Hierarchical Multi-Scale Molecular Conformer Generation
Authors: Jiapeng Hu
Weizhi Gao
Zhichao Hou
Xiaorui Liu
Venue: ICLR 2026 Poster (2026)
Keywords: Molecular conformer generation
Generative models
Primary area: generative models

Abstract: Molecular conformer generation is a fundamental task for drug discovery and material design. Although deep generative models have progressed in this area, existing methods often overlook the hierarchical structural organization inherent to molecules, leading to poor-quality generated conformers. To address this challenge, we demonstrate that capturing the spatial arrangement of key substructures, such as scaffolds, is essential, as they serve as anchors that define the overall molecular distribution. In this paper, we propose a hierarchical multi-scale molecular conformer generation framework (MSGEN), designed to enhance key substructure awareness by leveraging spatially informed guidance. Our framework initiates the generation process from coarse-grained key substructures, progressively refining the conformer by utilizing these coarser-scale structures as conditional guidance for subsequent finer-scale stages. To bridge scale discrepancies between stages, we introduce a molecular upsampling technique that aligns the structural scales, ensuring smooth propagation of geometric guidance. Extensive experiments on standard benchmarks demonstrate that our framework integrates seamlessly with a wide range of existing molecular generative models and consistently generates more stable and chemically plausible molecular conformers.

Published as a conference paper at ICLR 2026
HIERARCHICAL MULTI-SCALE MOLECULAR CON- FORMER GENERATION
Jiapeng Hu' Weizhi Gao'
Zhichao Hou'
Xiaorui Liu'
North Carolina State University jhu 88ncsu. edu. en, wgac23encsu. edu. en, zhou43ncsu. edu, xliu960ncsu, edu
ABSTRACT
Molecular conformer generation a fundamental task for drug discovery and ma- terial design. Although deep generative models have progressed in this area, ex- isting methods often overlook the hierarchical structural organization inherent to molecules, leading to poor-quality generated conformers. To address this chal- lenge, we demonstrate that capturing the spatial arrangement key substruc- tures, such scaffolds, is essential, they serve anchors that define the overall molecular distribution. this paper, we propose hierarchical multi-scale molec- ular conformer generation framework (MSGEN), designed enhance key sub- structure awareness by leveraging spatially informed guidance. Our framework initiates the generation process from coarse-graine graiied key substructures, progres- sively refining the conformer by utilizing these coarser-scale structures as condi- tional guidance for subsequent finer-scale stages. To bridge scale discrepancies between stages, we introduce molecular upsampling technique that aligns the structural scales, ensuring smooth propagation of geometric guidance. Extensive experiments standard benchmarks demonstrate that our framework integrates seamlessly with a wide range of cxisting molecular generative models and consis- tently generates stable and chemically plausible molecular conformers.
INTRODUCTION
Molecular conformer generation is fundamental task in computational chemistry with applications in drug discovery, material design, and property prediction, where we predict the 3D geometry of molecule from its 2D graph representation (Guimarães et al. 2012 Schütt et al. 2021 Axelrod 2023). Recent advances in deep generative models, such as diffusion mod- els (Dhariwal & Nichol 2021) and flow matching (Lipman et al. 2022), have demonstrated strong potential for generating accurate and diverse molecular conformers (Simm & Hernández-Lobarg
GGomez-Bombare
2019 Shi al., I Jing et al. (2022). However, many existing methods focus on generating atomic coordinates or local geometries, such as torsion angles, while neglecting higher-level struc- tural information. As result, they often fail to preserve essential substructures, such ring systems, leading to chemically implausible conformers (Zhou et al. 22033 Fan et al. 2024).
To improve the chemical validity of conformer generation, existing work incorporated different structural guidance. These efforts can be cat- egorized by the structural granularity they em- phasize. Methods like Subgdiff (Zhang et al. 2024). which preserve local structures by denois- ing within subgraphs, often disrupt global consis- tency. Fragment-level approaches aim to retain coarse molecular features. Early design (Qiang et al. 2023) captures fragment information at a sin- gle resolution but lacks coarse-to-fine generative process. Subsequent work (Park & Shen 2024) adopts a coarse-to- coasee-to-finn no-fine strategy based on equivariant blurring diffusion (Rissanen et al. 2022 Nev- ertheless, these figgeent-based methods generally treat all fragments uniformly, overlooking differ- ences functional importance and conformational flexibility. Therefore, is essential to effectively organize and utilize structural information in generative models.
Structure
Figure 1: Multi-Scale Molecule Representation.

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As shown in Figure 1 molecular structures naturally exhibit hierarchical organization (Zhang jet al. 2012; Gennis 2013). In chemistry, this hierarchical nature reflects how molecules are assem- bled from basic structural motifs, such rigid rings and flexible linkers, into complex functional architectures (Boninsegna et al. 2018). Consequently, generating valid molecular conformers re- quires more than just focusing coarse-grained structural features (Qiang et al.) 2023; Park & Shen 2024), they often fail to cap- ture the hierarchical assembly process that governs spatial dependencies across structural scales. In molecular systems, the arrangement of fine- the-ggrined components is typically conditioned on the positioning of key coarser substructures. In drug-like molecules, global scaffolds often constrain the spatial arrangement of flexible side chains, thereby ensuring chemically plausible geometries. This structural dependency motivates the incorporation of geometric guidance as an inductive bias to promote global consistency and chemical validity in conformer generation. To evaluate the ef- fectiveness and importance of structural guidance, we conducted a preliminary study comparing several generative models that incorporate structural features with a variant that leverages scaffold positional information in Section 3
single structural scale. Despite recent efforts to incorporate
The results demonstrate the strong power of substructure po- sitions enhance structural awareness, preserving geometric consistency and chemical validity during generation.
However, the geometric guidance is usually not available during generation. Based on these in- sights, we propose MSGEN, a general framework for molecular conformer generation that preserves chemical properties while supporting the multi-scale nature of molecular structures. Our framework adopts multi-scale generation process, progressively refining molecular structures from coarse to fine scale. The output of each stage serves as flexible and detailed structural guidance for the subsequent stage. To mitigate the structural scale mismatch between coarse and fine representa- tions, we introduce molecular upsampling. It can bridge the structural gap between stages, enabling more efficient and chemically valid propagation of structural guidance. To validate our framework, we conducted comprehensive experiments on standard molecular conformer benchmarks, including conformer generation and property prediction tasks. The results showed that the MSGEN framework consistently enhances existing bascline models in generation quality, leading to more chemically plausible conformers. In summary, our contributions are fourfold:
We conducted preliminary study to compare different structural guidance strategies to demon- strate the strong power of substructure positions enhancing structural awareness.
We proposed MSGEN, general framework for molecular conformer generation that explicitly reflects the hierarchical nature of molecular structures. This framework adopts a multi-scale gen- eration process from coarse scaffolds full-atom details.
We introduce molecular upsampling to address the structural scale mismatch between coarse and fine representations, facilitating the efficient and chemically consistent propagation of structural guidance across stages.
We perform extensive experiments and ablation studies to assess the effectiveness of our frame- work in enhancing the structural awareness of baseline models. The improved awareness leads to the generation of more stable and chemically plausible conformers.
2 RELATED WORK
Hierarchical Diffusion Model. Diffusion models (Ho et al. 2020 Song et al. 2020 Su al. 2024)
generate data by iteratively denoising samples from prior distribution and have achieved notable success vision tasks. To enhance the efficiency and structural fidelity of generative processes, hicrarchical diffusion models (Ho et al.) 2022a Saharia et al. 2022b; Ho et al. 2022b Wang et al.) 2024) decompose the generation task into multiple stages, progressively refining coarse structures into fine details. Motivated by the inherent hierarchical organization of molecules, recent works on molecular graph generation (Jin et al., 2020), protein structure (Yang & have shown the effectiveness of leveraging this multi-scale nature. In recent molecular conformer generation, EBD (Park& & Shen 2024) leverages a blurring diffusion (Rissanen et al.) 2022) to re- cover full atom details from fragments, but it lacks multi-scale structure awareness, as it treats all fragments uniformly. It remains unclear how to design hierarchical diffusion models for molecular conformer generation, and this work presents the first study to address this challenge.
2023)
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Molecular Conformer Generation. Various machine learning based approaches have been pro-
posed for molecular conformer generation (Mansimov et al. 2019. Simm & Hernández-Lobato
2019
Ganea al.] (021). Recently, the diffusion-basse method (Xu et al. 2022 Jing et al. 2022] Wang et al. 2023) has been introduced to this domain,
Xu et al.) 2021a, Shi et al.] 2021
showing great promise generating diverse and accurate conformers. GeoDiff (Xu al.) 2022)
introduced a diffusion model that learns to denoise 3D atomic positions conditioned on molecular graphs. TorsionDiff (Jing et al., 2022) designs a diffusion model on the torsion angles. MCF (Wang et al. 2023) trained a diffusion model over functions that map elements from the molecular graph to points in 3D space. However, most existing methods primarily focus on a single scale during molecular conformer generation, neglecting the inherent multi-scale organizations of molecules.
Equivariant Deep Learning. Recent work in Al for Science has focused on incorporating physical and chemical priors, particularly the rotational symmetries captured by the Euclidean group S3))).
In the past few years, various invariant and equivariant architectures have been developed for Carte-
sian (Schütt et al. 2017 Gasteiger et al. 2020; Satorras et al. 2021 Liao & Smidt 2022 Thölke
& De Fabritiis [2022] Wu et al. [023 Chen et al. (2224) and spherical basis (Thomas et al. 2018
Fuchs al.J [2200] Brandstetter et al., 2021: Frank et al.) 2022). For molecular generation, iniiial
methods like ConfGF Shi et al.] (2021) utilize invariant networks for inter-atomic distances, whereas
the equivariant Geometric GNNs have been used in GeoDiff and ET-Flow (Hassan al. 2024).
3 PRELIMINARY STUDY ON MOLECULE STRUCTURAL GUIDANCE
Problem Definition and Notations. Molecular conformer generation aims to produce chemically valid and novel molecular coordinates R. conditioned on a given molecular graph G. In this paper, we denote each molecule with n atoms as an undirected graph G = (V, E), where V= } represents atoms, & = {eey (i,5) S [V] x [V] represents inter-atomic bonds, and X encodes node features, e.g. the atom type. Each atom uy is associated with a coordinate r; E TR33 and the molecular conformer can be represented as a matrix R [ri..... rn] _R*
Limitations of Existing Works. Since vanilla diffusion-based approaches (Xu et al. 2022) fail to preserve essential chemical substructures during generation, several extensions (Zhang et al. 2012 Park & Shen, 2024) attempt to preserve key structures either through subgraph denoising or fragment-level features. These methods improve local connectivity and fragment consistency, but they neglect the dominant role of key structural components such as ring systems and heavy-atom backbones (Velkoborsky & Hoksza 2016; Manelfi et al. 202T). Without explicitly constraining these key substructures, generated conformers may satisfy local validity but still deviate from real- istic global arrangements. Therefore, we propose that incorporating geometric structural guidance, such the positions of key substructures, can serve a more chemically meaningful prior. These will help anchor the spatial arrangement of essential structural motifs. By integrating such informa- tion, we aim enhance structural awareness and reduce uncertainty in the generation
To validate our hypothesis, we designed preliminary study to evaluate model performance under four types of guidance. Specifically, we conduct a con- trolled experiment with (1) no structural guidance (Xu et al.) 2022). (2) subgraph-level struc- tural guidance (Zhang et al.)
Table 1:
Geometric Evaluation on GEOM-Drugs (8 = 1.25A). The results show that the geometric structural guidance effectively enhances the generation quality.
COVER (S) 1
MAT-R (.)
COV.P ($)
MAT.P (-4)
Guidance Type
Mcan Median
Mean
Median
Mcan Median
Mean Median
Noue Xu al. 2022]
64.12
75.56
1.1444
1.1246
43.16
42.02
1.3806
1.3314
Local Zinn dlei
74,30
86.43
L.0003
0.9905
Frigment (Par & Sien 2024
84.88
94.97
0.9118
0.8788
64.74 68.
1.1465 1.0876
Gesmetric((urs)
99.58
100.00 (0.5035
0.4899
88.56
94.70
0.7816
0.7515
2024], which enforces local connectivity, (3) fragment-level guidance (Park & Shen 2024). which emphasizes chemical motif identity, and (4) geometric structural guidance, which masks the posi- tions of light atoms while retaining the ground-truth positions of the heavy-atom backbone. All of those results use 500 diffusion time steps during generation.
As shown in Table I across all evaluation metrics compared to the other approaches, indicating that geometric structural guidance enables the model to capture substructures and their global arrangements more effectively than existing approaches. More implementation details are in Appendix C.4
incorporating geometric position guidance yields substantial improvements
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4 HIERARCHICAL MULTI-SCALE MOLECULAR CONFORMER GENERATION
In this section, we elaborate on our novel framework MSGEN, designed to enhance structural aware- ness during conformer generation while preserving chemical properties. The motivation for a multi- scale design arises from the inherently hierarchical nature of molecules (Manelfi et al., (021). Our framework adopts hierarchical molecular conformer generation process, where each stage incre- mentally refines the molecular structures based on the geometric structural guidance provided by the previous stage. We further introduce molecular upsampling reconcile the mismatch of structural scale across stages, enabling effective and efficient guidance propagation.
Finer Scale
Coarse Scaffold Generation Chenneet Prior
Stage Intermediate Structure Stractusal Guidance Chemical Frixt
Stage 3: Full melecular Guidance
DDPM
DDPM
Plansible
Matrcular ppangting
Modecator I ppampplnn
Molecular Upsampling D) Asiggn counse scale node
2) Extract fine scale structure and find upsampinng ender
3) Anchee-bmed Sampling
Cunfeemer Syberm
Coasse anchor
BFS level
Suxple Directee
Figure 2: Illustration of the MSGEN framework with three stages. (a) Each stage generates refined rep- resentation the molecular conformer, guided by previous stages, with the outputs connected via molecular upsampling. (b) In molecular upsampling, atoms the fine-scale representation are first ordered and then se- quentially assigned coordinates via anchor-base based sampling.
4.1 MULTI-SCALE MOLECULAR CONFORMER GENERATION
The results in Section 3 motivate the use of geometric structural guidance during generation. How- ever, such guidance is not directly accessible when only the molecular graph is available at in- ference time. To address this limitation, we propose multi-scale molecular conformer generation, where each fine-scale stage is conditioned on the structural representations generated at the previous coarse-scale stage. We define a K -leevel hierarchical representation of molecular structure based on its inherent multi-scale organization, as shown in Figure I which preserves chemical substructures across different levels granularity. Each level corresponds to a subgraph gå, forming a nested sequence G' C G
C g gk are denoted as R E R where 7ıg is the number of atoms in g We model the distribution po, (R* at the first stage and model po,
from coarse to fine. The atomic coordinates associated with
(R (R*Ig R for subsequent stages using individual generative models parameterized with 0z. where is the index of stages. While we use DDPM (Ho) al. 2020) as an illustrative example, our framework compatible with variety of generative
models, including flow matching (Lipman et al. 2022).
Forward Process. For each stage k 1, K, the forward process is defined as follows: (R R (RR Ra (1 I)), where E (1,..., Ti) the diffusion step and a denotes the cumulative noise schedule for stage k. Note that employ separate forward processes for each stage, allowing our framework the flexibility to construct stage-specific training data aligned with the corresponding distributions.
The reverse process generates conformers in guided, hierarchical manner, pro- gressively transitioning from coarse structural scaffolds to the complete conformer. At the first stage, the model generates the coarse-scale structure without guidance. The denoising process at this stage is formulated po, 19 RR) (R po, (G², R,, (), (1 of) where po,
Reverse Process.

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equivariant neural network parameterized by 81 and aj is the noise schedule. For subsequent stages k = 2.... the reverse process progressively refines the structure to R leveraging the coarse representation from the previous stage. The conditional denoising process is formulated Po, (R_ R. (emaa) = W(R Po, Po, (g*, R Roand- 1), (1 - (1-aş) )I), where no, (-) and of are the equivariant neural network and the noise schedule for stage k. Roond is a geometric structural guidance derived from the coordinates of the previous level R R = (R - where F& (-) is conditioning function. During sampling, we iteratively compute Reand based on the output sampled from the previous stage R effectively capturing and leveraging the multi-scale hierarchical structure of molecular conformers.
Algorithm 1: Topological Sort
Algorithm 2: Anchor- Upsampling
Input: Molecular graph = (V,E), Atom sets Zenarse, Itme
Input: Coarse-level positions Recarse Atom set Icoarse Molecular graph G.
Output: Upsampling Order O
Upsampling order O, Threshold
R = -ime'' Icoarse:
Output: Fine-level conditions Reoud
= {ili R, 3j E Icoarse: (i,j) E E}:
Assign positions: Reoud [ccarre]] f- Recarses Compute coarse centroid c <- mean Recarse
Initialize visited set V t BB
Initialize queue Q ;:
Initialize positioned set P I
while Q not empty do Pop from and append to O for each j (i) do
for each atom E O do .N(i) < n
i * 0 then
if j & V and j & Teoarse then if Any &s NN() in V then
Anchor a; +
1 [I
EJEN, Ry;
else
Add j into O and V
Anchor a; c end
end
end
Sample direction d; ~ ~O,11)
end end
d, Reond [o] +- a, Td.
Add into P end return Reond
Append any remaining atoms in R to O; return
4.2
MOLECULAR UPSAMPLING
In multi-scale framework, coarse-level outputs provide essential context for finer resolutions. However, directly using R k-1 as the guidance R R can lead to scale mismatch, hindering effeetive information transfer. While upsampling methods like interpolation common in vision tasks (Ronneberger et al.] 2013 Saharia et al.) (0022a), they are not well-suited to molecular conformers due the continuous nature of 3D molecular structures. To address this, we introduce a molecular upsampling strategy that respects chemical graph connectivity.
transposed convolutions
Formally, given coarse-level atoms Tsnarse with positions Recarse E Rmx33 and the full molecular graph G. our goal is to infer a chemically meaningful prior Reond E R for atoms in the fine set Iime- Since atoms are arranged the positions of already positioned atoms to achieve spatial consistency. To ensure valid ordering and avoid cycles, we perform a topological sort over G, yielding an upsampling sequence O (see Algorithm [T). Following this order, we iteratively assign positions to fine-level atoms using their placed neighbors as geometric anchors (see Algorithm To systematically track the atoms whose positions have been determined, we maintain a positioned set P, initialized with all coarse-level atoms. For each fine atom E O, we identify its positioned neighbors and compute:
-un-Euciid molecular graph G, some fine atoms rely on
Roond [ NO
Σ JENTONP
+ di, d, ~ N00,
(I)
where controls sampling radius and d, introduces spatial variation while preserving structural continuity. The atom is then added to P for subsequent steps. This process ensures that molecular geometry constructed progressively, respecting both chemical connectivity and spatial coherence.
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4.3 MODEL TRAINING
In this section, we introduce our conditional augmentation avoid the distribution shift problem and elaborate on the loss function for multi-stage training. We use DDPM following Section 4.1
Conditional Augmentation. In standard diffusion training, the forward process uses the ground- truth R to construct training data, while inference relies on the sampled approximation R*-1, which may deviate from the ground truth. This mismatch introduces distribution shift that can propagate errors across stages in our multi-stage framework. To mitigate this, we propose condi- tional augmentation, which perturbs the ground truth with controlled noise during training, improv- ing the generalization the guidance mechanism and enhancing robustness at inference time.
For the training of stage k, we denote the coarse structure obtained from ground-true truth data as R We perturb R to simulate the variability of coarse conformers encountered during inference. To be specific, we select a small timestep & and perturb R leveraging the noise schedule of the diffusion forward process:
R,
Ry
11-a ~0,,
(2)
where a ak-1 represents the cumulative noise schedule for stage k at timestep s. This formula- tion indicates that perturbed structure Rg is sampled from q 1 ( - Then the noisy coarse
structure is passed through the molecular upsampling module Fx to generate structurally aligned conditional input for the current stage:
R R = FR,,
This method ensures that the fine-scale model learns to condition on controlled variations of coarse- scale structures, thereby improving robustness to structural diversity during inference.
Training Loss. We derive an evidence lower bound (ELBO) for our hierarchical framework under conditional augmentation. For simplicity, we ignore the molecular upsampling, which does not affect the conclusion. The following proposition provides an ELBO:
Proposition 1. Assuming that the coarse-scale is unconditional and both the coarse and fine scale generative models use the same number of timesteps T, the ELBO of the combined model p is
- log (Ro) S (BT
Σ DKL(((RRR t>
[R], R/ (R)) RR, G')) + La (Ro R!), (4)
where RJ is the original coarse-scale output, Ro the fine-scale output, and Le (Ro Lo (,LTT(R)) + E.s1 DrL(a(R,-IBB, (R)] R 9)) + (Ro R 9).
R 9) =
Proof. Refer Appendix
F.
The above proposition formalizes the training objective of our framework. Optimizing the ELBO formulation in Eq. 4 decouples the training across different stages in the multi-scale generative process. This decoupling allows us to apply the diffusion loss at each stage independently. After adding the molecular upsampling, our loss function for stage is:
C(0x) = . - Eo, (R, Gk, R R I)]
(5)
For stage k = 1,, the loss reduces to the standard denoising loss without condition. In practice, we can search multiple possible timesteps for conditional augmentation to achieve the best sample quality. However, it would be computationally expensive to retrain the model separately for each choice. To enhance the training efficiency, we formulate a set T, and adopt the same method in Ho et al. (2022a). which avoids retraining models by amortizing the model over uniform random E T, at training. We detail the training and sampling procedures in Algorithm 3 and Algorithm
5 EXPERIMENT
In this section, we first describe the experimental setup, followed by comprehensive experiments that demonstrate the effectiveness of our framework. We also perform ablation studies to analyze the efficiency of our framework and the contributions of molecular upsampling and conditional augmentation. The visualizations of the generated molecules are provided in Appendix[]

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5.1 EXPERIMENT SETUP
Datasets. We conduct our experiments on the GEOM dataset (Axelrod & Komez-Bombarelll
2022]. which contains GEOM-QM9 and GEOM-Drugs. Each dataset comprises 40,000 molecules for training and 5,000 molecules for validation, each containing 5 conformers following data split of Xu et al. (022)) For the test set, we selected 200 molecules for each dataset, resulting in 22,409 and 14,324 conformers in QM9 and Drugs. The details of the dataset are in Appendix C.I
Evaluation. We adopted the widely used Coverage (COV) and Average Minimum (AMR) metrics proposed by Ganea et al. (2021) to assess the geometric quality of generated conformers within a til-meenn-quure deviation (RMSD), which is normalized Frobenius norm between two atomic coordinate matrices aligned using the Kabsch algorithm Kab-] sch (1976). We also evaluate chemical properties of generated conformers, including the average energy E, the lowest energy Emin, HOMO-LUMO gap e, the average gap Ze. the minimum gap Cmiin the maximum gap Acmaz- These properties reflect the chemical validity of a conformer.
threshold 6. These metrics are based
Baselines. We primarily evaluate framework by integrating it with GeoDiff (Xu et al. 2022
used as the main backbone in our experiments. To assess generalizability. we also apply it to ConfGF (Shi et al., 2021), ET-Flow, and EBD (Hassan et al. (202)). Models integrated with our framework are denoted by appending "+MSGEN" (e.g., GeoDifT-MSGEN) For broader compar- ison, we include RDKit (Landrum et al. 2013), which is arguably the most popular open-source software for conformer generation, and several such as GraphDG (Simm & Hernández-Looee Lobato 2019), CGCF (Xu et al. 2021a), ConfVAE (Xu
deep generative models for conformer generation,
et al. 202 and GeoMol (Ganea et al.] 2021). Baseline results are adopted from Xu et al. (022))
and /haag et al. (2024). We highlight the best results in tables using boldface.
Framework Setting. Our main experiment adopts a 2-stage framework, where each molecule is divided into heavy-atom backbones and hydrogen atoms. This design reflects chemical fact: heavy atoms provide structural scaffold, while hydrogens influence local stereochemistry and fine-tune conformers (Clabbers et al.) 2022 Thompson & White 2023). The backbone is generated first to serve as global spatial anchor, guiding hydrogen placement in the second stage. Beyond this setting, we also conduct experiments with a 3-stage framework that incorporates additional domain priors. The design guidance for incorporating additional stages is in Appendix F
5.2 PERFORMANCE EVALUATION
Table 2: Geometric evaluation on GEOM-Drugs using different baselines (6 = 1.25.4)
COV-R(%) f Median
MAT-R(Å)
COV-P(% t
MAT-P((A)
Models
Mean
Mean
Median
Mean
Median
Mean
Median
RDkit
45.74
31.75
1.5376
1.4004
54.79
59.48
1.3341
1.1996
GraphDG
8.27
0.00
1.9722
1.9845
2.08
0.00
2.4340
2,4100
CGCF
53.96
57.06
1.2487
1.2247
21.68
13.72
1,8671
1.8066
ConfVAE
55.20
59.43
1.2380
1.1417
22.96
14.05
1,8287
1.8159
GeoMol
67.16
71.71
1.0875
1.0586
ConfGF
62.15
70.93
1,1629
1.1596
23.42
15.52
1,7219
1.6863
GeoDiff
87.86
97.00
0.8686
0.8545
60.17
62.21
1,1871
1.1412
GeODIfT-MGGEN
90.41
98.16
0.8424
0.8248
66.26
70.65
1.1217
1.0682
Geometric Evaluation. The geometric evaluations of conformer generations are summarized in Table 2 and Table 9 (left in Appendix G As shown in the Table, GeoDiff-MSGEN significantly improves GeoDiff across all metries and achieves the best performance across different baselines. Specifically, the consistent improvement in precision metrics indicates that our model generates more accurate conformers. These results demonstrate that integrating our framework significantly enhances its structural awareness, enabling it to capture key substructures and generate high-quality molecular conformers, especially for complex molecules like those in GEOM-Drugs.
Chemical Evaluation. This task estimates the molecular ensemble properties (Axelrod & Gomez- Bombarelli 2022) over a set of generated conformers. This can provide a direct assessment of the quality of generated samples. Following (hh et al. 202I). we generated 50 samples for each of the 30 molecules, constituting a subset of QM9. Then we used the chemical tool PSI4 (Smith et al, 2020) to get the desired properties. The mean absolute errors (MAE) between calculated properties and the ground truth are reported in Table 3 We can observe that incorporating our framework significantly decreases the MAE across all evaluated ensemble properties for GeoDifff Furthermore,
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it shows that our framework enables the model to generate more chemically plausible molecular conformers, as evidenced by the significantly lower energy and HOMO-LUMO gap errors compared to baselines.
Table 3: MAE of different physical properties between generated and ground truth ensemble properties in eV.
Table 4: Geometric evaluation on GEOM-QM9 for domain generalization. (6 = 0.5A)
COV-R(SE) f
MAT-R(A) 1
Method
E
Easto
Z5z
Arouz
Model
Train Data
Mean
Median
Mean
Median
RDkit
0.9233
0.6585
0.3698
0.802.
0.2359
GraphDG
QM9
73.33
84.21
0.4245
0.3973
GraphDG
9,1027
0.8882
1.7973
4.1743
0.4776
CGCF
QM9
78.05
82.48
0.4219
0.3900
OGCF
28.9661
2,8410
2,8356
10,6361
0.5954
Con
QM9
77.84
88,20
0.4154
0.37.39
ConPVAE
8,2080
0.5100
160000
3
0.2429
GcoMel
QM9
72.00
CoIFGF
2.7886
0.1765
0.4688
2.1843
0.1433
SubgDiff
Drugs
83.50
88.70
0.3116
0.3075
GoaDii
0.25V7
0.1851
0.3091
0,7033
0.1909
GeoDil
Drugs
74.94
79.15
0.3492
0.3392
MSGGNN
0.1795
0.1019
0.2038
0.58998
0.1755
GeoDIffMMSGEN
Drigs
83.73
89.32
0.3055
Domain Generalization. We conduct domain generalization experiments to demonstrate the ef- fectiveness of our framework, following the design in SubgDiff (Zhang et al., 2024). Specifically, models are trained on GEOM-Drugs and evaluated on GEOM-QM9 without any fine-tuning. To ensure the validity of this setup, we systematically checked the two datasets at the molecule level using RDKit canonical SMILES and confirmed that they are disjoint. We also compare the results against several baseline models without domain transfer. As shown in Table our method, GeoD- iff+MSGEN, consistently outperforms the GeoDiff baseline across all evaluation metrics, achieving the best overall results in the table. Additional results for models trained on GEOM-Drugs and evaluated on GEOM-QM9 are provided in Appendix G.2 showing similar conclusions.
Multi-Stage Scalability. To demonstrate the scalability of our framework, we extend it to 3-stage generation process. We introduced new stage based on the MurckoScaffolds (Bemis & Murcko) 1996), which extracts core ring systems and linkers while removing side chains and decorations from the heavy-atom graph. The extracted subgraph provides stable and chemically relevant foundation that reflects key molecular geometry. We integrate the 3-stage design with ET-Flow and GeoDiff, that the 3-stage design
and evaluate on the GEOM-Drugs dataset. The result in j provides an improvement over the 2-stage case. These improvements suggest that incorporating more chemistry-informed structural decomposition can further enhance model performance. They also highlight the potential of our framework in conformer generation, particularly for biologically relevant molecules (Schuffenhauer et al. 2007).
Table 5: Geometric evaluation of applying MSGEN across different baselines on GEOM-Drugs
COV-R(%)
MAT-R(A) 1
COV-P(% f Median
MAT-P(Á) +
Backbone
Variant
8(A)
Mean
Median
Mcan
Median
Mean
Mcan
Median
baseline
87.86
97.00
0,8686
0.8545
60.17
62.21
1,1871
1.1412
GeoDiff
+MSGEN (2-stage)
1.25
90.41
98. 16
0,8424
0.8248
66.26
70.65
1.1217
1.0822
+MSGEN (3-slage)
91.05
98.33
0.8410
0.8265
66.87
71.40
1.1147
1.0579
buseline
74.47
81.00 1606
0,5514
0,5288
55.21
54.22
0.7855
0.7341
ET-Flow
+MSGEN (2-stage)
0.75
80.50
88.71
0.4579
0.4394
64.67
68.34
0.7342
0.6783
+MSGEN (3-stage)
81.91
89.24
0.4363
0.4280
66.12
69.51
0.7159
0.6573
Conf-GF
bascline
1.25
62.15
70.93
1.1629
1.1596
23.42
15.52
1.7219
1.6863
+MSGEN(2-stage)
65.51
73.31
1.1415
1.1328
27.72
20.21
1.6986
1.6765
baseline
92.10
98.40
0.8292
0.8391
65.97
67.95
1.1300
1.1026
EBD
1.25
+MSGEN(2-stage)
91.92
98.48
0.8257
0.8198
68.10
72.25
1.1013
1.0680
Framework Generalizability. To validate the generalizability of our framework in enhancing struc- tural awareness, we integrate our framework with different generative models, such as Score Match- ing (Song & Ermon, 2019) and Flow Matching (Lipman et af., 2022). For instance, ConfGF is based on Score Matching, while ET-Flow utilizes Flow Matching. In addition, we also evaluate our framework with Equivariant Blurring Diffusion (EBD) (Park & Shen 2024), which generates atomic details from coarse fragments. Following the 2-stage design, we integrate our framework and evaluate on the BEOM-Drugs dataset. Specifically, we use the data split and preprocessing method in ET-Flow since it utilizes a larger GEOM-Druge dataset. As shown in Table | our framework consistently improves the generation quality across various kinds of generative models, generating more diverse and accurate conformers. This improvement highlights the effectiveness of our multi-
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scale framework in enhancing structural awareness through progressive guidance, regardless of the underlying generative model. Further details of the implementation in Appendix D
5.3 ABLATION STUDY
Efficiency and Practicality We further investigate the efficiency of the multi-scale design. In particu- lar, we compare the performance of vanilla GeoDiff and our 2-stage MSGEN+GeoDiff under the same total number of diffusion steps. We set the same number of steps for both stages. From the result in Figure and Table [14] we observe that GeoD- iff+MSGEN consistently outperforms the baseline in most metrics, while spending shorter average generation time for each molecule. This highlights that our framework leads to more efficient utiliza- tion of the diffusion step, improving both effective- ness and practicality. Additional analysis is in Ap- pendix G.7
Model
1.30
Orocar
(AT-RRA)1
1.25
2000
1000
50
150
200
Avg Time per mol (s)
Figure 3: Comparison of GeoDiff and GeoD- iff+MSGEN in terms of average generation time and mean MAT-R. Marker size corresponds to the total number of diffusion steps
Impact of Hierarchical Design To evaluate the effectiveness of the hierarchical design, we conduct a stage-wise wise ablation on GEOM-Drugs at the coarse level. Specifically, we compare the generation quality of coarse-level atoms generated (1) from scratch, (2) after Stage 1, and (3) after Stage 2.
Stage
COV-R(%)
MAT-R(A)4
COV-P(*61
MAT-P(Á)L
From scratch
87.86
0.8686
60.17
1.1871
After Stage 89.17
0.8543
65.32
1.1293
After Stage
90.41
0.8424
66.26
1.1217
Table 6: Stage-wise geometric evaluation on coarse-level atoms of GEOM-Drugs.
As shown in Table we observe consistent improvement from generation from scratch to Stage and further Stage 2. These results demonstrate that the hierarchical design enhances structural awareness by providing meaningful coarse arrangement that can be further refined, rather than introducing rigid constraints.
Impact of Molecular Upsampling. To evaluate the effectiveness of molecular upsampling, we compare with two alternatives: random sampling. which assigns hydrogen positions from stan- dard Gaussian, and centroid sampling, which places them at the geometric center of the coarse structure without noise. As shown in Figure 44a our method yields more accurate conformers. This is primarily attributed to the effective utilization of chemical graph connectivity, which guides the fine-scale atomic placement based on anchor nodes, resulting better structural guidance.
Random
Centroid Molecular
1.2
CA CA
0.9
0.7
MAT-R(MAM)) MATT-R/medtaa) MAR-P(reca)) MAR-P(uecdtaa)
MAT R(ueau)
MAT R(median) MAR-P(uraa) MAR P(uredia))
(a) Impact of Molecular Upsampling.
(b) Impact of Conditional Augmentation
Figure 4: Ablation studies on Molecular Upsampling and Conditional Augmentation.
Impact Conditional Augmentation. To assess the performance of the conditional augmentation in our framework, we conduct an ablation study on GEOM- based on GeoDiff that removes the conditional augmentation in the training process. As shown Figure 4b conditional augmentation improves the performance of our framework, evidenced by lower MAT metrics. This demonstrates that this approach enhances the model's ability to adapt to varying structural scales and geomet- ric variations, leading to more robust and accurate molecular generation. We further evaluate the

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robustness and efficiency of MSGEN under reduced-step and equal-step settings. Additionally, present a failure case analysis and a complexity-awee aware evaluation to better highlight the framework's s strengths. We refer to Appendix G.3 to G.6 for detailed insights.
6 CONCLUSION
In this work, we introduce a novel multi-scale molecular conformer generation framework that lever- ages the inherent hierarchical structural organization of molecules to enhance structural awareness. Our framework generates coarse- grarseegriiedddeee substructures intermediate representations, which guide finer-scale generation through novel molecular upsampling strategy that bridges scales effectively. Through extensive experiments with competitive generative models, we substantiated the effective- ness and generalization ability of the proposed framework. Owing to its modular design, the frame- work flexibly incorporates domain-speciife priors and is well-suited for complex molecular systems, including proteins and polymers, where hierarchical structures are intrinsic. These results suggest promising direction for molecular conformer generation increasingly complex chemical spaces.
ETHICS STATEMENT
This work develops machine learning methods for molecular conformer generation using publicly available datasets (GEOM-QM9 and GEOM-Drugs), which contain only computational molecular structures. No personal, sensitive data are involved, and no human or animal subjects are included.
The research aims improve structural fidelity in conformer generation support computational chemistry and drug discovery. While generative models may carry potential misuse risks, this study limited standard benchmarks of organic molecules and does not involve generating or evaluating bioactive or toxic compounds.
No real- world synthesis performed, and no actionable synthesis protocols provided. All results are intended solely for scientific benchmarking. We encourage continued attention to responsible use and safeguards against misuse future work.
8 REPRODUCIBILITY STATEMENT
We have made extensive efforts to ensure the reproducibility of our work. All datasets used in our experiments are publicly available, and we provide detailed description of preprocessing steps in Appendix C.T The integration of our framework into baselines and the training hyperparameters
described in Appendix C.2 C.5 and D All theoretical proofs are provided in the Appendix F
Code is available at: Ittps:: //ggttttuu././.. com/ http///seriiaa//NN-P..llll
ACKNOWLEDGEMENT
Weizhi Gao, Zhichao Hou, and Xiaorui Liu are supported by the National Science Foundation (NSF) under grant number IIS-2443182. This work completed in full during Jiapeng Hu's visit visiting scholar at North Carolina State University (NCSU).
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A DENOISING DIFFUSION PROBABILISTIC MODEL
Denoising Diffusion Probabilistic Model(DDPM) (Ho et al. 2020 is defined by a forward process ((1r|2111) that gradually destroys data Ty ~ ((xo) for T timesteps and learned reverse process po(x (xy) that iteratively denoises to recover the original data distribution. In the setting of molec- ular conformation generation, the diffusion model adds noise to molecular coordinates R.
The forward process is defined fixed posterior distribution (RR:: RRo). Specifically, it is Markov chain according to a fixed variance schedule 31, 32,3.... Br:
(R.T (o)
II (R, R,-1,,
= = 1 BRR,
(6)
Let a₁ = 1- ā = 11 3t, so the closed form solution of the forward process (Ro) at arbitrary timestep is q (R, RR) (R a; (I) This indicates that with sufficiently large T. the whole forward process will convert R, to standard Gaussian distribution.
The reverse process is reverse Markov chain of the above forward process starting from Gaussian distribution pl (Rr) = NRRr; 0, I):
Po (Ro:T G, R
II Po R (R (R,, (R (. (7)
Here Ai are parameterized neural networks to estimate the means, and d, can be any user-defined 3 variance. In DDPM, no = 2, (R, ví-a S0 R2, ()), and Eg is the parameterized denoising network. Under the parameterization, we have the training objective of DDPM:
(()) E. Ro..oollle -oll,, Vo +VI ans, 1) |
(8)
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B EVALUATION METRICS
Geometric Evaluation We adopted the widely used metrics proposed by (Ganea et al. 2021 to assess the geometric quality of generated conformers. The metrics are based on root-mean-squae -quare deviation (RMSD), which is a normalized Frobenius norm between two atomic coordinate matrices aligned using the Kabsch algorithm (Kabsch 1976). Given the reference conformer set S, and the generated sample set Sg. then the Coverage and Matching metrics following the conventional Recall measurement can be defined
COV-R(S,, S,) = [S,] [fe S, s S,, s
(9)
MAT-R(S,, S,) |S,]
Σ min RMSD(C,C), CES. CES.
(10)
where 5 is the pre-defined threshold. The other two metrics COV-P and MAT-P is the Precision metric which can be defined similarly but with the generated and reference sets exchanged. In practice, S, is set as twice of the size of S, for each molecule. COV measures how well one set is covered by another within a threshold 8, MAT measures the geometric closeness between two sets using the average RMSD of one conformer set with its closest conformer in another set. The recall metric is focused on the diversity, while the precision metric measures the quality.
Chemical Evaluation we also assess chemical properties of the generated conformers, including:
Average energy E: Measures the mean stability of generated conformers, calculated as the aver- age the total potential energy values of generated conformers.
Lowest energy Emin Represents the minimum potential energy among all generated conform- ers, indicating the most stable structure.
HOMO-LUMO gap The difference between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), reflecting electronic stability.
The average gap >e: The mean value the HOMO-LUMO gap across all conformers.
The minimum gap Acmin: The smallest gap among generated conformers, indicating the most chemically reactive structure.
The maximum gap Amaa: The largest gap, indicating the most electronically stable structure.
These chemical properties provide insights into the stability and chemical validity of the generated conformers, complementing the geometric quality assessments.
C IMPLEMENTATION DETAILS
C.I DATASETS
We used GEOM-QM9 (QM9) (Ramakrishnan al. 2014) and GEOM-Drugs (Drugs) (Axelrod) & Gomez-Boone Bombarelli 2022) for analysis and comparison between molecular conformer genera- tion models. Each dataset comprises 40,000 molecules for the training set and 5,000 molecules for the validation set. We obtained the raw data, the pre-processed data and the data split at https://github..cm http://ichub.comm//inniiil commMinnaiXu/GeeDDiiii
Each dataset comprises 40,000 molecules for the training set and 5,000 molecules for the validation set, with each molecule containing 5 conformers. For the test set, we selected 200 molecules for each dataset, resulting in 22,409 and 14,324 conformers in QM9 and Drugs Our primary focus is GEOM- GEOM-Drugs Drugs, the most extensive and chemically relevant subset comprising many drug-like molecules.
Table summarizes the average number of nodes and edges per molecular graph in the datasets, both with and without hydrogen atoms. These statistics provide insight into the structural complexity and size differences between the two datasets.
C.2 MODEL ARCHITECTURE
For the main experiment the paper, we adopt the graph field netN) from (Xu al. 2022). Original GFN takes node embedding h R is the feature dimension) and corresponding
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Table 7: Dataset Analysis
GEOM-QM9
GEOM-Drugs
Avg-node
19.26
45.54
Avg-node w/o H
8.84
25.38
Avg-edge
38.75
95.13
Avg-edge w/o H
17.91
54.81
coordinate xt 12 as input. then produces h x follows:
myj 4m (h'. h², || x511² Egji 8m)
(11)
h [+1
Σ may 0%)
(12)
JEN(()
E SEN(()
d, (R. R,)0,
(13)
where & are feed-forward network and dy denotes interatomic distances. (i) denotes the neigh- borhood of inh node, including both connected atoms and other ones within radius threshold which enables the model to explicitly capture long-range interactions and support disconnected molecular graphs. To better adapt the network our hierarchical generation setting, we only maintain the orig- inal GFN formulation in the first stage unchanged and modify the architecture in the later stages. The main difference that we have extra information Zp E 1 from the previous stage as input, so we change the formulation of ma::
m 4 m (h,, My |xx] 1411.1112,, 2 eey; 6m,
(14)
C.3 EQUIVARIANCE
Equivariance with respect to 3D transformations is a crucial property in molecular conformation gen- eration, the predicted conformations should be consistent under global rotations and translations of the input. The original GFN is translationally invariant and rotationally equivariant. Similarly, we can prove that the modified GFN also has the same properties.
Proposition 2. Parameterizing our network Eq as composition of L modified GFN layers, and take the x after L updates as the output. Then the network is SE(3) equivariant w.rt the 3D system R.
Proof. Refer to Appendix F2
From Proposition we construct an equivariant Markov kernel. If we then sample from the invariant distribution, we can obtain an equivariant reverse generative process using Proposition 3 We can show that at each stage, the probability distribution of the output is invariant and the output is cquivariant.
Proposition 3. (Xu al. 2022) Let p(IT) be an SE(3)-invariant density function, T, be some roto- translational transformations from SE(3). If markov transition p(æ+-1/22) are SE(3)-equivarinn, ie p(z)-1/zz) = PTT, (TT
then we have the density po (Io) is also SE(3)-invarinn
More importantly, we can prove that if each stage the MSGEN framework is SE(3) -quivarran, then the output of the entire framework is also equivariant. This property is critical for ensuring that the generated molecular conformations remain physically meaningful under arbitrary global rotations and translations of the input.
Proposition Assume a MSGEN framework consist of N sequential neural networks fi..... fNr where each f; is equivariant under SE(3) transformations, then the full model F = IN also SE(3) equivariant.
Proof. Refer Appendix
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C.4 DETAILS OF PRELIMINARY STUDY
In this section, we provide the detailed setup and rationale for the preliminary study to evaluate the effectiveness of explicit geometric structural guidance molecular generation. The primary goal of this study is to investigate whether incorporating explicit substructure information can significantly improve molecular conformer generation compared to existing methods.
Model Selection To ensure a fair and comprchensive evaluation, we selected three representative generative models:
1. GeoDiff:
A standard diffusion- based model that generates molecular conformers without explic- itly leveraging structural features. It follows conventional denoising diffusion process where atomic coordinates are progressively refined through Gaussian noise perturbation and denoising steps.
2. SubgDiff: An extension of the GeoDiff approach that incorporates subgraph predictor for subgraph denoising, enhancing the model's 's ability to preserve subgraph patterns.
EBD: An Equivariant blurring diffusion model which generates atomic details from coarse- grained estimation of fragment structures, motivated by the blurring corruption of heat equation.
Additionally, we introduce our variant model, which builds GcoDiff by explicitly incorporating structural guidance. Specifically, our model leverages heavy-atom backbone positional information guidance during the diffusion process. We use the following strategies: 1) the heavy-atom back- bone positions are directly obtained from the ground-truth data; 2) the remaining atoms we placed the geometric center of the heavy atoms. This geometric center is calculated the centroid of the heavy-atom positions and is considered as a masked position to indicate the absence of accurate atomic coordinates for these atoms.
Experiment Setting To ensure fair comparison between the selected models, we maintain con- sistent settings across all experiments. Specifically, we set the number of diffusion steps to 500 for all models. For GeoDiff (Xu et al. (022) and our variant, we train both of them using the same hyperparameters in (Zhang al.] 2024). For SubgDiff (Zhang et al.) (024), we use the result from their paper. For EBD (Park Shen 2024), we train their model with 500 diffusion steps while keeping all other hyperparameters unchanged from the original implementation.
C.5 TRAINING AND TIME
We used single NVIDIA A6000 GPU for every training and generation task. For training, used a learning rate 10 for the first stage and 10-4 for the second stage with the Adam (Diederik) 2014) optimizer. For GeoDiff, the practical training time of the first stage is around 24 hours, and the practical training time of the second stage is around 50 hours. The conditional augmentation time 8 is set as [1, 10), and other hyperparameters of our framework are summarized Table & including highest variance level Br. lowest variance level $1, the variance schedule, number of diffusion timesteps T. radius threshold for determining the neighbor atoms т, batch size, and number of training iterations.
Table 8: Additional hyperparameters our framework
Task
81
9T
B Scheduler
T
T(A)
Batch Size
Training Iter.
QM9
1.e-7
2.e-3
sigmoid
5000
10.0
64
1.5M
Drugs
1.e-7
2.e-3
sigmoid
5000
10.0
32
2M
QM9
Le-7
2.e-2
sigmoid
500
10.0
64
1.5M
Drugs
1.e-7
2.e-2
sigmoid
500
10.0
32
2M
Drugs
1.e-7
9.e-3
sigmoid
1000
10.0
32
2M
Drugs
Le-7
5,e-3
sigmoid
2000
10.0
32
2M
Drugs
Le-7
4.e-3
sigmoid
2500
10.0
32
2M
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C.6 TRAINING AND SAMPLING ALGORITHM
go over the pseudo-code for the two-stage training and sampling procedure. Following prior work (Xu al. 2022), sample from the zero center-of-mass system to build the invariant to rotation and translation sample density for each stage.
The following algorithms 3
Algorithm 3: Training of a two-stage MSGEN framework
// Train base model repeat
(R' (V, (R(RRgg)) 1~ 7})
// Sample backbone of each molecule
(0,
R Va + - āgE
0 - to (R)
until converged
// Train second-stage model (in parallel with base model)
repeat
Sample molecule and get the (R, R X)) (R.
backbone
8 s~ T.). U((I,.,,T)
E ~ I0.
R a R
1 0,
R Vo,R + v 5+E
R cond + (psampling(R1101 9)
// Generate guidance
- 0 1/ ||e Eo (R,, R2 Rond 9
until converged
Algorithm 4: Sampling of two-stage MSGEN framework
Input: the molecular graph G(V, X), g' E,),)), the diffusion step Ti, T2 Output: the 3D molecular conformation R.
R ~ ,N
for i < Ti to do
Shift R! to zero CoM
R ~0 (R R G¹)
end
R + Upsampling [ g ,)
// Generate structural prior
Rí, I),
for i + T2 to do
Shift R, to zero CoM
R- po (R; R,, R2 Roond: 9)
end
return R
C.7 UPSAMPLING MODULES
In our framework, we design several modules for molecular structure refinement across scales, Be- low, we present the implementation details of both techniques.
C.7.1 CENTROID AND RANDOM UPSAMPLING
Centroid Upsampling. The additional fine-level atoms are placed at the geometric center of the entire coarse structure. This geometric center is calculated the centroid of the coarse structural positions. It is considered as a masked position to indicate the absence of accurate atomic coordinates for these atoms.
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Random Upsampling The spatial coordinates of the additional fine-level atoms randomly drawn from a standard Gaussian distribution. It a random initialization for unknown atoms.
C.7.2 MOLECULAR UPSAMPLING
This meth Algorithm reconstructs fine-level atomic coordinates by leveraging coarse- grained atoms as chemically meaningful anchors. For each fine-grained atom-such as hydrogen or side-chain atom-its spatial position is estimated relative to its covalently bonded atoms. Ran- dom directions are applied within chemically plausible radius to reflect local variations, ensuring the reconstructed geometries respect chemical bonding patterns and diree-dimensiona spatial con- straints.
C.7.3 GEOMETRIC UPSAMPLING
Our primary framework designed to leverage explicit substructural guidance as it provides superior accuracy in generating molecular conformers, as demonstrated by our study. However, we recognize that there are practical scenarios where coarse features are used. In such cases, it is beneficial to have a flexible alternative that can still provide reasonable geometric reconstructions.
To address this, we propose a (ete Algorithm 5) that reconstructs fine-level atomic coor- dinates by interpreting each coarse-level site as the centroid of chemically meaningful structural region, such as a rigid substructure center or functional group. We define a local conformational re- gion (ii, 7) around each coarse node modeled a sphere in 3D space, within which the fine-level atoms are distributed. To capture the spatial variability. atom positions are sampled along random directions on the surface of this sphere, with distances scaled by chemically plausible radius т.
Algorithm 5: Structure Aware Geometric Upsampling
Input: Coarse coordinates Recarse Rax3 mapping M from fine atom coarse index radius prior
Output: Fine-grained conditions Reond E Rmx3
Initialize Reond + O;
for Each fine-level atom j = -11,..,, 1,...,m.
......,77 m do Get parent coarse node 4 MM();;
Sample direction d, ~ (O,
Normalize d, d ||dy]];
Reond [j] + R Recare [1] + rr dd
end
return Reond
In practice, given the coarse feature positions Recarse E R and the coarse-to-fie -o-fine mapping M, our objective is to reconstruct the atom-level conditions Roond E R2x3 For each fine-level atom use the mapping M find its related coarse node i = MMj).. And then sample random direction d, ~ (0, I) and normalize it. The final conditional position of atom j is place on the tangent space of conformational sphere S(i, by:
Reond [j] Recarse +T
(15)
This method reflects the spatial flexibility atoms anchored around coarse node, and enables geom- etry reconstruction even without explicit bonding constraints. This idea is inspired by prior efforts in (Riniker Landrum 2015) Qiang et al. 2023), where spatial priors are incorporated to generate more realistic molecular conformers. We prove the effectiveness of this method in Appendix G.9
Coarse-to-Condd Coarse Conditioning via Atom-Level Reconstruction. While our primary approach emphasizes explicit substructure guidance, there may be scenarios where both the current and pre- vious stages operate at coarse resolutions (e.g., fragments or virtual nodes). When both current and previous stages operate at coarse resolutions, adopt a reconstruction-bused strategy to en- sure meaningful cross-stage conditioning. Specifically, the coarse output R (k-1) of stage k - 1 is first upsampled to recover an approximate full-atom structure using chemical and geometric upsam- pling. This full-resolution conformation is then re-coarsened using the same rules Mr as stage k, producing a resolution-alignedd structure that serves as
R (k) sond
((A--1),(81))
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This strategy preserves chemical plausibility while ensuring geometric consistency across coarse levels, acting as structural regularizer that improves alignment and stability throughout the gener- ative framework.
Unified perspective The rationale behind introducing the geometric upsampling and coarse-to- coarse conditioning strategies is enhance the flexibility and extensibility of our framework. While our primary framework is designed to leverage explicit substructural guidance, we recognize that practical scenarios often demand flexibility in handling diverse input types. In particular, there are situations where coarse features are the only available structural cues, or where transitions between coarse generative stages are necessary. The introduction of these methods highlights our frame- work's compatibility with both explicit and coarse features, showing its broad applicability.
D IMPLEMENTATION OF FRAMEWORK GENERALIZATION
Dataset. For ConfGF (Shi et al. 2021). J we use the same dataset as GeoDiff. And for ET-Flow (Has- et al., 2024), we the GEOM- Drugs dataset from (Axelrod & Gomez-Bombarell. 2022). which contains 304k drug-like molecules, each with an average of 44 atoms. For the data split, we use train/validation/eestt (243473/3433//000) split following the setting in their paper.
Baseline. For ConfGE, we use the result from their paper. And for ET-Flow, we download their code from the public Github http://ggghubb.com/hheemniii their model for our experiment. Notably, during the reproduction of ET-Flow (Hassan et al. 2024)) we encountered a critical issue when integrating the original source code. Specifically, the codebase utilizes the Kabsch algorithm to perform alignment during training and samples from harmonic prior. However, we found that this setup frequently results in NaN values during the initial training iterations, which renders the model training process. It is worth noting that the ET-Flow itself supports sampling from a Gaussian distribution as an alternative to the harmonic prior. Therefore, to ensure stable training, we used the Gaussian distribution throughout our experiments.
and use their config to train
To maintain the consistency and fairness of our evaluation, we preserved all other experimental described in the original ET-Flow implementation. The only ad-
settings and hyperparameters justments were the use of Gaussian sampling and the removal of the alignment step. As a result, the reported performance of ET-Flow in work may differ from the originally reported results. However, these results represent a reasonable and reproducible implementation the ET-Flow.
D.1 MODEL ARCHITECTURE
ET-Flow Original ET-Flow consists of two main components:()) representation module based on the equivariant transformer architecture from TorchMD-NET (Thölke & De Fabritiis| (002)) and (2) the equivariant vector output module (Schütt al.) 2017). We prefer readers to (Hassan et al.
2024 for more details.
For the conditional case, we modify the neighborhood embedding layer, which captures the local atomic environment:
embednbh [o(dij 9 I(E
(16)
where embed (2z) provides separate embedding for atomic numbers, dej is the distance between atom and j. di is the distance from the condition structure and 4j encodes edge features. The interaction function g combines distance and edge information. Then we change the attention weight formula:
A SiLU K, ))) o(dey, d()))
(17)
ConfGF ConfGF adopts a variant of Graph Isomrrphism Network (GINs) (Xu et al. 2018 to
compute node embedding based on graph structures. And then use an MIP as the noise conditional network to map edge embedding to scores. We prefer readers to (Shi et al. 2021 for more details.
For the conditional case, given molecular graph g = (V,E), wo modify the edge embedding layer: heg MT.P(eij dij, goond), Veig E
(18)
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Here Egj is the edge attribute and dignd is calculated from the condition.
D.2 EQUIVARIANT BLURRING DIFFUSION
Equivariant Blurring Diffusion (EBD) Park & Shen. 2024) has demonstrated promising perfor- mance in generating molecular conformers by progressively refining coarse molecular features into fine-grained atomic details. However, its reliance on fixed coarse fragments as starting points limits its structural adaptivity, especially when generating large and flexible molecules where maintaining spatial coherence among substructures is crucial. Our proposed multi-scale molecular generation framework provides an opportunity to address this limitation and enhance the performance of EBD.
Analysis of EBD. EBD leverages blurring schedule the core mechanism for generating molec- ular conformers. To understand why EBD can be seamlessly integrated with our multi-scale frame- work, it is essential to first analyze the fundamental principle behind its blurring process:
1. )=g + t M ( x1,)) (1-
(19)
T
where z is the ground-truth atom position, a5 the fragment feature, and M is the projection matrix between fragments and atoms. So M&Ø represents the positional prior derived from fragment features, serving as the initialization for the relative atomic positions within each fragment. In other words, EBD essentially propagates the atomic positions outward from the fragment centers, progressively reducing ambiguity to obtain the final conformer. Our framework can provide better structural prior for this process.
Experiment Setting for EBD. For the baseline model, we use the checkpoint from the public Github wttpghhhwwwoooonnnnaaaaaaeiee-ee
To enhance structural awareness of EBD, we design two-stage framework that integrates EBD with our method. In the first stage, we leverage GeoDiff (Xu et al.) 2022 to generate the heavy-atom backbone, which serves as a structural scaffold. We then apply our chemical upsampling method to build more informative condition for EBD progressively. Specifically, we utilize the backmapping matrix M transform the upsampled condition into a coarse-level feature &/. In the second stage, we employ EBD to refine the atomic coordinates from the coarse features obtained in the first stage. To ensure that EBD effectively leverages the new coarse features, retrain EBD with the same hyperparameters from scratch using these enhanced initial conditions. This retraining process adapts EBD to the improved structural priors, enabling to recover fineggrained atomic details more accurately.
Discussion Since the accuracy the initial atomic positions is critical to the final conformer quality, the frasee initialization in EBD can limit its performance, especially when generating large and flexible molecules. Our framework addresses this limitation by providing more accurate and chemically plausible priors through multi-stage generation process. Our framework can provide atomic coordinates that are better aligned with the underlying molecular structure, which shows the superiority.
E DETAILED GUIDANCE ON MULTI-SCALE DESIGN
We provide guidance on how MSGEN can be extended to multi-scale design. The MSGEN frame- work is designed to seamlessly integrate with existing molecular decomposition strategies and adapt to diverse application needs. particular, additional coarse stages can be introduced capture more fine- fine-grained structural priors. We outline two general approaches:
Graph-based extraction. Standard chemical decomposition methods such as Murck- oScaffolds (Bemis& & Murcko 1996), BRICS (Degen et al.) 2008), or other domain-specine motifs be applied to extract chemically meaningful substructures (e.g.. core ring sys- tems, linkers). These substructures then serve as coarse representations for the first stage of generation.
Structure-based construction. Coarse scaffolds can also be directly constructed from SMILES strings or RDKit (Landrum et al., 2013) RDMol objects, which is particularly useful when the scaffold is predefined by synthetic functional constraints.
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Our structure-aware molecular upsampling then bridges each stage by propagating geometry from coarse to fine levels. This modular design allows our framework to flexibly accommodate both data- driven and manually designed structural hierarchies, making it applicable across wide range of molecular generation tasks.
F
PROOFS
PROOF OF PROPOSITION
Proof. In the proposition, the ELBO is derived by conditioning directly on the coarse structure R which can be considered as truncating the reverse process of coarse scale at time 8, without apply- ing the upsampling module Fu. This simplification preserves analytical tractability while retain- ing the essential coarse-to-fine dependency. Recall that the original evidence lower bound(ELBBO) - log (xro < (o(r)) in DDPM, where
Lo(xo) F. (+ro)))+++++ E
DKL(((n-1114)) xo mo)o llg (o ~ g(xo) (20)
So when we generate a finer-scale molecule Ro involving first generating the coarser-scale molecule R/ from the previous stage model pe (R), G') and then feeding that result into (R. R 9). So this can be considered as using ancestral sampling from the latent variable model:
Pa (Ro) = (Ro 9)po (R))) g G')dRjs /
|Po Poo P)pa (Rirr G')dR'n 0.
(21)
So, using the conditional augmentation can be considered as truncating the previous generative process at timestep > i.e
Po (Ro) = / po Ro R (R (', g'yaR', = Po (Ro PR g'ddR
(22)
This can be considered as form data augmentation since we condition on noisy data, Then let us examine the ELBO for (Ro) in Eq. equation 22 Then treat p (Rg) VAE (Kingma al. 2013) with diffusion model prior, and the approximate posterior distribution:
(RooTT Ro. Ro. R =
(23)
t=1
which executes separate forward processes for two-stage pairs. So we have the new ELBO using Eq. equation 20
log p Ro) S (R)) Σ DKL( R() po Po (R. RR, g' po (Ro R 9) t>s
(24)
where (Ra) (R Ro)llp(Rr)) Note that the last term Po R R 9) can also consider as latent variable model conditioned on R.. So it has ELBO of the form log (Ro 9) < (Ro
Lo(Ro 9) EgILT (Ro) Σ DoL R. Ro)llo
R, R 9)
(25)
log R, R,,
Thus, taking Eq. equation into Eq. equation 24 we have the ELBO for the combined model and finish our proof.
F.2 PROOF OF PROPOSITION 2
Proof. We will prove that the output of our network defined in equation 12 13 and 14 is transla- tionally invariant and rotationally equivariant with the input R, which means
Rx h++ GFN(RX), RR g. h²)
(26)
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where 9 I& denote any translation transformations and orthogonal matrices R E R denote any rotation transformations.
Firstly, the node feature h is already invariant to SE(3) transformations, and the distances calculated between two atoms ||* - x3|| 112 || "py I| are still invariant to rotations. Because
Rx; - Rxyll 1| = (x; x²)² T R R(x, x,) = (x4 x,)² [(x; - xx) = |xx
(27)
So the mij in equation 14 is invariant. Similarly, be invariant.
prove that h from equation 12 will also
Next, prove that the position vector x updated by equation 13 preserves rotation equivariance and translational invariance. Given m,j already invariant as proven before, we have:
Σ
_ di
1
(RR +g- RR, - 22 (m;)) = Σ (EN(()
R, - (m).)
(28)
dij
JEN(()
Therefore, we show that +11 satisfies the properties and finish our proof. And by induction, it can be generalized to the L-layer case.
F.3 PROOF OF PROPOSITION 4
Proof. We can prove this by induction on the number of composed functions. Let Ra = R and define R² = f& (R;,1), so that F(R.) Rg. Then we proceed by induction:
1. n = 1, the conclusion is clearly established by the assumption.
2. Assume that the statement holds for n = k. i.e.,
fn o fn-1 o...o fi (RRO + g) = RR,, +
Then, using the equivariance of fn-1 and the inductive hypothesis, we have:
R R((gg) fn+ilfn 0... ff.. g)) I +9)
RRff+1 ) + RR +
Therefore, the equivariance property holds for k = n + 1.
By induction, the full model FN fNN fx-10 fi is SE(3) equivariant.
G ADDITIONAL EXPERIMENT RESULTS
G.1 CONFORMER GENER ON GEOM-QM9
We report the performance of GeoDiff-MSGEN compared to several basclines on the GEOM-QM9 dataset Table 9 GeoDiff-MSGEN achieves the best or highly competitive results across almost all evaluation metrics.
Results and Discussion Compared with the result in Table on GEOM-Drugs. we can see the improvement margins are relatively smaller compared to results on the larger and more flexible GEOM-Drugs dataset. This primarily because molecules in GEOM-QM9 are generally smaller, structurally simpler, and exhibit limited conformational flexibility. Due to the lower complexity of the underlying structures, even models with less sophisticated structural reasoning can achieve competitive results, thereby narrowing the observed performance gap.
In contrast, GEOM-Drugs contains larger and significantly more flexible molecules, with complex multi-scale conformational patterns that pose much greater challenge. In this regime, our frame- work strength the perception of both global molecular topology and local structural variations and achieve much larger improvements over baselines.
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Table 9: Geometric evaluation on GEOM-QM9 0.5.A)
COV-R(%) %)) t Mean Median
MAT-R(A) Mean Median
COV-P(%) t
MAT-P((A) +
Models
Mean Median
Mean Median
GraphDG
73.33 84.21
0.4245 0.3973
43,90 35.33
0.5809 0,5823
CGCF
78.05 82.48
0,4219 0.3900
36.49 33.57
0.6615 0.6427
ConfVAE
55.84 88.20
0.4154 0.3739
38.02 34.67
0.6215 0.6091
GeoMol
71.26 72.00
0.3731 0.3731
ConfGF
88.49 94.31
0.2673 0.2685
46.43 43.41
0.5224 0.5124
SuhgDiff
90.91 95.59
0.2460 0.2351
50.16 48.01
0.6114 0.4791
GeoDiff
88.20 91.20
0.2435 0.2211
51.16 50.53
0.5019 0.4413
GeoDiff-MSGEN
88.90 95.73
0.2428 0.2279
52.10 50.98
0.5014 0.4222
G.2 DOMAIN GENERALIZATION GEOM-DRUGS
We further evaluate the domain generalization ability of our method by training (M9 molecular with up to heavy atoms) and testing on Drugs(medium-sizee organic compounds).
Table 10: Results on the GEOM-Drup! dataset. (6 = 1.25.A)
COV-R(%)
MAT-R(Å)
Model
Train Data
Mean
Median
Mean
Median
GeoDiff
QM9
7.99
0.00
1.9722
1.9845
GeoDiff MSGEN
QM9
22.11
12.71
1.7924
1.7200
Results and Discussion work significantly improves the performance of the baseline model, demonstrating that our frame- work effectively enhances the robustness and generalization of the baseline model.
The results can found Table IO] As shown in the results, our frame-
G.3 FAILURE CASE ANALYSIS
While the MSGEN framework generally improves generation quality across datasets, we also ex- amined whether there exist systematic failure cases where the framework could degrade conformer quality compared to the vanilla baseline GeoDiff.
Results and Discussion As summarized in Table 11 identified small number of molecules (9 out of 200) where incorporating MSGEN yielded higher MAT compared to the baseline. Visualiza- tion and chemistry aware interpretation suggest these molecules often contain multiple ring systems connected by long and flexible linkers. Such configurations exhibit large conformational variability, and small deviations in positional guidance may lead to misaligned ring orientations distorted global geometry. Additionally, several failure molecules already displayed high MAT under the baseline, indicating that these cases are inherently challenging to generate for GeoDiff. Thus, the observed degradation may not primarily stem from the framework, but rather reflect fundamental challenges in modeling highly flexible, multi-ring structures. Despite these rare cases, the majority of molecules benefit from the structural prior introduced by MSGEN, and the overall performance trends remain consistently favorable.
AAA (A)
Count
Interpretation
157
Improvement
0 ~ 0.1
34
Slight degradation
> 0.1
9
Failure
Total
200
Table 11: Analysis of potential failure cases in generation.
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G.4 FEW STEP GENERATION
To evaluate the effectiveness of our multi-scale generation framework, we conduct an ablation study in which the whole diffusion process in each stage is replaced with few-step generation scheme. Specifically, we reduce the number of denoising steps in each stage 7 to a small fixed value and slightly adjust the noise schedule while keeping the overall architecture and training setup un- changed. For the above baselines, we reuse the results reported in (Zhang et al. (2024) for GefDiff. For Subgdiff, we train their model based on the config released on the public GitHub and get the result.
COV-R(%)
MAT R(A-)
COV-P(S)
MAT-P(-))
Dataset
Model
Mean
Median
Mean
Median
Mean
Median
Mean
Median
Drugs (T=1000)
GcoDiff
82.96
96.29
0,9525
0.9334
48.21
46.03
1.3205
1.2824
GeoDifI-MSGEN
90.05
98,00
0.8769
0,8641
56.73
57.32
1.2286
1.1797
GeoDiff
64.12
75.56
1.1444
1.1296
43.16
42.02
1,3806
1.3314
Drugs (T=500)
SubgDiff
74.30
77.87
1.0003
0,9905
GeoDiffMMGGEN
85.06
97.16
0,9538
0.9478
49,28
48.20
1.3140
1,2719
GeoDiff
87.80
93.66
0.3179
0.3216
46.25
45.02
0.6173
0.5112
QM9 (T=500)
Subg
89.40
94.39
0.2543
0.2601
49.21
47.45
0.5030
0.4724
GeoDiff-MSGEN
88.36
95,10
0.2606
0.2332
49,54
48.39
0.4934
0.4624
Table 12: Few Steps Generation
we can see that our framework still significantly outperforms baselines. The results demonstrate that the MSGEN framework maintains strong gener- ation performance in a few-step generation. This robustness stems from explicitly modelling struc- tural priors within the MSGEN architecture. In our framework, the first-stage model implicitly learns to capture coarse grained molecular structure, such as backbone layout and scaffold topology. This intermediate representation serves as a global geometric anchor for subsequent refinement. By conditioning the second-stage stage model on this guidance, MSGEN is able to focus its capacity local corrections and fine-scale geometry while maintaining the structural awareness of key substructures.
Result and Discussion. As shown in Table 12
G.5 COMPLEXITY-AWARE EVALUATION
We performed a complexity-aware evaluation by stratifying molecules based on the number of atoms and the number of bonds (edges). This analysis examines whether MSGEN's improvements are uniform across all molecules or more pronounced for more complex structures, which are typically more challenging to generate due to greater conformational variability.
Results and Discussion As summarized in Table 13 MSGEN achieves better generation quality across all complexity levels, but the gains are significantly larger for more complex molecules. This trend indicates that our framework becomes increasingly beneficial as molecular complexity increases, highlighting one of its core strengths.
Table 13: Performance comparison by number of atoms and edges.
Num of atoms
Model
COV-R (5) f
MAT-R (A) J
COV-P (%) Mean Median
MAT-P (A) 1
Mean Median
Mean Median
Mean Median
GeoDiff
98.23 100.00
0.6564 0.6615
85.67 91.12
0.8520 0.8998 0.8143 0.8025
40
GeoDiff+MSGEN
98.34 100.00
0.6482 0.6291
87.98 95.97
GeoDiff
85.23 96,00
0.9207 0.9118
53.37 52.31
1.2775 1.2407
40
GeoDiff'+ MSGENN
88.69 96.00
0.8826 0.8466
60.42 61.97
1.2023 1.1270
G.6 COMPARISON UNDER EQUAL TOTAL DIFFUSION STEPS
To further evaluate the effectiveness of our framework, we conduct an additional experiment on GEOM- where the total number of diffusion steps is kept the same across models. Specifically, we compare our two-stage framework against the standard model with the same total steps. We set the same number of steps for both stages. The results are shown in Table 14
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Num of edges
Model
COV-R (%) Mean Median
MAT-R (A) Mean Median
COV-P (%) Mean Median
MAT-P (A) 4
Mean Median
GeoDiff
98.31 100.00
0.6322 0.6262
86.64 94.44
0.8263 0.7719
< 40
GeoDiff+MSGEN
97.41 100.00
0.6319 0.6037
89.06 98.08
0.7930 0.7920
GeoDiff
93.62 98.75
0.8232 0,8158
69.27 71.33
1.0841 1.0562
40 50
GeoDiff-MSGEN
94.65 98.57
0.7945 0.8102
75.69 79.51
1.0084 1.0066
GcoDiff
77.91 85.73
1.0018 1.0010
40.09 36.87
1.4410 1.4000
50
GeDDiff+MSGEN
83.39 90.19
0.9583 0.9281
47.17 43.21
1.3690 1.3403
Under this setting, although each individual stage in our framework uses fewer diffusion steps, re- sulting in larger noise increments per step, our method still achieves superior geometric accuracy and coverage. This demonstrates that the coarse-to-fine structure and molecular upsampling intro- duced in our framework can effectively guide the denoising process even under more aggressive noise conditions.
Table 14: Comparison under equal total diffusion steps on GEOM-Drugs (5 = 1.25A).
(OV-R(%)+ Mean Median
MAT-R(A) Á)
COV-P(% t Mean Median
MAT-P(Á), + Mean Median
Total Step
Model
Mean Median
5000
GeoDiff
87.86 97.00
0.8686 0.8545
60.17 62.21
1.1871 1.1412 1.1474 1.1021
GeoDiff-MSGEN
91.12 98.52
0.8324 0.8207
63.26 66.69
2000
GeoDiff
87.82 97.13
0.8897 0.8696
54.35 56.65
1,2506 1.1949
GeoDiffMMSGEN
90.05 98.00
0.8769 0.8641
56.73 57.32
1,2286 1.1797
1000
GeoDiff
82.96 96.29
0.9525 0,9334
48.21 46.03
1.3205 1.2824
GeoDiffMMSGNN
85.06 97.16
0.9438 0.9378
49.28 48.20
1.3140
1.2719
This experiment highlights the strength of our generative framework in efficiently guiding the de- noising process through coarse-to-fine molecular upsampling. The ability achieve superior results despite fewer steps per stage demonstrates the robustness and flexibility of our method in maintain- ing geometric consistency under aggressive noise conditions. This property is particularly valuable when modeling larger and more complex molecular systems.
Baseline
Variant
Avg Time (s)
GeoDiff
Vanilla
223.43
+MSGEN (2-stage)
335.46
+MSGEN (3-stage)
395.78
ET-Flow
Vanilla
0.4243
MSGEN (2-stage)
0.6739
MSGEN (3-stage)
0.8154
ConfGF
Vanilla
42.87
MSGEN (2-stage)
75.10
Table 15: Average generation time per molecule in framework generalizability experiments.
G.7 EFFICIENCY ANALYSIS
Our primary focus in this work is to improve structural fidelity and chemical realism in molecular conformer generation. While computational efficiency is not the central focus, we include an anal- ysis here to characterize the trade-off introduced by our framework. This supplementary evaluation provides insights into the practical cost when plugging our framework into different models.
First, we report the average generation time in the framework generalizability experiments in Ta- ble [5] This increase is moderate and justifiable: As show in Appendix [.I the coarse molecular graphs used in earlier stages structurally simpler. Consequently, each generation step the coarse graph is substantially faster, which offsets part of the additional cost introduced by the multi- stage design.
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detailed coarse-level Published as a conference paper at ICLR 2026
Table 16: Statistics of model checkpoints used in GeoDiff+MSGEN,
Checkpoint
#Parameters
Size (MB)
Drugs 1.pt
1,366,810 (1.37M)
5.21
Drugs_2. 2tpt
1,597,724 (1.60M)
6.09
Drugs_3.pt
2,415,660 (2.42M)
9.22
Total (3-stage)
5,380,194 (5.38M)
20.52
Second, under the number of diffusion steps, MSGEN achieves better structural fidelity within a shorter average generation time compared to the baselines. This demonstrates that our framework only preserves efficiency but also leverages structural guidance reach higher-quality results without extra computational burden. However, we note that this efficiency gain comes at the cost of additional storage requirements(see in Toble which represent trade-off between computational efficiency and storage overhead.
G.8 STEP ALLOCATION ACROSS 2- STAGES.
To investigate the impact of diffusion step allocation between the first stage and second stage, we conduct an ablation study the Drugs dataset with fixed total number of 1000 diffusion steps.
COV-R(% f Mean Median
MAT-R(A) 4 Mean Median
COV-P(%) t
MAT-P(A) Mean Median
Allocations
Mean Median
300+700
82.24 96.00
0.9828 0.9655
45.61 44.67
1,3541 1.2930
400+600
83.43 95.72
0.9715 0.9676
47.56 46.49
1.3374 1.2816
500+500
85.06 97.16
0.9438 0.9378
49.28 48.20
1,3140 1.2719
600+400
83.57 95.95
0.9706 0.9643
49.09 49.50
1.3193 1.2559
700+300
83.97 96.85
0.9677 0.9488
48.79 48.44
1.3205 1.2789
Table 17: Step Allocation on Drugs (5 1.25.4)
Results and Discussion. The results are summarized in Table 17 The results reveal that bal- anced (llocation(50++500) achieves the best overall performance across most evaluation metrics, which suggests that equally distributing generation effort between coarse structure modeling and fine-gra fine-grained refinement leads the most faithful and diverse conformers. This ablation study dicates the importance of balanced hierarchical generation, where sufficient capacity is allocated each stage. It also supports our design choice of two-stage architecture and motivates further exploration of adaptive step allocation strategies.
G.9 FRAGMENT-LEVEE COARSE GENERATION
We design an ablation study based on fragment-level coarse structure generation to verify the effec- tiveness of our geometric upsampling strategy C.7.3
Fragmentation. We decompose the molecule G (V,8) E) into m non-overlapping fragments Fe. where F6 (Vis Ex) and V U V&. E U & using Principal Subgraph(PS) (Kong et al. 2022 The process begins by identifying all unique atomic fragments defined in the fragment vocabulary S. which serves the basis for baare-grainned representation. Following the progressive strategy in PS, the algorithm iteratively merges the neighboring fragments within each molecule. At each iteration, newly merged fragments are ranked by frequency across the dataset, and the most frequent one is added to the vocabulary. This process continues until the vocabulary reaches a predefined size |S]. By adjusting the target vocabulary size, we can control the granularity of the fragmentation: smaller vocabulary yields finer partitions and thus molecular structures.
Once the fragmentation is completed, each atom xa is assigned to its corresponding fragment Ze, and we construct a mapping matrix M that encodes the relationship between fine- grained atoms and Searse-grained fragments. This mapping is later used for geometric upsampling and coarse-to-fine
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Published as a conference paper at ICLR 2026
structural reconstruction. We use RDkit (Landrum et al. 2013) to set the initial coordinates of each fragment as the average of its constituent atom coordinates.
Table 18: Geometric evaluation on GEOM-Drigs (d 1.25.A)
COV-R(S)) 1 Mcan Median
MAT-R(AA)
COV-P(%)
MAT-P(A) 4 Mcan Median
Models
Mcan Median
Mcan Median
GeoDiff
87.86 97.00
0.8686 0.8545
60.17 62.21
1,1871 1.1412
GeoDiff-MSGEN
90.43 97.81
0.8530 0.8332
63.71 65.13
1.1515 1.0939
Experiment Setting. In our experiment, we set the fragment vocabulary size |S| = 50 to preserve more specific and chemically meaningful fragment structures. Then, decompose the confor- mation generation process into two stages. In the first stage, the model generates coarse-grained features at the fragment level. In the second stage. we apply geometric upsampling strategy to transform the fragment-level representation into fine-grained structural prior. This guidance is then used to condition the generation of the full all-atom 3D conformation, enabling the model to preserve global coherence while recovering chemically plausible local structures.
Result and Discussion. As shown the Table 18 our framework consistently outperforms the baseline model. These results highlight that geometric upsampling serves structurally meaning- ful bridge between coarse and fine representations. By leveraging fragment centroids and molecular topology to initialize the fine-level generation, the model can better preserve geometric consistency. This confirms the effectiveness our geometric upsampling design in our framework.
H LLM USAGE STATEMENT
In accordance with the ICLR 2026 policy on the use of large language models (LLMs), we re- port the following statement: LLMs were used solely as writing assistants to polish the language, correct grammar, and improve the readability of the manuscript. No part of the research design, data processing, experiment execution, results involved the use of LLMs. The authors take full responsibility for the entire content of this paper.
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Published as a conference paper at ICLR 2026
I
GENERATION RESULT
Molecular Graphs
Generated Conformer
Ground
Truth
Figure 5: Molecular graphs, generated conformers, and ground truth from GeoDiff+MSGEN (Part
1).
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Published as a conference paper at ICLR 2026
Molecular Graphs
Generated Conformer
Ground
Truth
Figure 6: Molecular graphs, generated conformers, and ground truth from GeoDiff-MSGEN (Part
2).
30


Reviews:
[{"review_id": "6Fy03C1J82", "reviewer": "Reviewer_6BnT", "summary": "This work propose a hierarchy molecule generation method. It first generate scaffold, than intermediate structure, and final full molecules. Each stages generations is upsampled and guide the generation of the next stage. Experiments and ablation studies on drug dataset reveals the effectiveness of the method.", "questions": "1. The three stage design are largely by heuristic rather than theoretical justification. Therefore, more empirical results should be provided to justify the reason for the hierarchy. Can we design finer hierarchy with more stages? Is there any alternative hierarchy tried?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Clear illustration of the method. Figure 1 shows 3 stages in the hierarchy. Figure 2 shows generation process in each stage and upsampling process between stages. Notations are also clearly defined in Section 3, 4.\n2. Strong experimental results. Table 2,3,4 shows significantly better score than previous generative models. Table 5 further shows that the hierarchical generation strategy can improve performance of different basemodels.", "weaknesses": "1. Missing related work. Similar to this work, which split the whole generation process into different stages, [1] also propose a hierarchical method. It first generate global representation, and then use the global representation as guidance to generate the full molecule. It also achieves good performance on Drug dataset. Therefore, it is necessary to compare hierarchy design and experimental performane between this work and [1].\n2. 3 stages needs 3 times inference time and parameters compared to vanilla 1 stage model with the same model size and diffusion inference strategy. Is the comparison with baseline conducted under fair setting? I think the experiments should keep the whole model size, total training time (controlled by training steps), and total inference time (controlled by diffusion steps) similar to the strongest baseline. \n\n\n[1] Zian Li, Cai Zhou, Xiyuan Wang, Xingang Peng, Muhan Zhang, Geometric Representation Condition Improves Equivariant Molecule Generation, ICML 2025.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "mAWclhsCH1", "reviewer": "Reviewer_iJvk", "summary": "This paper introduces MSGEN, a novel hierarchical multi-scale framework for 3D molecular conformer generation. The authors posit that existing deep generative models often fail by overlooking the inherent hierarchical structure of molecules, where key substructures like scaffolds act as anchors for the overall geometry. MSGEN addresses this by generating conformers in a coarse-to-fine process, first generating a coarse-grained structure and then using it as conditional guidance for subsequent, finer-scale stages. The authors demonstrate that this framework can be integrated with a wide range of existing generative models, consistently enhancing their ability to produce more stable, accurate, and chemically plausible conformers, especially for complex drug-like molecules.", "questions": "1. Can you show that MSGEN combined with other backbone models can solve (to some extent) Weakness 1?\n\n2. Did you examine different design choices of molecule upsampling? Current design could be heuristic.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. **Chemically-Grounded Motivation:** The paper's premise is strongly rooted in chemical principles. The preliminary study (Section 3) provides excellent justification by showing that a model provided with ground-truth geometric guidance (the heavy-atom backbone) dramatically outperforms other methods, confirming that substructure awareness is critical.\n\n2. **Novel and Necessary Technical Contributions:** The framework introduces two clever solutions to problems specific to this domain, e.g., Molecular Upsampling and Conditional Augmentation, which I find to be novel and reasonable.\n\n\n3. **8Thorough and Rigorous Experimentation**: The paper is supported by a comprehensive set of evaluations: Geometric and Chemical Evaluation, Generalization, Scalability. The evaluation is not limited to geometric metrics: The authors show that MSGEN-enhanced models produce conformers with lower (better) mean absolute errors on calculated chemical properties like energy and the HOMO-LUMO gap, indicating the generated structures are more physically realistic.", "weaknesses": "1. **Potential for Error Propagation:** As with any hierarchical system, errors from the initial coarse stage can be passed on and potentially amplified by the subsequent fine-grained stages. The paper's failure case analysis (Appendix G.3) acknowledges this, suggesting that for highly flexible molecules, small deviations in positional guidance may lead to misaligned ring orientations. Though they argue that this may be raised by GeoDiff, empirical evidence with other backbone models are not provided.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "412s0XJYGf", "reviewer": "Reviewer_P6Zq", "summary": "This work introduces MSGEN, a hierarchical multi-stage diffusion framework for molecular conformer generation that explicitly models chemical geometry across multiple spatial resolutions. MSGEN decomposes the molecular generation process into a coarse-grained diffusion stage and a fine-grained diffusion stage conditioned on the former, allowing for better-informed generation conditioned on structural geometry. Empirical evaluations demonstrate that this hierarchical approach significantly improves geometric accuracy, structural validity, and diversity over single-scale baselines, and that MSGEN represents a flexible extension for improving existing diffusion models.", "questions": "1. In table 5, are all methods also performed with the same total number of steps?\n2. Would it be possible to train on all stages with a shared backbone that receives the stage $k$ as additional conditioning information? Would this hurt performance significantly?\n3. The structural conditioning enabled by a 2-step generation process, starting with the backbone, seems to play a role somewhat similar to self-conditioning (see reference for implementation example), where the model conditions on its own ground-truth predictions. Do any of the baselines implement a similar ground-truth prediction conditioning mechanism? If not, it may be worthwhile to compare MSGEN to a baseline equipped with it.\n\nWatson, J.L., Juergens, D., Bennett, N.R. et al. De novo design of protein structure and function with RFdiffusion. Nature 620, 1089–1100 (2023). https://doi.org/10.1038/s41586-023-06415-8", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The work additionally provides a motivating study evaluating the performance gains achieved via geometric structural conditioning in conformer generation. \n2. The approach outperforms all baselines on conformer generation metrics, and the authors further demonstrate that the method outperforms GeoDiff alone using the same number of total diffusion steps, positioning MSGEN as a flexible extension for existing diffusion-based conformer generators. GeoDiff + MSGEN is also faster at sampling time than GeoDiff alone.\n3. The authors justify novel developments and implementation details, including conditional augmentation, molecular upsampling, and 2-stage step allocation with rigorous ablations.", "weaknesses": "1. The model requires different parameter sets for each stage in the molecular generation, meaning that model size scales linearly with the number of stages. For a more meaningful comparison, it may be best to scale the benchmark models to a proportional parameter count increase.\n2. Regarding the 2-stage framework adopted in the work's main experimental results, the use of anchor-based upsampling to place hydrogens given the heavy-atom backbone seems somewhat unnecessary. Wouldn't inferring their positions instead using any generic toolkit (e.g. RDKit, and perhaps adding some Gaussian perturbation) provide a stronger prior for the second stage?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "LhNbcBVO7T", "reviewer": "Reviewer_HeM6", "summary": "This paper introduces MSGEN, a hierarchical multi-stage framework for molecular conformer generation. The core idea is to generate molecular structures in a coarse-to-fine manner—first producing coarse substructures such as rigid scaffolds or heavy-atom backbones, and then refining them into full-atom conformations. Each stage is trained independently using a diffusion-based model, and the coarse-level geometry is passed to the next stage via an upsampling-based conditioning module.\n\nThe authors claim that this hierarchical design improves structural awareness, stabilizes generation, and produces more chemically valid conformers across standard benchmarks (GEOM-QM9 and GEOM-Drugs). Extensive experiments are presented across multiple backbone models (GeoDiff, ConfGF, ET-Flow, EBD), showing quantitative improvements in coverage (COV) and RMSD (MAT) metrics, as well as minor gains in ensemble property prediction.\n\nOverall, the paper proposes a conceptually clear extension of existing diffusion-based conformer generation frameworks by introducing multi-stage conditioning, with an aim to better preserve molecular substructures and enhance geometric fidelity.", "questions": "**Point 1. Scaffold Arrangement Correction:**\n\nHow is the relative arrangement of scaffolds in stage 1 adjusted once the full molecular context becomes available? Is there any mechanism allowing later stages to reposition or re-orient scaffolds when linkers are longer or more flexible than expected?\n\n**Point 2. Fair Comparison and Step Allocation:**\n\nCould you provide all main results under the *equal-total-steps* setting? Also, how were the diffusion steps allocated between stages, and why does the 5 k-step version yield higher recall than 10 k?\n\n**Point 3. Information Content of Upsampling:**\n\nWhat specific geometric information do the upsampled atom positions convey? Have you tested simpler or physically guided alternatives (e.g., random initialization, distance-geometry placement, or short MM relaxation) to quantify their effect?\n\n**Point 4. Novelty Beyond Coarse-to-Fine Tradition:**\n\nIn what sense does MSGEN offer conceptual or methodological advances beyond existing hierarchical generation schemes widely used in protein modeling? Is there any plan to automate hierarchy discovery or to demonstrate benefits in domains where coarse-to-fine design is already standard?\n\n**Point 5. Energy-Based Evaluation:**\n\nHow were energy metrics in Table 3 computed, and can you include a direct comparison with SOTA models under identical settings?", "limitations": "", "rating": 2, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "- **Clear Motivation and Formulation:**\n    \n    The motivation to introduce hierarchical structure into conformer generation is logical and clearly explained. The paper provides an organized mathematical formulation and schematic overview (Fig. 2) that makes the proposed pipeline easy to follow.\n    \n- **Compatibility with Existing Diffusion Models:**\n    \n    MSGEN is implemented as a general plug-in framework that can be attached to various diffusion or flow-based backbones (e.g., GeoDiff, ConfGF, ET-Flow). This design flexibility increases the practical usability of the method.\n    \n- **Comprehensive Experimental Coverage:**\n    \n    The authors evaluate their approach on multiple datasets (GEOM-QM9, GEOM-Drugs) and across different base architectures. \n    \n- **Empirical Consistency Across Models:**\n    \n    The hierarchical conditioning leads to modest but consistent improvements over several backbones, suggesting that the framework is stable and reproducible.", "weaknesses": "**Point 1. Ambiguity in Stage-1 Scaffold Arrangement**\n\nThe first stage determines the spatial arrangement of isolated scaffolds (Figure 2) without knowledge of the full molecular context (e.g., linker length or flexibility). This arrangement may dominate the final conformation but cannot adapt once the subsequent stages add connecting atoms.\n\nConsequently, the hierarchical formulation appears somewhat unsmooth, and it remains unclear whether this early arrangement can be corrected or whether it meaningfully affects overall generation quality.\n\n**Point 2. Unfair Comparison and Step-Dependent Behavior**\n\nThe reported improvements may partially stem from the increased number of diffusion steps rather than from the hierarchical design itself. While the appendix (Table 13) includes equal-total-steps results, the main text does not, preventing fair comparison.\n\nMoreover, GeoDiff + MSGEN with 5 k steps outperforms the 10 k-step version in recall (comparing Table 5 and Table 13), suggesting inconsistent behavior that is neither analyzed nor explained.\n\n**Point 3. Limited Informativeness of the Upsampling Scheme**\n\nThe upsampling process simply places new atoms randomly around their topological neighbors. Since these coordinates are only used as conditioning input, it is unclear what meaningful geometric information they (newly added atoms) provide. If the network must relearn local geometry from scratch, the added step contributes little beyond stochastic initialization.\n\n**Point 4. Limited Generality and Lack of Conceptual Novelty**\n\nThe hierarchical coarse-to-fine paradigm is well-established in biomolecular modeling. Diffusion-based protein generators (e.g., RFdiffusion, Chroma) already generate backbones first and refine them via side-chain packing or relaxation, while hydrogens are routinely added in post-processing. Therefore, separating hydrogens or rigid scaffolds into multiple stages offers little conceptual advancement. The independence of stage-wise training is also not unique—it parallels conventional coarse-to-fine pipelines widely used in structural biology.\n\n**Point 5. Unconvincing Energy-Based Evaluation**\n\nThe task of conformer generation is to recover physically valid ensembles, not to discover novel coordinates. Hence, energetic fidelity is central to evaluation. Yet the proposed method shows larger energy errors than comparable add-on methods (such as Woo et al. [1]), which operates on the same dataset. Although ET-Flow uses a different dataset, its much lower energy errors (0.18 / 0.02 kcal/mol) highlight the expected accuracy. Overall, MSGEN does not convincingly improve energetic realism relative to recent baselines.\n\n[1] Woo, J., Kim, S., Kim, J. H., & Kim, W. Y. (2024). Riemannian Denoising Score Matching for Molecular Structure Optimization with Accurate Energy. arXiv preprint arXiv:2411.19769.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)