Straight-Line Diffusion Model for Efficient 3D Molecular Generation
Authors: Yuyan Ni
Shikun Feng
Haohan Chi
Bowen Zheng
Huan-ang Gao
Wei-Ying Ma
Zhi-Ming Ma
Yanyan Lan
Venue: NeurIPS 2025 poster ()
Keywords: Molecule generation; Diffusion model
Primary area: machine_learning_for_sciences

Abstract: Diffusion-based models have shown great promise in molecular generation but often require a large number of sampling steps to generate valid samples. In this paper, we introduce a novel Straight-Line Diffusion Model (SLDM) to tackle this problem, by formulating the diffusion process to follow a linear trajectory. The proposed process aligns well with the noise sensitivity characteristic of molecular structures and uniformly distributes reconstruction effort across the generative process, thus enhancing learning efficiency and efficacy. Consequently, SLDM achieves state-of-the-art performance on 3D molecule generation benchmarks, delivering a 100-fold improvement in sampling efficiency.

Straight-Line Diffusion Model for Efficient 3D Molecular Generation
Yuyan Ni
Shikun Feng" Haohan Chi
Bowen Zheng
Huan-ang Gao*
Wei- Ying Ma Zhi-Ming Ma' Yanyan Lan
Academy of Mathematics and Systems Science, Chinese Academy of Sciences
Zhongguancun Institute Artificial Intelligence, China
Institute for AI Industry Research (AIR), Tsinghua University
University of Chinese Academy of Sciences
Huazhong University of Science and Technology
Beijing Frontier Research Center for Biological Structure, Tsinghua University Beijing Academy of Artificial Intelligence
Abstract
Diffusion-basee models have shown great promise in molecular generation but often require large number of sampling steps to generate valid samples. In this paper, we introduce a novel Straight-Line Diffusion Model (SLDM) tackle this problem, by formulating the diffusion process to follow linear trajectory. The proposed process aligns well with the noise sensitivity characteristic of molecular structures and uniformly distributes reconstruction effort across the generative process, thus enhancing learning efficiency and efficacy. Consequently, SLDM achieves statc-of-the-art performance 3D molecule generation benchmarks, delivering 100-fold improvement in sampling efficiency!
Introduction
3D molecular generation is an essential task in drug discovery, material science, and molecular engineer- ing. The goal computationally design 3D molec- ular structures that not only capture intricate physical and chemical constraints but also fulfill specific prop-
22
Nold So Sta biliityyyyyy (%)
erties.
Recently, diffusion models have been widely applied in this field, inspired by their remarkable success in
Motecaia
SLDM
GeoffNN EquiFM
image synthesis (Dhariwal and Nichol 2021 Rom- bach et al. 2022 Peebles and Xie 2023), J, and other domains Brooks et al. 2024 Abramson et al. 2024
78
EDM-arisge
200
320
2000
Methods like EDM Hoogeboom et al. [002] EDM-
NPE (in scale)
Bridge (Wu al.
2022] and GeoL.DM [Xu al.]
Figure 1: Comparison of molecule stability (t) across diffusion-basee molecular gener- ation models on QM9 unconditional gener- ation, evaluated by the number of function evaluations (NFE) during the sampling pro- cess.
2023) have demonstrated the potential of diffusion- based frameworks generate chemically valid 3D molecular structures. However, these direct appli- cations of diffusion methods usually require a large number of sampling steps to produce valid molecules.
"Equal contribution. Work done during Yuyan's internship at AIR. Corresponding author: Yanyan Lan (lanyanyan@air tsinghua.edd edu.
'The code is open-sourced at http://gthtuuccmmm///gegeiii/aat
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Taking EDM as an example, it requires approximately 1000 steps of function evaluations to generate molecules with around 82% stability, which is a key metric for assessing sample quality by quantita- tively measuring whether the molecule satisfies chemical constraints. Reducing the sampling steps significantly degrades the molecule stability.
To improve sampling efficiency, EquiFM [Song et al. 2024] and GeoBFN [Song et al. 2023a] have been proposed to utilize the Flow Matching (FM) framework [Tong et al. 2023 and Bayesian Flow Networks (BFN) Graves et al. [023] for molecular generation. The use of these advanced generative Al models enables a speedup of 5 x and 20x, respectively, compared to EDM. However, they still
require a large number of sampling steps (e.g. 1000) to achieve high molecule stability (e.g. 90%), as
shown in Figure !
To understand why existing methods suffer from low efficiency, we analyze the issue through the lens of truncation error in sampling. We begin by establishing unified perspective previous diffusion- based methods, including diffusion models, flow matching methods, and Bayesian flow networks. Specifically, their noise corrupting process can be generalized as 2, x where To represents the clean data, 2t is the noise corrupted data at time E [0, plt) and (t) define the schedule of the process. In this framework, all these processes can be equivalently framed as continuous-time Ordinary Differential Equations (ODEs), even though they may employ stochastic sampling practice. This viewpoint allows the sampling process to be interpreted as a numerical approximation of the solution trajectory of the underlying ODE. Crucially, most existing methods rely
~ (O, IN).
d'z(t) (At)?.
on first-order estimation, whose truncation error is governed by the second-order term
de=
We observe that d'z(() in these approaches can be large, requiring small step sizes At to reduce the de truncation error, which results in a large number of sampling steps.
To address this issue, we propose a novel diffusion process called Straight- Line Diffusion Model (SLDM). The key idea is to minimize truncation error by striving to achieve a linear sampling
trajectory, i.e. dit without sacrificing accuracy, leading to substantial improvement in sampling efficiency. Building on this objective, we theoretically prove that when v(l)= = 1 1/TT and a a small constant, the process guarantees a near-linear trajectory. Intuitively, this process features a linearly decreasing mean term and a consistently small variance term, representing a smooth linear progression from the origin point to the data.
df
This approach allows the diffusion dynamies to tolerate larger step sizes
Notably, SLDM strikes a good balance between efficiency and efficacy by using our straight-line schedule. Firstly, this strategy aligns well with the inductive bias of molecular generation, preventing the introduction of chemically implausible conformations. Unlike images, molecular structures are much more sensitive to noise, and even small perturbations can lead to unrealistic structures that violate chemical principles. This challenge requires a slower signal-to-noise ratio (SNR) decay during the noise-adding process, as suggested in Song et al.) (2023a]. By naturally satisfying slower SNR decay, our method maintains chemical information of the intermediate states, enhancing computational efficiency compared to traditional methods such as EDM. Secondly, our strategy achieves a more balanced generative process, significantly improving the model's learning efficacy. In methods like GeoBFN, minimal perturbations are applied in the later stages, shifting most of the reconstruction burden to earlier stages. Although this reduces the computational load the later stages, it creates an uneven distribution of effort, limiting the model's 's learning capacity. In contrast, our approach evenly distributes the reconstruction effort across the entire process, enabling the model to leam effectively at each stage. This balance results in more stable and efficient learning process, enhancing the robustness and accuracy of the generated molecular structures.
We conduct extensive experiments to demonstrate the potential of straight-line diffusion in 3D molecular generation and other domains. As shown in Figure [ using only at most 10 or 15 sampling steps, SLDM surpasses EDM EquiFM, GeoBFN with 1000 sampling steps, achieving up to 100- or 70-fold improvement in sampling efficiency. In terms of generation quality, SLDM with 200 sampling steps achieves 95% molecular stability, significantly outperforming the best baseline, GeoBEN, which requires 1000 steps to reach 90% molecular stability. We also observe that similar improvements can be achieved when applying SLDM to the conditional generation task, i.c. generating molecules with a desired property, highlighting its potential to enable more practical and controllable molecular design in future applications.
2

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2 Analysis on Sampling Efficiency
We begin by theoretically analyzing the underlying factors contributing to the sampling efficiency issue. In particular, we first present a unified framework for diffusion-baset methods, including their SDE and ODE formulations. We then examine the sampling truncation error from the ODE perspective, highlighting the critical role of the process's second-order derivative in improving sampling efficiency.
We denote the data as x E IR-,, where for molecules, it refers to a 3D point cloud comprising atomic coordinates and potentially other atomic features. According to Karras al. [2022]. Xue
et al. 2024a], various diffusion-basel models, including DDPM (Ho et al.
2020]. DDIM [Song et al.
2021a VE [Song et al. [0211], FM (Lipman et al. [023]. and BFN [Graves et al. 2023 can be
formulated as unified form with the noise corrupting process defined as:
F2 p(t) + (t)e, c (O IN), e [0.
(I)
where æo, 2a are clean data and noise corrupted data respectively, μu(t and (t)) define the schedule of the process. Specifically, as detailed in Appendix [A.2]
[..2) the schedule parameters are summarized
as follows: (DDPM(EDM) 1- (L/T)², JVR MADIM 1, MAPN
minn
2(1-1/7)
MPM = 1-1177
ODDPMEDM) = (1- (2) VVE = Vt, DDIM = t, dury = V /BBN (BBN). FFM +
t/T) mmins where DDPM(EDM) uses the approximated DDPM schedule given in EDM [Hoogeboom
al. 2022]. miin are defined small constants to ensure p(0) and (0) es 7 is typically chosen to be sufficiently large so that Øт approximates a known distribution.
Extending a similar theoretical technique from Karras et al. 2022 to the unified form, we can prove that equation iis the solution to the following linear stochastic differential equation (SDE):
dx, = ju(t) pstt
x, + v (t)) - (t) ju(t)
dw
ju(t)
where p(t) and ot)) should adhere to the constraints p(0) 1. (0) O. with v(t) 2 being monotone non-increasing, (t) > O, and (t)/p((t being monotone non-decreasing. The detailed proof is elaborated in Appendix A.I
According to the ODE and SDE relations revealed in Song et al. (0011), we can derive the equivalent ODE that follows the same marginal probability densities the above SDE:
dæ,-
(u(t) (u(t)
(o(t)o(t)
ji(t) pu(t)
V ao (x, dt.
Consequently, most existing diffusion-basee methods widely used in 3D molecular generation, including DDPM, DDIM, VE, FM, and BFN, can be interpreted from continuous perspective, where their diffusion process can equivalently be viewed as an SDE in equation [ or an ODE in equation 3 Therefore, we can use the ODE formulation as valuable perspective to analyze the sampling efficiency issue.
Specifically, the diffusion sampling process can be interpreted as a numerical approximation of the backward solution trajectory of the underlying ODE in equation 3 This numerical approximation inherently introduces truncation errors at each step. For example, under Euler's method, the simplest and widely-used numerical scheme, sr(t - At) is approximated by sx(t) da(t) At when solving the ODE backward in time. However, the true value can be derived from the Taylor expansion:
dr(t) x(t - At) = xt)) di At +
dd
d/2 (At)² + ((()))))
where At is small step size. Thus, the truncation error primarily arises from the second-order term, governed by dFx(t)
dr
To minimize this truncation error, traditional numerical analysis primarily focuses developing higher-order solvers to estimate higher-order terms, assuming the ODE structure is fixed. EquiFM
Connections between sampling of baseline methods and first-order ODE sampling are discussed in appendix
A.3
3

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inherent ODE exhibits minimal truncation error without resorting to complex solvers that require multiple function evaluations. This [Song et al. 2024] applied such a technique in 3D molecular generation to speedup the sampling process. However, higher-order solvers face trade-off between the number of function evaluations (NFF) and accuracy, limiting their ability to achieve even smaller NFEs. As result, the issue remains partially unsolved. In contrast, we take a different approach: rather than focusing on the sampling algorithm, we reformulate the diffusion process itself to minimize the truncation error, which directly influences the sampling efficiency.
However, our unified formulation of diffusion introduces key flexibility; the ability to modify the schedule of the process, thereby altering the ODE structure itself reduce these errors. Therefore, we emphasize that the key to minimizing the truncation error is to reduce the second-order derivative of the process.
3 Straight-Ling Diffusion Process
To reduce the truncation error of the sampling process, we aim to design a new diffusion process whose ddx(t) dia By achieving this, even basic Euler iteration can deliver low
novel perspective advances the Pareto frontier of efficiency and accuracy in diffusion sampling.
3.1 Derivation of SLDM
Given the intractability of the score function for general data distributions, we begin by examining a simplified case where the initial distribution is delta distribution. This setup provides tractable backward ODE, enabling a more straightforward analysis. In this case, the only solution that ensures de remains constant is for (t) be constant and (tt) be linear function of Details can be found in Appendix A.4
Given the boundary condition that ao approximates data distribution and 2:-1 approximates known distribution, the only feasible choice is to set (1) to a small poistive constant and (u(t) = =1 I/T, which satisfies all the schedule constraints noted under equation 2 except (0) 0. This is because setting (0) = 0 exactly would result in trivial denoising objective, To avoid this, we instead approximate o (0) by a small value, which keeps the denoising task non-trivial while still satisfying the boundary condition approximately. In practice, we use a value for that is two orders of magnitude smaller than the data scale, yielding good results.
Rescaling time to the interval [0, 1]. the resulting diffusion process is:
x = (1 - t) + ac, E [0, 1],
(5)
which ensures straight-line trajectory under the delta data distribution assumption.
We then analyze the above diffusion process for general data distribution and demonstrate that a near-linear trajectory could be achieved by setting a small constant value for a, as shown in the following theorem.
Theorem 3.1 (Near-linear Trajectory of SLDM). For a general data distribution and the schedule in equation the following inequality holds for each dimension
dx (1)
(3)
P(I
8)
(6)
dt
82 t)2
When a + 0, dm, + The Zi is that I-t
dt ? converges to zero in probability, where the solution to equation da, is constant, which corresponds to a linear trajectory.
m 1-f
0
dt
In other words, for / E 10, 2.1. (a)
holds with probability at least - e indicating that setting a to small value ensures that the trajectory remains close to a straight line for most timesteps. This aligns with our empirical observations shown in Figure the overall SLDM trajectory is straighter compared to other diffusion processes, and the sampling trajectory deviates more from straight line in the initial steps but becomes increasingly linear later.
I-1
To further improve the sampling quality, we aim to mitigate the impact of early-stage truncation errors. This achieved by our stochastic sampling design, which exhibits robustness to such early- stage errors. supported by empirical evidence provided in Appendix C.6 As a result, despite the

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EDM
Goo- BFN
SLDM
Figure 2: The diffusion process of atomic coordinates in EDM, GeoBFN and SLDM.
challenges in minimizing the early-stage truncation error, our method manage to achieve high-quality samples with few sampling steps.
3.2 Comparisons with Previous Diffusion Processes
Previous studies Nichol and Dhariwal 202 Rombach 2022] have underscored the importance of amortizing the generative difficulty through the diffusion process to improve sample quality. These works typically customize signal-to-noise ratio (SNR) schedules adapt varying data characteristics. Notably, as pointed out in Song et al. [2023a]. the point cloud representation of molecular structures is far more sensitive to noise compared images. Consequently, for molecular generation, the SNR needs to be decreased at a much slower pace than in the image generation domain.
Our proposed method aligns remarkably well with these insights. As shown in Figure the SNR in our method decreases significantly slower than that of previous methods. In ad- dition, our schedule results a smoother and more stable generative process for molecules, as demonstrated in Figure 2 The process un- folds uniformly from the origin, preserving the relative spatial relationships of atomic co- ordinates in intermediate states and retaining critical chemical information throughout the generative trajectory.
3.0
10
10
0.0
0.8
0.0
PM RPN
8
SLDM
0.4
0.4
0.
0.0
60
02
06
03 1.0
0.0
02
0.4 0.6
0.8
3.0
02 0.0
04
0.5
1.0
Figure 3: Comparison of schedule parameters across diffusion-based models. Here SNR refers to (μ/a)², For SLDM, we set 0.05, consistent with experi- ments. The cutoff values mmax and m used in other baselines are provided Appendix A.2
In contrast, traditional diffusion models like EDM, which follows the process schedule of DDPM, also known as VP, involve noise variance that increases monotonically during the forward process. This increasing noise quickly disrupts the spatial structure of molecules, rendering much of the diffusion process chemically meaningless. result, many computational resources are wasted on steps that attempt reconstruct signal-less states, leading severe reduction in overall model efficiency. Though GeoBEN adopts low-noisc regime for most timesteps, it keeps Re and o for over half of the process, making little changes to the molecular structure. Therefore, GeoBFN shifts the majority of reconstruction difficulty to the earlier stages, creating severely unbalanced generative process. More illustrations of generation process are provided in Appendix C.7
To sum up, compared existing diffusion-bars based models, the new schedule in SLDM aligns better with the inductive biases of molecular data. The approach of distributing the reconstruction difficulty more evenly across the entire diffusion process helps facilitate more effective learning, improving the model's 's efficiency and efficacy.
3.3 Sampling Strategy of SLDM
Here we introduce the specific sampling strategy of SLDM. First, we need derive the reverse process of equation or equivalently equation Notably, previous work [Xue et al. (024b) (Equation 6) provides a unified form of the reverse-time stochastic differential equation (SDE) which shares the same marginal distribution the forward process, thereby ensuring that the sampling process has the ability reconstruct the data distribution. We apply the parameter μ and of SLDM this equation and then discretize it using the Guler-Maruyan method, yielding the following discretized
5

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Algorithm SLDM Training
Algorithm SLDM Sampling
Input:
data zo with dimension N, neu- ral network 6i, constant o
Input: trained network 6, same N and as training, sampling steps T, temperature annealing rate Sample -N(O.
repeat
Sample / (0, 1])
for = T to 1 do
Sample ~ .V (0, (N)
I - T-(i-1)
(æ a. o(x,t))
Compute 2; = (1 - t) + de
T-
if then
Minimize || o(x,, t]ll
E, 1(0.In) x x + (2
V2
until converged;
end
return 6
end return x²
reverse-time SDE:
At
At At o² V log los 2 (xr o log +3 +3(t), t
Zt +
1 oc,
ODE Sampling equation 3
Langevin dynamics
where B()) is any non-negative bounded function.
Similar to previous works [uue et al. 2024b Karras et al. 2022 this sampling can be interpreted as a combination of ODE sampling and Langevin dynamics. The ODE component drives the denoising process along deterministic trajectories. while the Langevin dynamics introduce stochastic corrections. Specifically, we choose (t) = (1 - t)/ At, to ensure the expectation of the RHS of equation 7 iis B[æ;-A1 At Tz). it provides the optimal estimation of -t in terms of minimizing the mean squared error (MSE). Details are demonstrated in Appendix A.5
Thus the sampling algorithm becomes:
1 (L Al) (x, +
(8)
Note that t appears in the denominator of the above equation. To avoid potential division by zero, we choose to skip the L = 1 sampling step and find that works well. This might be because the early sampling steps are less critical to the final result, likely due Langevin dynamics not strictly requiring specific prior, which imparts an inherent error-correction capability to the algorithm. This property is further supported by our observation that the sampling process remains relatively robust even when initialized from an invalid distribution, as demonstrated in Appendix C.6
The above sampling algorithm provides principled way approximately sample from the input data distribution, but its practical application in molecular datasets presents unique challenges. In particular, the molecular stability of the dataset is not guaranteed to be 100%. To enhance stability and mitigate the impact of data noise, low-temperature sampling can be employed to generate samples that preserve essential properties of the input data. For Langevin dynamics, the sampling temperature can be controlled by scaling the stochastic term with a constant. However, prior work has demonstrated that such temperature control in diffusion model sampling often fails to effectively balance diversity and fidelity [Dhariwal and Nichol, 2021]. Additionally, as observed in our toy dataset experiments, conventional low-temperature sampling suffers from mode collapse, failing to fully cover all high-density regions of the target distribution.
To address these limitations, we propose a time-annealing temperature schedule:
1 - (t At) (x; + o2 log (xx)) t" 22E,,
Bt-At
where controls the decay rate of temperature over time. This approach allows for higher stochasticity during the initial stages of sampling, enabling the exploration of distant probability density maxima. As the temperature decreases in later stages, the process converges to the probability density maxima. Further theoretical explanations and toy data illustrations are provided in Appendix A.6] With the help of the temperature control, our sampling strategy can reduce the impact of noise the data, and generate molecules with enhanced stability. The complete training and sampling procedure of

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straight- Straight-line diffusion are given in algorithm and We further discussed the relation to relevant techniques used in general generative approaches in Related Work (Appendix D.2). including noise scheduling. flow-based methods, other straight-line trajectory models, etc.
Table I: Unconditional molecular generation results on QM9 and GEOM-Drugs datasets. For all diffusion-base models, denotes sampling steps. Metrics are calculated with 10000 samples generated from each model. Higher values indicate better performance.
QM9
GEOM-Drie Drugs
#Metries
Atom sta %)
Mol sta( %))
Valid(%)
V*U(%)
Atom sta ())
Valid(%)
Data
99.0
95.2
97.7
97.7
86.5
99.9
E-NF
85.0
4.9
40.2
39.4
G-Schnet
95.7
68.1
85.5
80.3
EDM (T=1000)
98.7
82.0
91.9
90.7
81.3
92.6
GDM (T=1000)
97.6
71.6
90.4
89.5
77.7
91.8
EDM EDM-Bridge (T=1000)
98.8
84.6
92.0
90.7
82.4
92.8
GcoLDM (T=1000)
98.9
89.4
93.8
92.7
84.4
99.3
EquiFM (T=200)
98.9
88.3
94.7
93.5
84.1
98.9
GeoBEN (T=1000)
99.08
90.87
95.31
92.96
85.60
92.08
END (T=1000)
98.9
89.1
94.8
92.6
87.0
89.2
SLDM (T=1000)
99.43
95,42
97.07
90.42
88.30
99.95
SLDM (T=50)
99.30
93.37
96.24
93.63
89.03
99.57
Experiment
To validate the advantages of our method in molecular generation, evaluate its overall performance and sampling efficiency in both unconditional and conditional generation scenarios.
4.1 Setup
Datasets We evaluate our model using two widely adopted datasets for unconditional molecular generation, with all dataset splitting strictly following baseline settings (Hoogeboom al. 2022
Song et al., 2024 2023a QM9 [Ruddigkeit et al. 2012 Ramakrishnan et al.) [2014] contains approximately 134.000 000 small organic molecules with up nine heavy atoms. split into training (100K), validation (1 (18K), and test (13K) sets. GEOM-Drugs [Axelrod and Gomez- Bombarelli 2022 focuses on drug-like molecules, comprising around 430,000 molecules with sizes ranging up to 181 atoms and an average of 44.4 atoms per molecule. Its larger size and greater diversity make it more challenging for generative models. The dataset is randomly divided into training, validation, and test sets using 8:1:1 ratio.
For conditional molecular generation, we adopt the QM9 dataset with the same setup prior work HHoogeooom et al. 2022 Song et al. 2024 2023a]. The QM9 training partition is split into two halves, each containing 50K samples. Specifically, the QM9 training set divided into two halves of 50K samples each. The first half is used to train classifier for ground-truth property labels, while the second half is used train the conditional generative model.
Implementation The molecule is represented by atomic coordinates and atom types, z = (xx, h). where E R&M denotes the atomic coordinates, M is the number of atoms, and h encodes the atom type information. Thus for molecule generation, the model needs to generate both coordinates and atom types. A key requirement for the molecular coordinates ensuring the SE(3) invariance of the probability distribution, meaning that the probability of generating two molecular conforma- tions should be identical if they only differ by translation or rotation. Following Xu et al. (2022), Hoogeboom et al. [2022], we tailor the SLDM algorithm to satisfy equivariance. Specifically. to ensure translation invariance, we constrain the coordinates to a zero Center of Mass (CoM) space, while rotation invariance is preserved by employing equivariant neural networks to predict the noise.
To ensure a fair comparison of generative algorithms, we use EGNN (Satorras et al., 2021 as
backbone model, consistent with the baseline methods [Garcia Satorras et al. 2021
Hoogeboom al. 2022 Wu et 2022 Xu al. 2023 Song et al. 2024 2023a prove that the generated data distribution satisfies SE(3) invariance, provided in Appendix A.7
For atom types, follow

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(Botr*)
(mnvy]
[m
LUND (meV)
(cal(mot
3.14
Notorss
CwoBYN
MAE
MAE
PAAE
MALE
SLOM
scalel
Figure 4: Comparison of performance in conditional generation (L), with respect to the number of function evaluations (NFE) during the sampling process.
UniGEM Feng et al. 2024 to predict atom types based on the generated coordinates. The SLDM
algorithms tailored for molecular generation are provided in Appendix B Hyperparameters An introduction to the baseline models is included in Section[D]]
summarized in Appendix E
Baselines To ensure a fair comparison, we select competitive baselines that also focus on generative modeling and have the same neural architectures (e.g., EGNN) and input information (e.g., coordinates and types). We also acknowledge that some studies propose strategies orthogonal to generative algorithms, as discussed Related Work (Appendix [DI]]. However, these strategies are not directly comparable to work, as they do not aim to replace improve the generative model itself but rather focus on improving network expressivity and augmenting input information. We believe that substituting their generative algorithms with ours could lead to improvements, and exploring this will be part of the future work.
For conditional molecular generation, Two additional basic baselines: Random and Natoms follow EDM [Hoogeboom et al. 2022]. The Random baseline involves shuffling property labels in the training data and evaluating the property classifier on the shuffled data. The Natoms baseline uses the property classifier network to predict the property based solely on the number of atoms in the molecule.
Metries It is important note that evaluation protocols differ across the literature, as discussed in section D.I To ensure consistency, we strictly adhere to the evaluation methods used in our baselines. We sample 10,000 molecules and predict the bond type (single, double, triple, or non-existent) based on the distances between each pair of atoms, as in Hoogeboom et al. [2022). Atom stability is computed the proportion of atoms with correct valency, and molecule stability is the fraction of generated molecules in which all atoms are stable. Validity is evaluated using RDKit by checking whether the 3D molecular structures can be successfully parsed into SMILES format. Uniqueness is determined as the ratio of distinct molecules among all valid samples, indicating the diversity of generated molecules. V means the ratio of valid and unique molecules.
4.2 Unconditional Molecular Generation
Unconditional generation assesses the model's ability to learn the underlying molecular data distribu- tion, aiming to generate chemically valid and structurally diverse molecules. The results, summarized in Table [ show that our method significantly outperforms the baselines across both quality and diversity metrics for the generated molecules. Of particular note is the substantial improvement the stability of the generated molecules, indicating that our method better satisfies chemical constraints. This validates our hypothesis that our low-noise dynamic is well-suited for generating molecular data. A visualization of generated molecules is provided in Appendix C.7
Additionally, our approach can achieve superior results with significantly fewer generative steps. A detailed comparison, presented in Figure]] [ demonstrates that SI.DM achieves 100x faster sampling than the baseline EDM and around 70x speedup compared to GeoBEN and EquiFM. Since the methods use the same network architecture, NFE can reflect practical sampling time. It is important to note that EquiFM utilizes variety of advanced and efficient ODE solvers, and the results we present correspond to the best performance reported in their paper for different step sizes. Besides, EDM and EDM-Bridge results are from the EDM- PeM-Bridge paper while GeoBFN results are reproduced from the official codebase. This observation underscores the advantages of our straight- diffusion process: while sophisticated solvers play an important role, the diffusion process design may offer more fundamental improvements. Furthermore, our handcrafted sampling strategies can be seamlessly combined with modern sampling techniques, such as optimal time discretization and advanced solvers, which we leave for future exploration.
8

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Table 2: Conditional generation QM9 dataset, evaluated by MAE(J) between the property con- dition and the properties of generated molecules predicted by a pretrained EGNN classifier. SLDM uses T 1000.
Table 3: Comparison of generative methods within the UniGEM framework, marked by for un- conditional generation on the QM9 dataset using small sampling steps T', Higher values indicate better performance.
Property Units
As
HOMO
JUMO
C
T Model
Atom sta (6) Mol (E) Valid(%) V*U(%)
Bohr'
mcV
meV
meV
D
50 EDMr:
98.55
85.73
93.29
91.78
Data
0.10
64
39
36
0.043
0.040
50 GenBFNg
98.28
87.
93.97
91.82
Random
9.01
1470
645
1437
1.616
6.857
50 SLDM
99.30
93.37
96.24
93.63
Na
3.86
866
426
813
1.053
1.971
30 EDM
97.58
78.75
89.39
87.96
EDM
2.76
655
356
584
1.111
1.101
30 GeoBFNg
96.74
81.05
90.93
87.47
GeoLDM
2.37
587
340
522
1.108
1.025
30 SLDM
99.30
93.02
96,20
92.76
EquiFM
2,41
591
337
530
1.106
1.033
GeoBFN
2.34
577
328
516
0,998
0.949
SLDM
1.46
440
320
348
0.797
0.745
4.3 Conditional Molecule Generation
Conditional generation evaluates the model's capability to produce molecules with desired properties. Following the baseline approaches, we incorporate property values as additional inputs during training and sample them from prior distribution during inference. Descriptions of baselines and evaluations are provided in Appendix E
The results in Table demonstrate that our method consistently outperforms baseline models across all metrics. Moreover, illustrated in Figure 4 our approach achieves a 20-fold acceleration over previous tatt--oftthealle methods. This result showcases its potential for application in wide range of controllable generation scenarios.
4.4 Ablation Study
Previous approaches differ in their methods for atom type generation. EDM represents atom types one-hot vectors, generating them simultaneously with coordinates through diffusion. In contrast, GeoBFN treats atom types as atomic numbers and generates them by the BFN algorithm for dis- cretized data, which incorporates binning technique convert continuous probabilities into discrete probabilities. UniGEM, on the other hand, generates only coordinates via diffusion and predicts atom types based on the generated coordinates. We adopt the UniGEM framework due its superior performance.
To compare generative methods without the influence of atom type generation differences, integrated EDM and GeoBEN coordinate generation algorithms into the UniGEM framework and compared them with our approach. The results, shown in Table with sampling steps T 50 and T 30, demonstrate that our method exhibits a clear advantage in smaller steps, confirming the efficiency benefits of our generative algorithm. A stability comparison w.r.t various sampling steps is provided in Figure 7 Besides, we also conduct an extensive comparison of sampling efficiency of SLDM under joint dilfusion framework, i.e. without UniGEM framework with GeoBFN and EDM in C.3 and an ablation study about temperature control in Appendix C.5
Appendix
4.5 Evaluation on Toy Dataset
We provide toy dataset experiment in Appendix C.4 ities of our approach faithfully modeling complex data distributions. These results suggest that SLDM holds promise for generalization to broader application domains.
demonstrating the superior generative capabil-
5 Conclusion
This paper proposes the Straight-Line Diffusion Model (SLDM),, a novel generative method that ensures near-linear diffusion trajectory, effectively reducing truncation error during sampling.

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The proposed process schedule naturally aligns with the characteristics of point cloud molecular data and effectively balances the generative difficulty. As result, our method achieves significant improvements in both sampling efficiency and quality, as demonstrated in both unconditional and conditional generation settings, paving the way for large-scale, controllable molecular generation in practice.
Several challenges remain open for future investigation. On the theoretical side, refining the schedule to fully satisfy the boundary conditions, establishing principled guidelines for selecting the noise level a, such as analyzing its impact on training stability and sampling accuracy, and deriving optimal discretization schemes could strengthen the theoretical foundations of our approach and further accelerate sampling. On the practical side, applying our method diverse controllable molecular generation tasks and extending it to broader domains presents exciting opportunities for future advancements.
Acknowledgment
This work is supported by Beijing Academy of Artificial Intelligence and Beijing Frontier Research Center for Biological Structure Fundings.
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Appendix
Table of Contents
A
Supplementary Theoretical Results
22
The Derivation of A Unified Formulation of Diffusion Models
22
A.2
A Summary of Previous Process Schedules
23
A.3
Connection Between Sampling of Baseline Methods and First-Order ODE Sampling
24
A.4
Diflusion Schedule with Straight-ling Trajectory
24
Supplementary Proof for Sampling
26
A.6
Temperature Control .....
27
A.7
Modeling Invariant Probability Density for 3D Coordinate Generation
27
B
Molecular Generation Algorithms
29
Supplementary Illustrations and Results
29
Sampling Trajectory of Diffusion- based Models
29
Sampling Efficiency Comparisons under UniGEM Framework
30
C.3
Sampling Efficiency Under Joint Diffusion Framework
30
C.4
Toy Data Results
31
C.5
Ablation Study for Temperature Control
32
C66
Rohustness to Different Prior Distributions
33
Visualizations Generated Molecules and Sampling Process
34
D
Related Work
34
D. An overview of 3D Molecular Generation
34
D.2 Related Diffusion Based Studies and Key Differences with SL.DM
35
Implementation Detail
36
21

---

A Supplementary Theoretical Results
A.1 The Derivation of Unilied Formulation of Difflusion Models
The stochastic process of the diffusion-based model can be formulated in general as a linear stochastic differential equation (SDE):
dx, = f(t) + g(t)dW
(10)
where X, characterizes the noise corrupting process of the data in a diffusion model, f(t) S
and g(t) 2 0 are measurable functions defined on the interval [0, oo), W, represents standard
Wiener process. The process X; is a time rescaled Ornstein- Uhlenbeck process whose law converges exponentially fast to the standard Gaussian distribution [Chen et al. 2023).
Under the assumption that all the relevant integrals exist, the solution of the above SDE is given by:
X₂ Xo exp( Sh S(E)dE) + f cxpp/. S. ((S)dE)(())dW, (0)
(11)
quick proof is as follows:
Proof. By Itô's formula, and applying equation I0:
d(X. exp(- S/¹ ((S)d)) = (dx; - XxJ(()d() expl- S.' r(E)d))
(12)
cxx((- L/¹ S(E)dE) g(t)aW,
Next, we integrate both sides and obtain:
X, exp(- S. ((E)dE)) - Xo = f exp( - [- S(E)dE) g(s)dW,
(13)
Finally, we multiply both sides of the equation by exp(_" f (())d)) and rearrange the terms to obtain equation II
By the property that the stochastic integral with respect to a Wiener process, X, is normal random variable. The mean and variance of X, are calculated as follows:
E[X = Xo xxxeepp L/' f(E)dE)
(14)
COV[x.] = EIL expls. S. f())))) (eapll J: f(E)d))(()))dds..
(15)
The covariance computation uses the Itô isometry property.
To fit the process X, to the general form
Zt = p(t)xo + (t))), ~ NOO, I),
(16)
we compare it with our solution in equation
I1 and identify that
( ((E)E)) a(t) o(t)=((0))) ml0 S 9(s))/
m(t) (00)=crpfff
(17)
Furthermore, we can express f(t) and g(() terms of p(l) and a(): f(t) d(Inu(t) ju(t)
di
m(()'
(18)
.(1)2 = ())((( d(o(t)/u(t)) 20 (1))((( - (1))2(()) ((()
dt
jutt)
(t) and (t) satisfy the initial conditions μ(0) 1, (0) = 0 and additional conditions: v(t) 2 is monotone non-increasing, (tt) 0, d(a(t)/k(1)) > 0, i.c. (tt//((t) is monotone non-decreasing.
22

---

Substituting the expressions for f() and g(t) back into the original SDE equation 10 we obtain:
dX, ju(t) (t))
ii(t) 11)
X,dt + V (2(()))(() - (0(
dw.
(19)
This result is conceptually equivalent that presented in Appendix B Karras et al. [2022] But they adopt different definition of the schedule compared to equation [6 which leads to a different outcome compared to equation 19
A.2 A Summary of Previous Process Schedules
This section provides a comprchensive overview several widely adopted liffusion-based models and their corresponding process schedules.
The Denoising Diffusion Probabilistic Model (DDPM) [Ho al. 2020 is characterized by process schedule that exhibits the Variance Preserving (VP) property. which can be expressed as: u(t)²= (t)² 1. This formulation ensures the preservation of variance across timesteps, assuming the data has unit variance. In the domain of 3D molecular unconditional generation, the Equivariant Diffusion Model (EDM) (Hoogeboom et al., [0022] employs a schedule closely resembling the cosine noise schedule introduced by Nichol and Dhariwal [202T). albeit with simplified notation:
Ft P) + V1 (1 t. E [0, 1],
(20)
The Variance Exploding (VE) schedule, initially proposed the context of Denoising Score Matching [Song and Ermon 2019). can also be categorized denoising diffusion model with distinct process schedule Karras et al. 2022 Song et al.) 2021b), which given by:
3t = æb + Vie, [0, Tman]
(21)
By rescaling time, we derive the following expression:
3; xo + Viamaze, E [0,
(22)
where omer needs to be sufficiently large to ensure that x approximates a uniform distribution. For illustrative clarity, set Omax 10 for Figure and Fmaz = 20 for Figure 6
Denoising Diffusion Implicit Model (DDIM) offers an accelerated sampling process for diffusion models. As proved by Karras et al. [2022] DDIM employs the following schedule:
C; 2g + tc, E [0, manz).
(23)
By rescaling time, we derive the following expression:
I; 3g + m, [0, 1].
(24)
For illustrative clarity, we set dmar = 10 for Figure and Figure
Flow Matching (FM) (Lipman al. 2023 proposes linear interpolation between the data distribu- tion and a standard Gaussian, with a neural network learning the corresponding vector field. This noise-adding process also be viewed as defining a diffusion process schedule, given by:
aC = (1 l)zg + (1. + (1 t) ),, E [0, 1].
(25)
where Omin needs be sufficiently small to ensure that æc aligns with the data distribution t t= O. Besides, smaller values of a min have been reported to yield better results for FM (Tong et al.
2023 Accordingly, we set min 0.001 for both Figure 3 and F
Bayesian Flow Networks (BFN) [Graves et al., 2023 is a generative model based on Bayesian inference that accommodates both continuous and discrete variables. For continuous variables, the data is parameterized by Gaussian distributions. this scenario, the generative algorithms can be interpreted denoising diffusion model [Xue et al. 2024aa with process schedule given by:
Kt a min )x0 + V (1 mi )
2t
Tiin E, E [0, 1],
(26)
where Tniin needs to be set small to satisfy x, align with the data distribution when = Following the settings in (Song et al. 2023a we miin 0.001 for Figure 3 and Figure
23

---

A.3 Connection Between Sampling of Baseline Methods and First-Order ODE Sampling
In section we analyzed the error of first-order ODE discretization methods and claim that the baseline sampling methods primarily rely on first-order ODE discretization. In this section, we provide further clarification on this matter: EquiFM [Song al. 2024] directly utilizes ODE-based sampling which includes first-order ODE discretization. For methods like EDM [Hoogeboom et al. 2022] and GeoBEN [Song et al., 2023a]. while they resemble first-order methods due to requiring only single function evaluation per iteration, they incorporate random sampling. This distinction necessitates an explanation of how their random sampling processes relate ODEs. Specifically,
EDM uses the same sampling method DDPM, which is proved first-order discretization to the reverse-time SDE of DDPM in Appendix of Song et al. [2021b]. Similarly, GeoBEN adopts the same sampling method as BFN [Graves et al. 2023 which is proved as first-order discretization to the reverse- reverse-time SDE of BFN in Proposition 4.2 in Xue et al. [202aa]. Please note that both of the reverse-time SDE can be decomposed an ODE and langevin dynamies:
dæ. [f(t)=: V log ((gg((x)))dd + g())dw,
[f(()) g²)
V- logp(x;))dtt gt())
V log ((g)(x))dt g(t)
(27)
reverse time ODE equation B
Langevin dynamics
where [tt = A(t
μ(t) and g())² 20 (t) (x) p(r)- Thus the sampling processes of EDM and
p(()
GeoBFN can be effectively upproximated first-order discretization of an ODE augmented by Langevin dynamics. By isolating and analyzing the discretization error of this ODE component, gain valuable insights into the limitations of methods with first-order discretization, including baseline approaches in molecular generation such as EDM, GeoBFN, and EquiFM.
A.4 Diffusion Schedule with Straight-line Trajectory
As illustrated in section 3.1
we aim to reduce the truncation error during sampling by minimizing the second-order derivative of the trajectory. To this end, we first consider a simple case where the initial distribution is delta distribution zo 8a (z), E RR In this scenario, 2; follows normal
distribution (I((ta, IN) according to equation I The score function is then tractable as
v log (x,,) = x (tt) Substituting this into equation
3 gives:
a()
dx o(t) x + ji(tta
(t) a(t) u(t)a
(28)
dt att)
We aim to keep de constant, which requires must be constant, and p(t) must be a linear function of t.
* 0 and ju(t) be constant. This implies that (t)
Given the boundary condition that oy-o approximates data distribution and x:-T approximates
known distribution, the only feasible choice is to set (() to a constant u(t)= = 1 - 1/T. This satisfy all the schedule constraints noted under equation except (0) O, which is approximately satisfied by choosing a to be a small value. In practice, we use value for a that is two orders of magnitude smaller than the data scale, yielding good results. We can also rescale the time in [0, 1]. leading to the following schedule for the diffusion process:
St = (1 1)aso + oe, L E 1),
(29)
which ensures a straight-ling trajectory under a special data distribution assumption.
Next, we extend the result into general data distribution in the following theorem.
Theorem A.1. For a general data distribution and our process schedule x; = (1 - -))x0 + oe.
t e [0, 1}, the following inequality holds for each data dimension i:
dar
(8)
(6)
P(
8)
(30)
dt
82 (1-
As σ + O, the term das, 2;
dt 1-1 converges to zero probability. More specifically, the solution to the
dm, equation db I-t 0 is that 1-r becomes constant, which corresponds to linear trajectory.
24

---

In practice we set o to be small and when t < 1 - the trajectory is approximately linear. The proof of the theorem needs two following lemmas.
Lemma A.2. For a general data distribution f(x)) and a general diffusion process with the schedule 2, = p(t)zo + (t)e, its ODE description in equation 3 can be rewritten dæ. di
o(t) o(1) (at (((l)_[(oo)
ju(t)E(aol x
(31)
Proof of Lemma A.2 For the general schedule, we have pt xx)) ~ ((u(t)xx, We start by considering the score function
S f(x)) (((co)Von aa 2d00
Va (xx))
p(zx)
f (( |20)
(ff))m
)doo
a(1)²
(32)
p(x))
3t
(t)2 1 2 22)]
pu(t)
(t)²
Substituting this into the ODE form of the diffusion process, we have:
dæ ia(t)
((())(()) a(1)2((1)
m(()
B B(
dt m(t)
(()) (t)2
o1))
(33)
((1) (at ((t) (x:))
+ (t)
Lemma A.3. Let be random variable that satisfies p(a(z)) ~ N(xx, N( (tt)//u(tt))
Define
E[zo [a] a. Then, the following properties hold:
1. Ely] 0,
2. Var (
S (11)//11/))00))) i = 1,..,, N, where yf is the 8th component of y-
Proof of Lemma A.3
The expectation of y given by:
Ey
[[E[xo|a] al = E, |a [2o a) = Ex, Eajao (x0 a) = 0
(34)
For the i-th component (1) compute the variance:
Varly(0) = (1) y²] = La( 119
(35)
I E|Ezz/ (1x ( Ezg Balzo (= -
The inequality follows from Jensen's inequality.
Proof of A.
For our specific schedule (t) = a and put) = 1 - t, equation 31 in lemma A.2
reduces to:
dæ, E E xz).
(36)
dt
For simplicity of notation, we define a = x, (t),, satisfying p(a[xo) ~ ~ (1))/((())) By applying lemma A.3 the defined y E|zo a) a satisfies E[y] and (ar < (1)2//((13 (I-1y2, for all dimension i.
Applying Chebyshev's 's inequality, get:
Var (1)
(( 2 5)
(37)
82
82 t)²
Thus, we have the inequality:
da
d²
P(
(-x)) /11 f))
2 8) <
(38)
dt
82 - -1
25

---

A.5 Supplementary Proof for Sampling
Lemma A.4 (Conditional Expectation Minimizes MSE). The conditional expectation م [xe]
is the estimator of -t-At At given that minimizes the mean squared error (MSE). That is, for any estimator h(xs), the following inequality holds:
Ei(x:-1 - h(<x))² - -()))))
(39)
Proof. We begin by expanding the conditional squared error term. By the properties of conditional expectation, the cross-term vanishes:
B((x; - 1 h(==))) 2 |ars] ((( 1B(xr- B(a, am) h(xx))) (
[2xe] B|(x)-- - (B --n-hhhe h(x:)))
E|
[[((((2)-21/ [22]] ((x)))) |ao]]
(40)
[E( 1(x [ [[E (xx
[ -f xx (= [
Taking the expectation with respect to Hi, we get equation 39 which completes the proof.
Proposition A.S. For the straight-line diffusion schedule defined in equation 5 we have
1 (t
At) 1 (x: + o V'V pr
E E(x
(41)
Proof. From equation 32, the conditional expectation æo given a; is:
((() (ær + ( log (x:))
(42)
For æt- Ar, the conditional expectation is derived as follows:
E E(2;
[x]]
- s1-Aip(ta-s//xdddee
f x1- / p( ( )dzddx
/ (J (1--IDlEe- bo p(zool
(43)
/ plt (t)
p(t At) xz).
Here, we make the assumption that x, and zt-At at are independent given Zn- In fact, the specific temporal dependency among the random variables (rx)r=o,....) defines a particular sampling method within the broader family of valid samplers characterized by equation 7 Our subsequent work Ni) eet al. [2025] generalizes this framework beyond the SLDM schedule and introduces the sampling approach used in this paper as StoMax (maximal stochasticity) sampling, which corresponds to the variant with the highest degree of randomness within the family.
Substituting [x from equation 42 we have:
E E(x;-At Ait [a:]] = p(t At)
(ar + ( (or))
(44)
((t)
Adopting the straight-line diffusion schedule defined in equation 5 we produce equation 41
As proved in (Xue al. 2024b), ) there are family of reverse processes that share the same marginal probability distributions as equation 2 and equation 3 When applied to our process schedule and apply MarernMa method discretization, obtain the following iterative algorithm:
1 - t At
At V (x.)) + 28(1) 1- oze.
At
+ (1 8(0)
(45)
1
26

---

uses a Gaussian distribution to model the backward probability p(z xz), whose expectation is 1-++A/ 2n + (1 + B(t)) ~ log (xt). According to Lemma A.4 the optimal iterative step that that minimizes MSE should satisfy I-t x2 + (1 ((t)) Al Vo Log (())) 1- Ar)
equation 45
(Et + V log (x:)). This result in B(L) A7
1-t
A.6 Temperature Control
For Langevin dynamics, the sampling temperature can be controlled by scaling the stochastic term with constant T.
1 () log) + Tg(()dw'.
dæ
(46)
where T is the temperature parameter. Then ((x) & + the stationary distribution of the process in equation 46, as proved as follows:
Proof. The marginal probability density (e(x) evolves according to Fokker-Planck equation
@m((x) V a
1 (000) (og 2.. (x()))
(47)
For the stationary distribution, the probability density becomes -im-inndeenndet,, i.e. Thus, we solve:
0x,(w)
8t
V V log (0)
2 - v.v. v 1 ((((()((()))
(48)
We can easily validate that (xx) x p(x)) + satisfies the stationary equation.
Thus, higher temperatures oo) increase diversity, with (<))) approaching uniform distribution. Conversely, lower temperatures (7 -> 0) enhance fidelity, with (x: converging to a 5-distribution at the global maximum of p(a:).
However, prior work has shown that this low-temperature sampling in diffusion models fails to effectively balance diversity and fidelity in image generation, often resulting in blurred and overly smoothed outputs [Dhariwal and Nichol 2021]. We also notice that this low-temperature sampling approach applied to straight- line diffusion fail to fully cover all high-density regions of the target distribution, as shown in Figure(5:.
To address this, we propose an empirically designed time-annealing schedule that introduces higher stochasticity during the initial stages sampling, allowing for the exploration of distant probability density maxima:
36. A: 1-t+ At (æ, + - = t" V20e,
(49)
where v controls the decay rate of temperature over time. Larger // enhances molecule stability and enables the model to cover all modes, shown in Figure The low-temperature sampling helps mitigate the disturbance caused by noise in the training data, and thereby can improve the quality of the samples. The default value of for molecular generation is analyzed through an ablation study in Section C.5
A.7 Modeling Invariant Probability Density for 3D Coordinate Generation
In this section, we aim to prove that the probability density of the generated atomic coordinates. as produced by Algorithms 3 and is invariant to both translations and rotations. Formally, we aim to establish that for the Probability Density modeled by our model p(æc)) = p(æ. + t) and p(z0)) = pl Rao), where t is translation vector and R is an orthogonal matrix representing rotation. The proof in the same spirit of that in Hoogeboom et [2022], Xu al. 2022]. The
result can be seen special case of Theorem and Theorem 2 in Zhou al. 2025
27

---

T
0.1
Servate
u=0 v=
0.5
Figure 5: a. Generation results from straight-ling line diffusion with vanilla temperature control as defined in equation 46 Training data (black points) represent 2D Swiss roll coordinates with added Gaussian noise, where the highest density region lies in the interior the spiral. Diffusion-generant samples (yellow points) exhibit reduced diversity regions, favoring those near the origin. b. Generation results from straight- diffusion with our annealing temperature control as described in equation 49 Faster temperature decay (larger v) leads to concentrated samples in high-density regions, successfully covering all local maxima.
decreases but fail to cover all high-density
To ensure translation invariance, the generative process defined in the quotient space translations, specifically the zero Center of Mass (CoM) space. This is achieved through two key operations. First, noise is sampled from a CoM-restricted Gaussian distribution
NCuM (0, J3as ), which sampled by first sampling a standard Gaussian noise vector c' ~ NO, JaM ) and then subtracting its center of mass:
3MT EiM é Second, the network's output at each step projected into the zero CoM space. These operations ensure that all intermediate coordinates ær f = 0,... remain strictly within the zero CoM space throughout the generative process.
é' +
To establish rotation invariance, we rely on two key properties. First, we use an equivariant neural network satisfying (()) (Ræ,, 1) = RØ(*)) (ax, l). Second, we utilize the rotational invariance of zero mean isotropic Gaussian distributions. This property can also be extended to the CoM-restricted Gaussian distribution, whose probability density function is: JNNw (x) (2m) 1172 exp( |
Then, we can verify SNa (x) [Neas (Rx) for any orthogonal rotation matrix R. Now we prove the rotation invariance the generation probability as follows:
At each iterative step of the generative process, we have:
1 (t - At)
Rt-At ~ -NcMM
(x) - o. ((((,,,))) 262" Isad
(50)
1-t
1-(1- (1-At)
(Ra, - a 1b( 18(Ret,
p( (.Ra,-Ai (xx;) = fNNw
1-t
V2t"o
B1-A1
(-(t-Ar)
(I) (C((I)))))
= -Noa
(51)
V2t"a
It-At
1-(1-At) 1-1
(x,
p(z,,))))
= -Nea
p( ((00--)ll
/28hg
In the second equality, we apply the equivariance property the neural network. The third equality follows from the rotational invariance of the isotropic Gaussian distribution.
28

---

Additionally, the initial distribution p(z1) NCM o IsM) is rotation invariant. Combining these facts, we can propagate rotation invariance across the generative process. Thus, for the final generated distribution:
p p(Rxo) = f ...
II pl p (Re,) dx 1=Ar
(52)
f / II p(=:-a/lm)ppaa)dda ... drst = p(æa). t=Ar
In the second equality, we use the rotational invariance of both the transition probabilities and the initial distribution. This proves that the final probability density of the generated data is rotation invariant.
B
Molecular Generation Algorithms
Algorithm 3 Straight-Line Diffusion Training for Molecules (UniGEM)
Input: 3D molecular data 2 [ ho] with M atoms, 2o CoM space, neural network ø. variance constant nucleation time in E [0, 1]
repeat
Sample t { fn,) ((t,, 1])
Sample 6 ~ NCaM 0, 73M)
Compute 2; = - t) + oE
Minimize |c (b() (x,,, t)l Legr. ho (xI,
until converged:
Algorithm 4 Straight-Line Diffusion Sampling for Molecules (UniGEM)
Input: Number of atoms M, neural network c, variance constant nucleation time tn, sampling steps T, temperature annealing rate
Sample 21 .Ncum JaM)
for = T to 1 do
f t= i/T, A1 1/T
0t-A1 (t)
(xt - o o(x,, t))
1-t
Projecting Øt-Ar into zero CoM space ifi 1 then
Sample NCoM (0, Isas)
Ft- 2t- + 1" V 6
end
else
ho 6(h) (xo: 0)
end
end
return 20
Fo, ho]
C Supplementary Illustrations and Results
C.1 Sampling Trajectory of biffusion-basee Models
To evaluate whether SLDM exhibits year-linear trajectory under general data distributions, as suggested by Theorem 3.1 we visualize the ODE trajectory and compare it with other diffusion- models. We consider scenario where the data distribution as this setup offers tractable score function and serves as representative example of general distributions. The resulting trajectories are shown in Figure Our method maintains trajectory that is consistently closer to straight line compared to other approaches, leading to smaller truncation
one-dimensional Gaussian mixture,
29

---

-00
62
8FN SLOM
Figure 6: The trajectory of the ODE equation of various diffusion-based models. The data distribution is mixture of Gaussians in one dimension, defined as zo ~ 0.5 (2, 1/4) + 0.5 (- 1/4). The background color indicates the value of probability density p(z+). During sampling, the timestep decreases from 1 to O.
errors under first-order numerical discretization. As suggested in Theorem 3.1 the trajectory slightly deviates from a straight line the early stages (i.e., as t approaches 1), which could increase the sampling error in theory. However, our empirical observations indicate that the sampling process is relatively robust to such errors during these initial steps. This robustness may be attributed to the incorporation of Langevin dynamics in the sampling algorithm, which can mitigate the impact of early-stage inaccuracies. As result, SLDM achieves an overall lower sampling error.
In comparison, DDIM demonstrates a relatively linear trajectory during the early stages but exhibits significant curvature in later stages, where accurate sampling more crucial. This could potentially degrade its performance, For the FM method, where the initial distribution is Gaussian and no explicit dator-data joint distribution is predefined as Tong al. [2023]. it fails achieve a straight- line trajectory. Other methods also demonstrate an evident curved trajectory. Besides, we can also see from the illustration that the distribution remains nearly static during the later stages of BFN sampling, which aligns with the discussion in section 3.2 and suggests potential inefficiencies of time scheduling.
C.2 Sampling Efficiency Comparisons under UniGEM Framework
We complement the ablation study in Section 4.4 by incorporating results with diverse sampling steps. The results demonstrate that while UniGEM enhances the performance of EDM and BFN when sampling step is abundant, these methods still struggle to achieve satisfactory molecular stability in the low sampling step scenario.
C.3 Sampling Efficiency Under Joint Diffusion Framework
A natural question arises: can SLDM achieve strong performance without the UniGEM framework? Specifically, can it jointly generate atom types and coordinates using diffusion models, following the approach of EDM and GeoBEN? To address this, introduce variant called SLDM (joint diffusion) as a preliminary exploration, where atom types are generated using an SL.MM model tailored for discretized data, replacing UniGEM's prediction strategy. The training and sampling procedures of SLDM (joint diffusion) for molecule generation are detailed in Algorithms and respectively.
Unlike UniGEM, follow GeoBEN represent atom types by their atomic numbers (positive integers) and adopt the binning technique from BFN to handle discretized data. Here, Pp.,.r, (b) denotes the discretized cumulative distribution function (CDF), defined F(6) [[ /22 with Pbb F(b) for - < < P(b) = 0 b < 1,, and P(b) for 2 "NearestCenter" decodes the generated continuous continuous normalized atomic number into atomic type by
b- (-=)),
30

---

100
Molecular Stability (%)
60-
40
SLDM
GeoBFN
20
GeoBFN(U)
EDM
EDM(U)
500
NFE (in log scale)
Figure 7: Comparison of molecule stability (1) across diffusion-hars based molecular generation models with UniGEM framework on QM9 unconditional generation, evaluated with respect to the Number of Function Evaluations (NFE) during the sampling process.
finding the nearest center of bins (normalized atomic numbers), then unnormalizing it and mapping the corresponding atomic type. Notably, there remains substantial room for further exploration regarding the handling of diffusion for discretized data, which we leave for future work.
Algorithm 5 SLDM Training (joint diffusion)
Input: Number of bins K E N (equal to the largest atomic number), fixed variance a, 3D molecular data = [zo ho] with M atoms, 2n E CoM space, normalized atom number ho
(2 * ho 1)/K E * 1,
repeat
Sample / (1(0, 1), 5Th No. JaM), eh ~ NO, JaM)
Compute 2, = (1 t) + 06
Compute h = (1 h + oc
Network output: 2td fb, of D( ht,t)
h 6n exp((nod]) 1-t
Minimize denoise loss:I|e" = ēn1 (pth -E") - Ei Pa (b:)(b: (bi)((b - (mn)l?
until converged;
Results are presented Table 4 The proposed temperature annealing mechanism endows the model with greater flexibility, enabling a balanced trade-off between stability/validity and unique- ness/diversity. We manually tuned the temperature annealing rates for coordinates numbers (P)), and empirical observations indicate that setting > Vy, yields better performance in achieving such balance. should be noted that we did not conduct exhaustive hyperparameter searches, leaving room for potential further improvements. ...M (joint diffusion) outperforms EDM across all metrics and exhibits superior stability and validity with slightly lower uniqueness compared to GeoBEN when the number of sampling steps T is limited. Notably, the overall performance of this variant still inferior the original SL.DM which employs the UniGEM framework. This finding reaffirms that the UniGEM framework may be more optimal approach for generating small-molccule.
(v²) and atomic
C.4 Toy Data Results
We evaluate our model several 2D toy data to test its generality. The datasets included swissroll and moons to represent continuous data distributions, as well as chessboard simulate discretized data distributions. The dataset consists of 100,000 samples. For the moons and swissroll datasets,
31

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Algorithm 6 SLDM Sampling (joint diffusion)
Input: Number of atoms M, neural network o, variance constant o, nucleation time sampling steps T. temperature annealing rate V2 and P/
Sample 21 a. a.NcoM (0, JaM), , h1 (N IsMs)
for = T to do
t i/T, At 1/T
Network output: 6", p Iu o - = ( h1,
jin - P Oh gexp(lnok)
1-1
Update 01- (-(t-Ar) (20-- -
I-1
Update ho Ei P. os (bi)bs, he- (1 (t ho) Projecting D1-Ai into zero CoM space if >>>> 1 then
Sample 6" ~ (C TaM), 6" ~ ~O. 3MM)
Dt- Dt-At + t 2 60
h : h1- + 1h v 6
end
return
Bg. Neares)Center(ho))
Table 4: Sampling efficiency comparison under joint diffusion framework QM9 dataset uncondi- tional generation task.
Method
Atom Sta (%)
Mol Sta (%)
Valid (%)
VxU (%)
EDM (T=1000)
98.70
82.00
91.90
90.70
GeoBEN (T=1000)
99.08
90.87
95.31
92.96
SLDM (joint diffusion) (T=1000)
99.01
91.57
95.20
91.57
GeoBEN (T=500)
98.78
88.42
93.35
91.78
SL.DM (joint diffusion) (T=500)
99.89
90.57
94.00
91.11
GeoBEN (T=100)
98.64
87.21
93.03
91.53
SLDM (joint diffusion) (T=100)
98.92
90.31
93.84
89.36
GeoBEN (T=50)
98.28
85.11
92.27
90.72
SLDM (joint diffusion) (T=50)
98.70
88.09
92.84
89.53
performed training with 40 diffusson steps and 100 epochs, and for the chessboard dataset, the training was extended to 600 epochs with 100 diffusion steps to ensure convergence for all models. Other settings are kept the same for all datasets: The model is 5-layer MLP. The batch size set to 2048, and the optimizer used was Adam with a learning rate of 0.001. No temperature control is used during sampling. These settings follow consistent across all generative algorithms to ensure a fair comparison. The results are provided in Figure The data generated by our model show the closest alignment the original distributions. Specifically, our model produced samples with fewer outliers and achieved good coverage of the data distributions. These findings highlight the superior generative capabilities of our approach in faithfully modeling complex data distribution.
comm/albarittoy-ddfffussono/ and are kept
C.5 Ablation Study for Temperature Control
Theoretically. a larger temperature annealing rate corresponds to faster cooling scheme. The results in Table 5 demonstrate that adjusting the temperature can effectively enhance molecular stability. However, extreme values of may significantly reduce diversity. We selected v = 0.5 the default setting for our model, as it achieves the highest x V score, indicating the optimal balance between diversity and fidelity.
32

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DOPM
Figure 8: Generative performance comparison of Straight-Line Diffusion, DDPM, and BFN on 2D toy datasets.
Table 5: Impact of temperature annealing rate evaluated on the QM9 unconditional generation with sampling step T 50. Higher values indicate better performance.
Atom sta(%
Mol sta (%)
Valid(%)
U V(%)
SLDM(v==)
99.14
91.74
95.40
92.76
(LDM()=0.5))
99.28
93.03
96.20
93.28
(LDM(v=1)
99.27
93.46
96.08
92.42
SLDM(v=3)
99.37
94.34
96.84
90.08
SLDM(v=5)
99.42
94.83
97.33
86.10
SLDM(v=10)
99.53
96.06
97.98
75.13
C.6 Robustness to Different Prior Distributions
Empirically, find that SLDM maintains high validity and stability even when the initial distribution during sampling deviates from the theoretical valid one, i.c., Gaussian distribution with standard deviation a, as shown in T This robustness helps mitigate the impact of deviations from the ideal linear trajectory in the early sampling steps, as discussed under the Theorem 3.1
Table Robustness of SLDM to different initial sampling distributions on the QM9 unconditional generation task. Here N00 7) and the sampling step 50.
Prior Distribution
Mol_stab (%)
Atom_stab (%)
Validity (%)
V*U (%)
(theoretical)
93.43
99.32
96.10
93.31
2.
91.81
99.12
95.00
92.79
0.1 .
93.31
99.29
96.07
93.46
0.01 o
92.94
99.24
96.06
93.06
33

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Figure 3D molecules generated by the SLDM model trained on the QM9 dataset.
EDM
Geo-
BFN
SLDM
Figure 10: The generative process of EDM, GeoBEN and SLDM models.
C.7
Visualizations of Generated Molecules and Sampling Process
We have randomly generated 40 molecules using trained models on QM9 in the unconditional setting, and visualized their 3D structures using py3Dmol, as shown in Figure 9 These examples clearly demonstrate the structural plausibility and diversity of our generated molecules.
In Figure we compare the diffusion processes different models by adding noise to the same molecule s coordinates, enabling straightforward visual comparison of their noise schedules. As a complement, we present a comparison of the practical sampling processes of the three models in Figure Although generating the exact same molecule across models is not feasible this case, still arrive at conclusions analogous to those in Section of the main text. Additionally, unlike the noise-adding process, the generation process of SLDMM demonstrates more pronounced randomness.
D Related Work
D.1 An overview of 3D Molecular Generation
Recent advances 3D molecular generation can be categorized based on the underlying generative algorithms: Autoregressive methods generate molecules step by step, progressively connecting atoms or molecular fragments. G-SchNet [Gebauer et al.) [019) and G-SphereNet [Luo and Ji 2022) are early examples that use this strategy to model 3D molecular structures. Building
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these, Symphony (Daigavane et al., 2024] incorporates higher-degree E(3)-equivarian features and cessage-passingg to improve the modeling of molecular geometries. Normalizing flow [Chen et al.
2018 has also been applied to molecule generation by E-NF [Garcia Satorras et al. [202] and Köhler
et al. 2020]. Diffusion-based models have gained prominence for 31) molecular generation recently. Hoogeboom et al. (Hoogeboom et al., [022] introduced the Equivariant Diffusion Model (EDM)
to jointly generate atomic coordinates and atom types. Extensions of EDM include EDM-Bridge Bridge
[Wu et al. 2022). which enhances performance through prior bridges, and GeoLDM [Xu et al.
2023]. which performs diffusion in a latent space using an autoencoder. EquiFM [Song et al. [2024]
employs flow matching for efficient molecule generation, and GeoBFN [Song et al. 2023a combines Bayesian Flow Networks with distinct generative algorithms tailored for discretized charges and
continuous coordinates. END [Cornet et al. 2024 proposes learnable data- and time-dependent
noise schedule of the diffusion process, and achieves improved sampling efficiency.
Several studies offer complementary strategies to generative algorithms. First, representation- conditioned generation has been explored by MDM [Huang et al.) [023a], which conditions on VAE representations, and GeoRCG (Li et al. 2024]. which extends EDM by leveraging pretrained
molecular representations. Second, some studies propose advanced network architectures to enhance
2024a| Irwin et al. 2024 [Huag et al. 2023b Mercatali et al. 2024]. Third, additional input information has been introduced to the
molecular generation (Hua et al. [2224] Huang et al. 2023a Le et al.
molecular generation. For instance. MolDiff [Peng al. 2023 explicitly predicts bonds during
generation. MiDi [Vignac et al. 2023]. JODO [Huang et al. 2023b]. EQGAT-diff [Le et al.) [024a]
and SemlaFlow [Irwin et al., 2024] further extend generation to include bonds and formal charges, enriching the molecular inputs. Twigs [Mercatali et al. 2024), on the other hand, integrates additional molecular properties into the diffusion training process, leading to enhanced conditional generation.
worth noting that evaluation strategies for 3D molecular generation can be different across methods. EDM [Hoogeboom et al. 2022] employs strict rules for bond definitions based on in- teratomic distances, implicitly enforcing constraints on bond lengths and steric hindrance. In this framework, the metric "stability" is rigorously defined, requiring correct valency and neutral atomic charges. Our method, along with most diffusion- approaches [Wu et al. 2022 Xu et al. 2023 Song et al. 2023a], J adheres to this evaluation standard. In contrast, another line of studies VVignac ct al.) 2023 Le et al.] 2024a Irwin et al. 2024) infers bonds and formal charges directly from the model, allowing atoms to have non-zero formal charges. Therefore, the bond inference imposes no constraints on bond lengths or steric hindrance. Further, the "stability" metric is defined more loosely, permitting discrepancies between valency and the number of covalent bonds. This relaxed evaluation framework makes it challenging to directly compare these two approaches. We advocate future efforts to establish more equitable evaluation methods to fairly assess the strengths of both paradigms.
D.2 Related Srlfuusion-Base Studies and Key Differences with SL.DM
Noise Scheduling It is important to note that our proposed process schedule differs fundamentally from previous works on noise scheduling [Karras et al. 2022 Nichol and Dhariwal 2021 Le et al.)
2024b). time discretization strategies [Xue et al. 2024c Li et al. 2023], J. and adaptive step size methods [Lu et al. 2022). These approaches can be interpreted as applying a time transformation T(-) that jointly rescales the diffusion process schedule as (T(()) and (T()). Notably, such transfornations preserve the monotonicity and endpoints of the schedule functions. In contrast, our method decouples μ and σ, and fundamentally alters the monotonicity and endpoint of ot)).
Linear (t).. Although our method is derived from the diffusion perspective, its process schedule shares some similarities with flow matching (FM) algorithms, such as FM-OT (Lipman et al. 2023 and conditional flow matching (CFM) [Tong et al. 2023]. Both methods employ a linear (t).. and CFM introduces additional low-scale noise. From this perspective, our process schedule can also be interpreted as a variant of CFM. Specifically, our approach employs a prior distribution modeled as a small-scale Gaussian distribution centered at the origin. However, FM and diffusion differ fundamentally in their perspectives: FM models the generative process as an ODE, while diffusion models stochastic process. This core distinction leads to differences in both the learning targets and sampling methods. Therefore, unlike FM that learns the velocity field and relies on ODE-based sampling, our approaches focus learning the noise and employ stochastic sampling.
Straight Sampling Trajectory. Moreover, similar to our approach, flow-based methods also aim at flows with straight trajectories. However, these methods rely on predefined prior-data joint distribution
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to produce straighter paths. This distribution is procured by solving an optimal transport (OT) problem during the training of flow matching [Tong et al., 2023 Song et al., 2024), and solving such problem is often challenging. When the OT solution is unavailable, achieving straight-line trajectories typically requires additional distillation steps or solving optimization problems, as studied in rectified flow
[Liu et al., 2022]. progressive distillation [Salimans and Hoj [022]. consistency model [Song et al. 2023b Luo et al. 2023) and optimal flow matching [Koriilvv et al.) 2024]. NFDM [Bartosh et al.
2024] proposes learnable forward process to adapt to align with the reverse process and introduces penalties on the curvature of the reverse process's 's trajectories. In contrast to these existing methods, approach offers more straightforward solution for achieving straight trajectory, by designing novel diffusion process that minimizes the rccond-order derivative of the trajectory.
E
Implementation Detail
The hyperparameter settings for molecular generation are detailed in Table Settings follow UniGEM
[Feng et al., 2024), with two additional tunable hyperparameters introduced by our generative algorithm: the noise variance and the temperature annealing rate
Table Network and training hyperparameters.
Network Hyperparameters
Value
Embedding size
256 for unconditional generation, 192 for conditional generation
Layer number
9 for QM9, 4 for Geom-Drugs
Shared layers
Training Hyperparameters
Value
Batch size
64 for QM9, 128 for Geom-Drugs
Train epoch
3000 for QM9, 32 for Geom-Drugs
Learning rate
1.00 x 10-1
Optimizer
Adam 10 1000
Sample steps T
Nucleation time
10
Oversampling ratio
0.5 for each branch
Loss weight
for each loss term
Generative Algorithm Hyperparameters
Value
Noise Variance
0.05 for unconditional generation, 0.1 for conditional generation
Temperature Annealing Rate v
0.5 for unconditional generation, for conditional generation
Non-uniform Discretization
False if > 13
For QM9, it takes approximately 10 days on a single A100 GPU, For GEOM-drugs, it takes approxi- mately 16 days on four A100 GPUs. While the training epochs are longer compared to the baselines, this extended duration is not mandatory. Although longer training leads to slight performance gains, found that the mol stable and atom stable metrics remain stable around 2000 epochs on QM9 and 13 epochs on DEOOMDruuge during the training process.
For sampling steps greater than 13, the geometric straight-line diffusion use uniform time dis- cretization like GeoBEN and EDM. However, according to theorem 3.1] our trajectory exhibits a larger second-order derivative at the beginning of sampling. Therefore, more efficient discretization strategy is to use fine- fine-grained discrctization for larger values. We manually set an empirical dis- cretization strategy that yields a 1% to 10% improvement in Mol Stable when T S 13. For sampling steps greater than 13, the impact on the results is less significant (< 1%). We leave the exploration of the optimal discretization strategy for straight-line diffusion for future work.
For conditional molecular generation, we follow the baseline approaches [Hoogeboom et al. 2022 by incorporating property values as additional inputs during training and sampling them from prior distribution during inference. The results are measured using property classification network trained on the first half of the QM9 dataset, while the remaining portion is used for training the generative model. addition to generative model baselines, we include two basic baselines: Random and Natoms The Random baseline involves shuffling property labels in the training data and evaluating
36

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the property classifier on the shuffled data. The Natoms baseline uses the property classifier network to predict the property based solely on the number of atoms the molecule. The property classification network is used for evaluating the generated molecules. The metric is the MAE between predicted properties and target properties. As the conditioning machanism, property values are concatenated to atom types as EGNN's invariant input, which is in line with other baselines.
37


Reviews:
[{"review_id": "5aWzFyvXRu", "reviewer": "Reviewer_e8rx", "summary": "This paper proposes Straight-Line Diffusion Models (SLDM) for 3D molecular generation. The authors design a special noise schedule for training diffusion models, which then can be used to generate novel molecules with time-annealing sampling. Experiments shows SLDM has noticeable advantage in generation speed and quality compared with other models.", "questions": "See Weaknesses.", "limitations": "N/A", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. Research on fast molecule generation is important for the AI for Science area as the generation speed of existing methods is still unsatisfactory.\n2. The paper is clearly written and easy to follow.\n3. SDLM indeed shows strong performance when sampling with only 50 steps. It seems very promising. Overall, SDLM could be a valuable contribution to the molecule generation community.\n\nWeaknesses:\n1. Recently, fast sampling from diffusion models usually relies on ODE solvers instead of directly sampling from the SDE trajectory. Could the authors add ODE results for the experiments in Section 4? I wonder if using ODE solvers can get even faster generative model.\n2. It would be better if the authors can provide some qualitative examples of the generated molecules to help readers intuitively understand the generation quality.", "was_revised": true, "final_justification": "After reading the other reviews and rebuttals, I would like to thank the authors for their additional results and detailed explanations. They further solidify the methodology proposed in the paper, especially on the faster sampling speed. Therefore, I agree with the other reviewer's consensus opinion on accepting the paper. I changed my rating to 'accept' accordingly.\n\nI encourage the authors to include these new discussions and quantitative results in the revised version as promised.", "rebuttal": "We sincerely thank the reviewer for the thoughtful comments. We will carefully revise the manuscript to address these points.\n\n> Fast sampling from diffusion models usually relies on ODE solvers instead of directly sampling from the SDE trajectory. Could the authors add ODE results for the experiments in Section 4? I wonder if using ODE solvers can get even faster generative model.\n\n**Reply:** Indeed, ODE-based sampling is a common strategy to accelerate generation in diffusion models. In our work, this is taken into account when comparing with the baseline EquiFM, which leverages flow matching with multiple efficient ODE solvers. We have already compared with the best result of EquiFM in Figure 1 and Table 1, showing that SLDM outperforms in both efficiency and sample quality under comparable settings.\nTo further address your suggestion, we will include EquiFM results in Table 2 for the conditional generation task on QM9. The following table summarizes the results (lower is better): \n\n| Property | α    | ∆ϵ  | ϵHOMO | ϵLUMO | μ     | Cv    |\n|----------|------|-----|--------|--------|-------|-------|\n| EquiFM   | 2.41 | 591 | 337    | 530    | 1.106 | 1.033 |\n| SLDM     | **1.46** | **440** | **320**    | **348**    | **0.797** | **0.745** |\n\n\n>  It would be better if the authors can provide some qualitative examples of the generated molecules to help readers intuitively understand the generation quality.\n\n**Reply:**  Thank you for your constructive suggestion. We have randomly generated 40 molecules respectively using our trained models on QM9 and GEOM-Drugs in the unconditional setting, and visualized their 3D structures using PyMOL. These examples clearly demonstrate the structural plausibility and diversity of our generated molecules.\nUnfortunately, the rebuttal phase does not allow us to include external links or supplementary material. However, we will definitely include representative visualizations in the appendix of the revised manuscript to provide readers with a more intuitive understanding of the generation quality.\n\n---\nWe sincerely thank the reviewer for recognizing the strong performance and contribution of our work. We've open-sourced our code at submission, as we genuinely hope our method can be useful to the community. We’d greatly appreciate a more favorable score to help increase the visibility and impact of this contribution! If there are any further questions or clarifications needed, we would be glad to continue the discussion."}, {"review_id": "4LEA5ChO0v", "reviewer": "Reviewer_1HAM", "summary": "This paper presents a straight-line diffusion method (SLDM) for generating molecules in 3D. A unified framework is presented to understand truncation errors that lead to low-efficiency sampling across existing diffusion, flow matching, and Bayesian flow methods for molecular generation, framing them all as continuous-time ODEs. It is observed that large-magnitude second-order terms in the numerical approximation to the backward solution trajectory of the ODE necessitates small step sizes that limit efficiency. To mitigate this, SLDM is proposed to enforce a near-linear sampling trajectory that minimizes second derivative magnitudes and allows for larger step sizes and higher efficiency generation. Additional algorithmic components are introduced to improve sampling fidelity, including a Langevin dynamics term with time-annealing temperature control. Results are included and compared to baseline methods for unconditional and conditional generation with QM9, GeomDRUGs, and toy datasets.", "questions": "1. Could you clarify why the SLDM formulation results in the deviation from linear sampling at early steps, ultimately correcting to linear? \n2. After line 154 (and in Figure 2), it is said that “the process unfolds uniformly from the origin, preserving the relative spatial relationships of atomic coordinates in intermediate states.” How is this so? I also don’t quite understand how the process unfolds from the origin, is this in the forward or reverse process? And is this a part of the implementation? Please clarify. One could imagine with a set of atom types initialized from the origin, there are many ways to unfold the process; if chemical/spatial relationships are preserved, you might observe that the final structure is essentially determined in early time steps. Is this what you see and is this beneficial? Or, do you observe that the initial unfolding involves some structure (atom-type) rearrangements, ie, that the early states have a degree of flexibility or stochasticity before the final molecule is largely determined? One might imagine this would be beneficial to prevent forms of memorizing from the initial origin (ie v high noise state), which seems to be what plagues other methods. Is this what's taken care of by time-annealing temperature schedule for the Langevin in (9)?\n3. Is Figure 2 showing forward or reverse processes? Sort of confusing matching with the main text description in 155–162 and noise scheduling in Figure 3", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The approach taken “[to] understand why existing methods suffer from low efficiency, we analyze the issue through the lens of truncation error in sampling” is well-posed \n2. Establishing a unified framework for assessing this over diffusion, flow matching, and BFNs is nice, framing them all as “as continuous-time Ordinary Differential Equations (ODEs)”\n3. The observation (line 31) that second-order terms can be large and connecting this why small step sizes are required to reduce truncation error (ie readout) is very useful. Proposing a solution to “strive for a linear sampling trajectory,\" i.e., to push the 2nd-order term toward 0 is also well-posed\n4. Introducing a linearly decaying $\\mu(t) = 1 - \\frac{t}{T}$ w/ small constant $\\sigma$ to guarantee a near-linear trajectory is really nice\n5. Strong results in Figure 1 (speedup w/ high stability) and Table 1 (generation quality metrics). In particular, it’s quite impressive to achieve rather high validity & stability while predicting bond types from coordinates, meaning that the geometries generated are quite good\n6. Introducing the Langevin component to mitigate observed initial deviations from linear sampling in the stochastic sampler is really nice\n7. Figure 2 demonstrating the slow diffusion, stable process, unfolding uniformly from the origin appears quite realistic and elegant (though see questions)\n\nWeaknesses:\n1. Predicting atom types based on coordinates instead of diffusing jointly. I see that UniGEM is referenced as having \"superior performance,\" but I don’t understand the chemical feasibility of generating 3D coordinates without knowing atom types, as the same set of atoms could have any number of reasonable geometries depending on what the atom types of each node are. Or, ie, the atom identities are typically thought to \\textit{determine} the geometry (and they do). So is the idea here to learn geometries of nodes that sort of implicitly contains atom-type information and purely from data, then fill in atom types post-hoc? This seems like it would limit generalization capacity, but I could be wrong. Please comment\n2. The baseline ($N_{atoms}$ classifier) for conditional generation is not super relevant – better would be to train a real classifier from, e.g., 2D graphs or 1D strings. Or, more realistically, to compare to other conditional or classifier-guided generation methods from the literature, particularly given that the conditioning mechanism here seems to be fairly simplistic, though it is barely formulated or discussed. More information on how the conditional generation is done would be nice, as this is an important setting\n3. Molecular stability is not included in GeomDRUGs, which leads one to assume it’s fairly low. This might be reasonable given it is challenging, but there are methods that can achieve strong performance (e.g., VoxMol). Please comment, strong results here would increase significance of the work\n4. Very slow training (Appendix E, line 822)", "was_revised": true, "final_justification": "During the rebuttal process, the authors added clarification to the questions raised by this reviewer. This reviewer's rating for clarity of the manuscript was already high at \"good\" and remains there, reserving the maximum score \"excellent\" for exceptionally clear manuscripts. However, it is because of the clarifications that this reviewer's confidence in the work's acceptance has increased, and the confidence rating has been increased accordingly.", "rebuttal": "Thank you for your recognition and the thoughtful suggestions, many of them inspired us to improve both the clarity and completeness of the paper. We will carefully revise the manuscript to address these points.\n\n> Unigem rationale discussion\n\n**Reply**: We would first like to clarify that the generation framework used in UniGEM is not the contribution of our paper, but we are happy to share our understanding on this design choice.\nWe fully agree with the reviewer that atom identities typically determine the geometry. However in de no vo molecule generation, both atom types and coordinates need to be modeled. Typical way is to jointly iteratively refine atom types and coordinates by diffusion model. However, modeling discrete variables (like atom types) in diffusion models is challenging. UniGEM notes that atom type generation by diffusion can lead to instability and mode oscillation (i.e., frequent switching between atom types during denoising), which in turn negatively impacts coordinate generation. By first generating continuous coordinates alone, UniGEM avoids this problem and achieves more stable and accurate 3D predictions.\n\nThis idea has been further supported by [1] (Sec. 4.3 last paragraph), showing that generating positions first and predicting bonds/types later leads to better results.\n\nRegarding:\"Is the idea here to learn geometries implicitly contain atom-type information and purely from data, then fill in atom types post-hoc?\"\nYes, that is indeed the key assumption behind UniGEM. \n\n[1] Piloting Structure-Based Drug Design via Modality-Specific Optimal Schedule\n\n> Conditional generation and baseline (classifier).\n\n**Reply**: The classifier is not part of the generative model, but is used post hoc to evaluate whether generated molecules match the conditioning properties. We apologize for any confusion and will clarify the setup in the revised version. All settings follow baselines. Briefly:\n\n- QM9 is split into two halves: one to train the property predictor 𝜙, the other for training the conditional generative model.\n\n- 𝜙 evaluates generated molecules; conditional accuracy is measured by MAE between predicted and target properties.\n\n- Conditioning machanism: property values are concatenated to atom types as EGNN's invariant input. Though simple, this method performs well and even outperforms recent guidance-based approaches:\n|Category|Model|Cv|µ|α| ∆ε|HOMO|LUMO|\n|-|-|-|-|-|-|-|-|\n|Classifier-guidance|EEGSDE|0.941|0.777|2.50|487|302 |447|\n|Training-free guidance|TFG-Flow|1.750|0.817|2.32|804|364 |941  |\n|Conditional generation|Cond-Flow|1.520|0.962 | 3.10|805|435|693  |\n|\"| Cond-EDM|1.065|1.123|2.78|671|371|601|\n|\"| SLDM|0.745|0.797|1.46|440|320|348|\n\nClassifier-guided methods require an auxiliary classifier at each diffusion step, which incurs high computational cost. Since our goal is to evaluate the ability and efficiency of the generative model, we chose not to include such methods, but we’re glad to include them if you find it helpful.\n\n> Lack of mol_stable metric in GeomDrugs and compare to VoxMol\n\n**Reply**: We agree that molecular stability is a key metric for 3D molecule generation.\nFor GEOM-Drugs, the low molecular stability is not specific to our method. As noted in [1], EDM-style bonding heuristics are not well-suited for this dataset due to its diverse structures. Even ground truth molecules score poorly under this evaluation (see Table below). Despite this, our model (SLDM) shows clear improvement, validating the benefit of our low-temperature sampling (lines 189–193).\n|| Mol Stab(%) | Atm Stab (%) | Valid (%) | Unique (%) |\n|-|-|-|-|-|\n|Data|2.80|86.50|99.90|100.00|\n|SLDM (T=50)|7.09|89.03|99.57|99.97|\n\n[1] GEOM-Drugs Revisited: Toward More Chemically Accurate Benchmarks for 3D Molecule Generation\n\nRegarding VoxMol, we appreciate the reference. VoxMol adopts a different evaluation setup, which use via OpenBabel to get bond and allows charged atoms. We re-evaluated SLDM under their protocol and outperforms VoxMol: \n||Mol Stab(%) | Atm Stab (%) | Valid (%) | Unique (%) |\n|-|-|-|-|-|\n|Data|99.90 |99.90|99.80|100.00|\n| EDM |40.30|97.80|87.80|99.90|\n| VoxMol|75.00|98.10|93.40|99.10|\n| VoxMol (Oracle)|81.90|99.00 |94.70|97.40|\n| SLDM (T=50)|84.88|99.26|99.58|**99.99**|\n| SLDM (T=1000)|**89.40**|**99.71**|**99.97**|99.92|\n\nAllowing charged atoms may overly relax evaluation and may be unsuitable for models that do not generate charges. Thus, we did not adopt this metric, but we’re happy to include the comparison in the appendix if you find it helpful.\n\n> Slow training\n\n**Reply**: Thanks for point this out. We compared the performance of SLDM with other baselines, such as UniGEM.\n||QM9||||GEOM-Drugs(GD)|||\n|-|-|-|-|-|-|-|-|\n|Model (Epochs, T=1000)|mol_stab|atm_stable|Validity|V*U|atm_stab|Validity|\n| UniGEM (QM9:2000, GD:13)|99.00|89.80| 95.00|93.20|85.10|98.40|\n| SLDM (QM9:2000, GD:13)|99.41|95.32|97.04| 90.48|87.87|99.55|\n| SLDM (QM9:2980, GD:32)(reported in our paper)|99.43|95.42|97.07|90.42|88.30|99.95|\n\nWe observes SLDM already achieves state-of-the art result under the same number of training epochs and training time. While longer training leads to slight performance gains, we find that the mol_stable and atm_stable metrics remain stable around 2000 epochs on QM9 and 13 epochs on GEOM-Drugs during the training process. This indicates that extended training time is not a necessary factor for SLDM’s performance advantage.\n\n> Clarify why the SLDM formulation results in the deviation from linear sampling at early steps, ultimately correcting to linear?\n\n**Reply**: The deviation from a linear trajectory in SLDM primarily originates from the trajectory crossing phenomenon in the forward diffusion process, as discussed in Rectified Flow. Although the forward noising process is defined as $x_t=(1−t)x_0+σ(t)ϵ$ which suggests a linear mean trajectory $(1 - t) x_0$, the added stochasticity introduces crossing between diffusion paths, i.e. the same $x_t$ can be derived from multiple different $x_0$. This crossing effect becomes especially pronounced at early sampling steps ($t\\to1$), where the noise dominates and leads to high uncertainty in the reverse mapping.\n\nSpecifically, in SLDM, the reverse ODE is given by Eq.36 $\\frac{dx_t}{dt} = -𝐸[x_0|x_t]$. If $𝐸[x_0|x_t]$ were constant, the sampling path would be linear. But due to forward crossing, $𝐸[𝑥_0∣𝑥_t]$ shifts over time: from dataset average at high t to more precise $𝑥_0$ as t→0, resulting in curved trajectories. This behavior is quantitatively characterized in Theorem 3.1 and visualized in Figure 6.\n\nNotably, this issue is common in diffusion models. SLDM controls this deviation within initial generative steps (Theorem 3.1), and our stochastic sampling helps mitigate error accumulation over time. Other approaches often rely on costly training techniques like distillation or solving OT to overcome this issue, as discussed in Appendix D.2.\n\n> Clarify Figure 2 setup, memorizing issue\n\n**Reply**: We apologize for the confusion. Figure 2 shows the forward noising process. We used this for consistent comparison across EDM, GeoBFN, and SLDM on the same target molecule.\nThat said, we also visualized the reverse sampling process (not shown here due to rebuttal limitations on links/media), and found consistent trends:\n- EDM produces very noisy intermediates throughout;\n- GeoBFN quickly settles, barely changing after early steps, underusing later steps;\n- SLDM shows less noisy and more progressive refinement. \n\nImportantly, our sampling method is stochastic, allowing flexible rearrangement of atomic positions. This helps avoid rigid memorization and promotes diversity. To demonstrate this, we show that SLDM trained on GEOM-Drugs can generate novel molecules not seen in the training data:\n|| SLDM (T=1000) | SLDM (T=50) |\n|-|-|-|\n| Valid&Unique&Novel|99.72%|99.93%|\n\nAs for annealed Langevin temperature in Eq.9, it enables fine-grained control over the stochasticity by injecting more randomness early (for exploration), and reducing it later (for precision), which we found helpful for balancing chemical stability and diversity.\n\n---\nIf there are any further questions or clarifications needed, we would be glad to continue the discussion. If our responses have addressed your concerns or contributed to a clearer understanding of our work, we would deeply appreciate your consideration for a higher score or confidence, which would greatly support the visibility and potential impact of our work within the community!"}, {"review_id": "UYwqroxfPX", "reviewer": "Reviewer_QjdK", "summary": "The paper proposes a type of diffusion model specifically designed for 3D molecular generation. The core idea is to choose a schedule \\mu and \\sigma of the SDE to achieve a near straight-line backward process. This allows the model to produce high quality samples with \na significantly smaller number of steps.", "questions": "1. In Appendix A.4, I don't quite understand the logic jumping from the toy example of delta distribution to a general data distribution. Is it just a loose analogy, without rigorous mathematical reasoning or intuition?", "limitations": "I do not have much to say about it.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n\n1. The goal of the method is to generate molecular samples with a small number of steps. The technical motivation for that is natural and clear: To reduce the error of first order discretization, choose a schedule of the diffusion SDE so that the backward process is linear. The exposition revolves around this motivation and is clear and self-contained.\n\n2. The experimental results shown in Table 1 and Figure 4 are impressive, demonstrating that the method achieves significant speed up over the baselines with better quality.\n\n**Weaknesses:**\n\n1. There are plenty of works focusing on accelerating diffusion generation for other domains such as image generation (e.g. [1]). In the experimental setup (Section 4.1) the authors only seem to compare to methods already used in molecular generation. I suspect directly adapting the progress in fast image generation to molecular generation can already give significant speed-up. More such experimental comparison is favorable.\n\n\n*References:*\n[1] Consistency Models, Song et al.", "was_revised": true, "final_justification": "Overall I think the paper is well motivated and the solution is on point. I keep my original positive rating.", "rebuttal": "We sincerely thank the reviewer for the thoughtful comments and for recognizing the motivation and contributions of our work.\nIf deemed useful, we will include the discussion in our paper. \n\n> I suspect directly adapting the progress in fast image generation, e.g. consistency model, to molecular generation can already give significant speed-up. More such experimental comparison is favorable.\n\n**Reply**: Thank you for this insightful suggestion. We fully agree that progress in accelerating diffusion models from other domains is potentially inspiring for molecular generation. We have discussed their connection with SLDM in Appendix D.2.\n\nHowever, in practice there are nontrivial domain-specific challenges that limit their effectiveness in molecular settings. For example, a recent blog post “Equivariant Diffusion for Molecule Generation in 3D using Consistency Models”  attempted to apply the Consistency Models to the 3D molecule generation task using the EDM framework (which shares the same architecture and evaluation protocol as ours). Although they achieved up to 24× speed-up, the best atom stability reported was only 19%, which is significantly lower than both our method and most baselines (>95%). In our view, this reflects a domain gap that requires careful adaptation, beyond simple plug-and-play transfer, underscoring the importance of developing specialized and efficient generative models for molecular data — which is precisely the goal of our work. We believe transferting image domain techniques to molecules is an important future research.\n\n> In Appendix A.4, I don't quite understand the logic jumping from the toy example of delta distribution to a general data distribution. Is it just a loose analogy, without rigorous mathematical reasoning or intuition?\n\n**Reply**: The delta-distribution case in Appendix A.4 serves as an intuitive starting point, not a derivation for general data distributions. Specifically, our goal is to design a schedule that yields as linear sampling trajectories for any data distributions as possible. As a necessary condition, the linearality should holds for any special data distribution. Due to its analytical tractability, we use the delta distribution case to derive a necessary condition for the forward process: a constant noise level (σ = const) and a linearly interpolating mean (μ(t) linear).\nThe necessary condition and boundary condition(line85-86) determine the SLDM schedule.\n\nWe then move to the general case and provide a separate, formal analysis (Thm3.1) showing that SLDM produces approximately linear trajectories even for realistic data distributions. So the toy case is used only as motivation, while the general result is proved independently. We believe this structured approach, from intuition to formal validation, provides both interpretability and generality. \n\n---\nIf there are any further questions or clarifications needed, we would be glad to continue the discussion. If our responses have addressed your concerns or contributed to a clearer understanding of our work, we would deeply appreciate your consideration for a higher score or confidence, which would greatly support the visibility and potential impact of our work within the community!"}, {"review_id": "osCdI3DJkv", "reviewer": "Reviewer_fZs4", "summary": "The paper introduces a novel diffusion process characterized by near-straight-line trajectories. To achieve this, the authors propose a new perturbing forward kernel designed to minimize the second-order truncation error in ODE discretization schemes (e.g., Euler’s method) for sampling. The primary application is 3D molecular generation, where the authors argue that the proposed diffusion process is particularly suitable due to the high noise sensitivity of molecular data. The model achieves faster sampling speeds without sacrificing performance.", "questions": "1) Line 43: It remains unclear why straight-line diffusion aligns well with molecular data (see Weaknesses)?\n\n2) Could the authors incorporate the novelty metric employed in the GeoBFN [2] framework? Specifically, how does the novelty of the generated molecules compare to the training set? It would also be helpful to discuss whether the low stochasticity in the straight-line constant-variance diffusion process increases the risk of the model memorizing training samples.\n\n3) Line 154: Could you clarify what is meant by a “more stable generative process” and why your proposed schedule leads to such stability? It is not immediately clear what Figure 2 is illustrating—is it showing the generative process or merely the forward noising process? If it is indeed the generative process, how is the same conformation generated across all three models? Based on the figure, it appears that GeoBFN achieves a stable conformation as early as step 4, whereas your method reaches a similar conformation only at the final step. This seems counterintuitive and appears to contradict the claim that your method accelerates the sampling of stable conformations.\n\n4) Table 1: Did you evaluate the results using multiple random seeds? It would be important to report the standard deviations as in the baselines to ensure a fair and meaningful comparison.\n\n5) How does the model's performance vary when using different constant sigma values? It would be helpful to understand the sensitivity of your method to this important parameter.\n\n6) Lines 145-147: It is unclear why the use of Langevin dynamics would make the trajectories to appear more linear or straight at the initial steps. Could you elaborate on the underlying mechanism or intuition behind this observation?\n\n7) Equation 7: It is unclear where the parameter $\\beta_t$ comes from? Although Appendix A.5 discusses it, the choice of $\\beta_t$ remains non-trivial and would benefit from further explanation. Moreover, in Algorithm 2, the purpose of the inner condition is not clearly stated. Could you elaborate on its role?\n\n8) It would be helpful to include a discussion comparing the training processes of UniGEM and the proposed framework. Specifically, why does the straight-line diffusion approach result in significantly longer training times than the standard diffusion method—10 days versus 7 days on QM9, and 16 days versus 6.5 days on GEOM-Drugs—despite both methods using the same architecture, learning rate, and optimizer? Intuitively, one might expect that using a simpler straight-line constant-variance diffusion would accelerate training as well. Clarifying this discrepancy would strengthen the experimental section.\n\nReferences\n\n[1] UniGEM: A Unified Approach to Generation and Property Prediction for Molecules. In ICLR25.\n\n[2] Unified Generative Modeling of 3D Molecules via Bayesian Flow Networks. In ICLR24.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\nThe paper proposes a reverse-engineering approach in which the authors first analyze the truncation error of first-order ODE solvers in sampling. They then design a diffusion process that cancels out higher-order terms, enabling the use of larger step sizes and accelerating the sampling process. To the best of my knowledge, this idea appears novel and interesting.\n\n**Weaknesses**\n\n- In terms of molecular generation, this paper heavily relies on the existing framework of UniGEM [1], adopting its model architecture and generation process for molecular atom types and coordinates. However, it remains unclear whether the proposed method offers substantial improvements, given that UniGEM [1] has already achieved strong performance. Notably, Table 1 lacks a direct comparison with UniGEM [1], making it difficult to assess the claimed advantages.  As a result, the paper appears to contribute only marginal novelty to the field of 3D molecular generation.\n\n- The paper does not provide sufficient justification for why a straight-line diffusion process is particularly suitable for molecular data. The explanation given between lines 41 and 45 is unconvincing. Intuitively, whether the diffusion follows a straight-line trajectory or not, small perturbations would still introduce noise and potentially generate invalid molecular conformations. In addition, one could argue that a straight-line diffusion process might struggle to adequately explore the whole molecular conformation or atom coordinate distributions, while standard stochastic perturbations could achieve better coverage, leading to more reliable and smoother data score estimation. Although the authors attempt to address this by introducing a small constant variance, they do not include an ablation study to show how varying this variance impacts model performance.\n\n- I remain skeptical about the novelty of molecules generated by the model trained under the small-variance straight-line diffusion process, particularly since the authors have not reported any novelty metrics to address this important aspect.\n\n- Furthermore, several key theoretical derivations lack clarity (see Questions).", "was_revised": true, "final_justification": "The paper introduces a straight-line diffusion model (SLDM), which is mathematically grounded, and performs well on 3D molecular generation. During the rebuttal, the authors have addressed my primary concern about SLDM relying on the previous UniGEM model. However, the authors still need more clarification on Figure 2 in the revised version. During the rebuttal, the authors have added several new results, which I believe will further strengthen the paper. It's thus that I recommend accepting the paper.\n\nBest,", "rebuttal": "Thank you for the detailed and thoughtful feedback, which has greatly helped improve the clarity of our work. We carefully incorporated these explanations in the revised version. We also sincerely appreciate your recognition of our novel contribution.\n\nWe believe the main concern may stem from misunderstandings, and we hope our responses will clarify the key points. If so, we would be truly grateful if you would consider increasing the score. \n\n> Lack a direct comparison with UniGEM\n\n**Reply**: We believe there may be misunderstandings of the experimental results. Contrary to the reviewer’s impression, our work does include direct comparisons with UniGEM. Specifically:\n- The EDM(U) baseline used in Table 3 and Figure 7 corresponds to the main model proposed in the UniGEM paper, which uses EDM as diffusion model.\n- Our results demonstrate that under the UniGEM framework, SLDM achieves significant improvements in sampling efficiency over UniGEM. For example the mol_stable: \n  - T=1000:  unigem 89.8 v.s. SLDM 95.42\n  - T=50: unigem 85.73 v.s. SLDM 93.37\n  - T=10: unigem 23.56 v.s. SLDM 87.46\n\nWe intentionally did not include UniGEM in Table 1, because Table 1 is focused on comparing different diffusion models (e.g. BFN, DDPM/EDM, Flow matching), while UniGEM is not a new generative model, but rather a change in the generation paradigm—i.e., decoupling the generation of coordinates and atom types. In fact, UniGEM can be combined with EDM, BFN, or other diffusion models. But if you believe it would improve clarity, we are happy to include UniGEM in Table 1 as well.\n\n\n> Why a straight-line diffusion process is particularly suitable for molecular data (lines 41-45) ? Small perturbations would still introduce noise and potentially generate invalid molecular conformations.\n\n**Reply**: We believe there might be some misunderstanding of the role of noise scheduling and our statements. For any diffusion-based model, the forward (noising) process inevitably drives the signal-to-noise ratio (SNR) toward zero, resulting in fully corrupted, uninformative structures regardless of whether the trajectory is straight or not.\n\nOur contribution is not to avoid this endpoint, but to control the rate at which information is destroyed. In particular, SLDM employs a slower decay of SNR, meaning the molecular structure is not immediately and completely perturbed at early steps. Instead, noise is introduced more gradually, preserving meaningful spatial relationships longer and enabling a more uniform and gradual reconstruction in the reverse process. This contrasts with EDM schedules that overly disrupt structure early, potentially making recovery harder.\n\nWe also note that the brief explanation between lines 41–45 is expanded in detail in Section 3.2 for the benefits of such a schedule in the molecular setting.\n\n>  Incorporate the novelty metric? SLDM may struggle to explore the whole distributions\n\n**Reply**: We thank the reviewer for raising this important point regarding novelty metric in molecular generation.\n1. On novelty evaluation in QM9:\nAs discussed in the EDM paper [1], novelty metrics are not meaningful on QM9 dataset because it is an exhaustive enumeration of small organic molecules within a narrow chemical space. EDM further observes that novelty consistently decreases as training progresses. Following standard practice, we thus omit novelty reporting on QM9, as do prior works including EquiFM, END, GeoLDM, and UniGEM.\n2. On meaningful novelty in broader chemical space:\nTo provide a more meaningful evaluation, we report novelty scores on GEOM-Drugs, a dataset sampling sparsely from a much larger chemical space. As shown below, our method (SLDM) achieves higher novelty than EDM (implemented in UniGEM), while maintaining strong validity and uniqueness:\n|| Atm_stab(%)|Validity(%)|Valid&Unique(%)|Valid&Unique&Novel(%)|\n|-|-|-|-|-|\n|EDM (U) (T=1000)|84.93|98.36| 98.31|98.18|\n|SLDM (T=1000)|88.19|99.96|99.91|**99.72**|\n|SLDM (T=50)|89.16|99.52|99.49|99.42|\n\nThese results confirm that SLDM can generate novel, valid, and unique molecules in realistic settings. We appreciate the reviewer’s concern regarding the potential risk of memorization in a low-variance setting. However, SLDM does not remove stochasticity, rather, it carefully balances stochasticity over time through the annealed temperature scheme eq. 9. This ensures early-stage exploration while gradually refining the structure in later stages for stability. This design prevents collapse into duplicated training samples while maintaining high fidelity.  As shown in Appendix C.4, we tune the annealing rate to maximize the unique & valid score, achieving an effective balance between diversity and chemical stability.\n\n> Ablation of constant sigma\n\n**Reply**: We find our method remain robust across a reasonable range of σ values. We only trained for 1280 epoch due to time limitation of rebuttal.\n|epoch=1280| Atom_sta(%)|Mol_sta(%)|Valid(%) |V*U(%)|\n|-|-|-|-|-|\n| σ=0.03|99.3 |92.6|95.2|92.4|\n| σ=0.04|99.2|93.3|95.3 |92.7|\n| σ=0.05|99.3|92.7|95.7|92.8|\n\n> Random seeds for tab.1\n\n||QM9||||GEOM-Drugs|||\n|-|-|-|-|-|-|-|-|\n||mol_stab|atm_stable|Validity|V*U|atm_stab|Validity|\n|Seed 1|93.35|99.30|96.33|93.38|89.15|99.45|\n|Seed 2|93.22|99.28|96.31|93.41|89.12|99.53|\n|Seed 3|93.46|99.31|96.29|93.61|89.01|99.56|\n|Mean ± Std|93.34±0.10|99.30±0.01|96.31±0.02|93.47±0.13|89.10±0.06|99.51±0.05|\n\n\nOther issues on clarity:\n> Line 154: What is “more stable generative process” and why?\n\n**Reply**: By “more stable,” we refer to the fact that our schedule injects significantly less noise/slower SNR drop compared to EDM in the early steps of the reverse process, resulting in a more gradule, less chaotic trajectory. This stability avoiding large, abrupt shifts that lead to invalid conformations in small t.\n\n> Fig2 : backward or forward process?\n\n**Reply**: It is the forward noising process. Showing the reverse sampling process would not ensure that all models generate the same molecule, making comparison unclear. \n\n> Fig2: GeoBFN faster?\n\n**Reply**: We respectfully note that this is a misinterpretation. The figure shows the diffusion process from t = 1 to t = 0 (left to right). In diffusion models, even when using fewer sampling steps, all methods simulate the full trajectory from t = 1 to t = 0 by discretizing it differently, not by stopping early (e.g., at t = 0.5). GeoBFN’s schedule allocates many steps focusing on fine-tuning geometry, while compressing the challenging structure formation phase into very few early steps, limiting model capacity.\n\n> L145-147: why Langevin dynamics make trajectories linear?\n\n**Reply**: We clarify that Langevin dynamics does not directly make the sampling trajectory more linear, but rather mitigates the impact of deviations from the ideal trajectory at early steps, i.e. improving robustness to initial sampling error. We agree the wording in the original text may be misleading and will revise the manuscript accordingly.\n\nAn ablation support this, SLDM maintains high validity and stability (surpassing other baselines in paper) even when the initial distribution deviates from the theoretical optimum(ε: standard Gaussian noise) (Sampling steps T=50)\n|Prior distribution|Mol_stab (%)|Atom_stab(%)|Validity (%)|V*U (%)|\n|-|-|-|-|-|\n|σ·ε (ideal)|93.43|99.32|96.10|93.31|\n|2·σ·ε|91.81|99.12|95.00|92.79|\n|0.1·σ·ε|93.31|99.29|96.07|93.46|\n|0.01·σ·ε|92.94|99.24|96.06|93.06|\n\n> Eq.7: where does β come from?\n\n**Reply**: Eq.7 approximates the true reverse transition distribution p(xt−Δt∣xt) using a Gaussian with a mean and variance parameterized by β. While the true reverse distribution is intractable and non-Gaussian, it is possible to derive its true conditional mean analytically under our forward process assumptions, as shown in Eq.41. To ensure our Gaussian approximation is as faithful as possible, we determine β s.t. the mean of Eq.7 matches this true conditional mean. \n\n> Alg 2: why inner condition?\n\n**Reply**: The inner condition (i.e., skipping noise injection at the final step) is a common technique used to improve the final sample quality. Intuitively, this step allows the sampler to directly apply the learned score function to denoise the sample, effectively implementing Tweedie’s formula.This idea has been adopted in previous works such as DDPM [1, Alg.2] and improved score-based models [2, Alg.1], and we follow the same practice here. \n\n[1] Denoising Diffusion Probabilistic Models\n\n[2] Improved Techniques for Training Score-Based Generative Models\n\nWe additionally provide ablation results below to show the effect of this choice:\n|T|if last step has noise|Mol Stable (%)|Atom Stable (%)|Validity (%)|U×V|\n|-|-|-|-|-|-|\n|50|w/ noise|90.91|98.99|94.94|92.25|\n|50|w/o noise (ours)|**93.37**|**99.30**|**96.24**|**93.63**|\n|1000|w/ noise|94.97|99.36|96.91|**90.55**|\n|1000|w/o noise (ours)|**95.42**|**99.43**|**97.07**|90.42|\n\n> Why longer training time?\n\n**Reply**: Thanks for the suggestion. We confirm that SLDM already outperforms UniGEM under the same number of epochs and training time:\n||QM9||||GEOM-Drugs(GD)|||\n|-|-|-|-|-|-|-|-|\n|Model (Epochs, T=1000)|mol_stab|atm_stable|Validity|V*U|atm_stab|Validity|\n| UniGEM (QM9:2000, GD:13)|99.00|89.80| 95.00|93.20|85.10|98.40|\n| SLDM (QM9:2000, GD:13)|99.41|95.32|97.04| 90.48|87.87|99.55|\n| SLDM (QM9:2980, GD:32)|99.43|95.42|97.07|90.42|88.30|99.95|\n\nWhile longer training leads to slight performance gains, we found that the mol_stable and atm_stable metrics remain stable around 2000 epochs on QM9 and 13 epochs on GEOM-Drugs during the training process. This indicates that extended training time is not a necessary factor for SLDM’s performance advantage.\n\n---\nIf there are any further questions or clarifications needed, we would be glad to continue the discussion."}]

Author rebuttal:

Meta-review: The paper proposes (near) straight-line diffusion models for 3D molecule generation. The approach relies on a relatively small modification to the forward process compared to e.g. EDM that adds a small amount of noise that is constant across time. The method performs competitively at both high and moderate number of function evaluations across a range of datasets.

The reviewers found the approach interesting and reached a consensus recommending acceptance, and I find no reason to disagree. When revising please pay attention to add the new discussions, results, and request for clarifications that were brought up by the reviewers.
Decision: Accept (poster)