generation, we believe our work offers insights into **essential elements** for modeling 3D molecular structures. Specifically, our findings suggest that a strong inductive bias towards geometric priors can significantly simplify the inherent complexity of 3D geometric generation.\n\n## Weaknesses 2. Out-of-distribution generation\n\nThanks for your insightful comment! Current generative framework, training objectives and even evaluation metrics tend to **encourage models to fit the empirical distribution** of the training data. As you mentioned, we also recognize the importance of generating novel and valid molecules beyond the training set, which holds significant practical value. However, **evaluating generalization ability remains an open challenge**. To address this, we conducted additional empirical studies and highlight three key points to demonstrate the model’s generalization capabilities:\n\n### Novelty and Validity as proxies for generalization:\nAs shown in Table 1 in the original manuscript, our method achieves state-of-the-art novelty metric: **V&N&U (67.03% on QM9)** while maintaining high validity: 96.04%. This suggests that the geometric prior does not collapse to memorizing training structures but instead guides exploration of chemically plausible regions beyond the training distribution.\n\n### Generalization across datasets:\nAs mentioned in the main text, we have explicitly tested the prior’s transferability to external dataset. We constructed the geometric-prior using training data from GEOM-DRUG, and plugged it into the generative model trained on QM9—a dataset with distinct structural patterns (smaller molecules, different atom types). Using 12 NFEs we generated 10000 molecules and evaluated on QM9: \n\n| Model| Atom stability%| Molecule stability%| Validity%|\n|-|-|-|-|\n|GeoBFN | 100 | 98.6| 87.2| 93.0|\n| MolTD with QM9 prior| 99.4| 92.53| 96.04|\n| MolTD with **DRUG prior**| 99.2 | 90.22 | 94.58 |\n\nAs the results show, using prior constructed from external dataset achieves the same accelerating effect as original MolTD, demonstrating that the prior captures fundamental geometric structures shared across datasets.\n\n### Generalization to a number of atoms that do not exist in the training set\n\nFor this task, we generate molecules with **28 atoms**, as QM9 does not contain a molecule with 28 atoms and most of its molecules have less than 28 atoms (>99%). We construct prior using GEOM-DRUG dataset and plugged it on MolTD trained and evaluated on QM9,  and generated 1000 molecules with 28 atoms. The results show competitive performance with SOTA using only 12 steps:\n\n| Method                | NFE | Atom stability% | Molecule stability% | Validity% |\n|-----------------------|-----|-----------------|---------------------|-----------|\n| EquiFM                | 200 | 99.0            | 83.5                | 90.2      |\n| GeoBFN                | 100 | 98.8            | 80.5                | 91.2      |\n| MolTD                 | 12  | 98.5            | 78.1                | 89.1      |\n| MolTD with DRUG prior | 12  | 98.8            | 80.3                | 90.9      |\n\n## Q1: How the novelty metric is computed?\n\nWe appreciate the opportunity to clarify the computation of the novelty metric, which consist of three steps: First, we select the generated molecules that can be successfully transformed to SMILES strings using RDKit. Then, among these valid SMILES strings, we select a subset of the unique SMILES strings. Finally, if a unique and valid SMILES strings did not appears in the training set, then we called it a novel molecules, and the novelty metric is the percentage of novel molecules among the uniqe ones. Furthermore, the metric V&N&U is the percentage of novel molecules among all generated molecules. \n\n## Q2: How rotational symmetry evolves throughout the generative process?\nThanks for the insightful question! As detailed in the main text, the geometric-informed prior utilized the EOT algorithm to align the geometric structures of molecules, which **eliminates the variations due to rotation and translation**. As a result, the geometric-informed prior simplifies the multiple rotational symmetries to just one orientation, and accelerates the stabilizing of rotation. Proposition 4.5 formalizes the intuition as the rotational-equivariant property of generated molecules. \n\nFurthermore, we provide empirical evidence for the acceleration effect. Similar to the main text, we analyzed the **generative trajectory with respect to rotation**, by measuring the rotation angle between intermediate molecules to the final molecules. The result shows that MolTD requires less rotational transform to align intermediate molecules to the final molecules, demonstrating the acceleration effect of our method in terms of rotation symmetry.\n\nFollowing the NeurIPS 2025 instruction, we can not upload image in this stage of rebuttal. We will update the results in the revised manuscript.\n\n## Reference \n[1] Equivariant Flow Matching with Hybrid Probability Transport for 3D Molecule Generation. In NeurIPS23.\n\n[2] Unified Generative Modeling of 3D Molecules via Bayesian Flow Networks. In ICLR24."}]

Author rebuttal:

Meta-review: This paper introduces a method to accelerate 3D molecule generative models by analyzing and decomposing the generative process into two phases: permutation reordering and atomic feature adjustment. The authors propose a geometric-informed prior to reduce inefficiencies in the permutation phase and a consistency parameter objective to improve the adjustment phase. Empirical results on QM9 and GEOM-DRUG show that MOLTD achieves state-of-the-art performance with large speed-up compared to baselines. 

The reviewers were mainly concerned about limited comparisons to baselines, terminological ambiguity. These were largely addressed in the rebuttal. Overall, the reviewers agree that the paper is technically solid and impactful. I recommend acceptance.
Decision: Accept (poster)