Organizations: Institute for Artificial Intelligence, Peking University · School of Intelligence Science and Technology, Peking University · Department of Electronic Engineering, Tsinghua University · Institute for AI Industry Research, Tsinghua University · State Key Laboratory of General Artificial Intelligence, Peking University
Generative modeling of three-dimensional (3D) molecules is a fundamental yet challenging problem in drug discovery and materials science. Existing approaches typically represent molecules as 3D graphs and co-generate discrete atom types with continuous atomic coordinates, leading to intrinsic learning difficulties such as heterogeneous modality entanglement and geometry-chemistry coherence constraints. We propose VecMol, a paradigm-shifting framework that reimagines molecular representation by modeling 3D molecules as continuous vector fields over Euclidean space, where vectors point toward nearby atoms and implicitly encode molecular structure. The vector field is parameterized by a neural field and generated using a latent diffusion model, avoiding explicit graph generation and decoupling structure learning from discrete atom instantiation. Experiments on the QM9 and GEOM-Drugs benchmarks validate the feasibility of this novel approach, suggesting vector-field-based representations as a promising new direction for 3D molecular generation.
Figures & tables
Figure 1: Overview of the proposed neural field framework for 3D molecular modeling. The figure illustrates two tightly coupled pipelines that share the same neural field decoder and reconstruction module. Top: Latent neural field encoding and reconstruction. A 3D molecule, represented by its atomic coordinates and types, is first encoded by a neural field encoder Eϕ into a grid-based latent field z∈RL3×d , where each spatial location stores a local latent code. Given a set of spatial query points Q={qi}i=1m∈Rm×3 , a neural field decoder Dψ maps the latent field to a continuous molecular vector field V=Dψ(Q,z) . Bottom: Latent field diffusion and molecular generation. A denoising diffusion probabilistic model is trained in the latent field space. Starting from Gaussian noise zT∼N(0,I) , the diffusion model progressively denoises the latent variables to obtain a sampled latent field z0 . Th neural field decoder and the reconstruction module then converts the sampled vector field into a discrete molecular structure through iterative gradient-based ascent and merging operations (see Section 3.4 ).
stable
stable
valid
unique
valency
atom
bond
bond
bond
mol %↑
atom %↑
%↑
%↑
W1↓
TV↓
TV↓
len W1↓
ang W1↓
data
98.7
99.8
98.9
99.9
.001
.003
.000
.000
.120
EDM
97.9
99.8
99.0
98.5
.011
.021
.002
.001
.440
GeoLDM
97.5
99.9
98.0
.005
.017
.003
.007
.435
VoxMol
89.3
99.2
98.7
92.1
.023
.029
.009
.003
1.96
FuncMol
89.2
99.0
92.8
.021
.012
.006
.005
1.56
Table 1: QM9 generation results w.r.t. the test set for 10,000 samples per model. ↑ / ↓ indicate that higher/lower is better. The row data corresponds to randomly sampled molecules from the validation set.
stable
stable
valid
unique
valency
atom
bond
bond
bond
mol %↑
atom %↑
%↑
%↑
W1↓
TV↓
TV↓
len W1↓
ang W1↓
data
99.9
99.9
99.8
100.0
.001
.001
.025
.000
.05
EDM
40.3
97.8
87.8
99.9
.285
.212
.048
.002
6.42
GeoLDM
57.9
98.7
.197
.099
.024
.009
2.96
VoxMol
75.0
98.1
93.4
99.6
.254
.033
.024
.002
0.64
FuncMol
69.7
98.8
100
95.3
.245
.109
.052
.003
2.49
Table 2: GEOM-drugs generation results (standard metrics) w.r.t. the test set for 10,000 samples per model. ↑ / ↓ indicate that higher/lower is better. The row data corresponds to randomly sampled molecules from the validation set.
single
median
ring sz
atms/mol
QED
SA
logP
frag %↑
energy ↓
TV↓
TV↓
↑
↑
↑
data
54.5
.011
.000
.658
.832
2.95
EDM
42.2
951.3
.976
.604
.472
.514
1.11
GeoLDM
51.6
461.5
.644
.469
.497
.593
1.05
VoxMol
82.6
69.2
.264
.636
.659
.762
2.73
FuncMol
70.5
109.7
.427
1.05
.713
.811
3.09
Table 3: GEOM-drugs generation results (additional metrics) w.r.t. the test set for 10,000 samples per model. ↑ / ↓ indicate that higher/lower is better. The row data corresponds to randomly sampled molecules from the validation set.
Figure 2: Element-specific gradient magnitude on a planar cross-section of a representative molecule. High-gradient regions surround atomic nuclei, with element-dependent spatial extent and intensity.
Figure 3: Atomic coordinate reconstruction from neural vector fields. Left: reconstruction from raw neural field codes; right: reconstruction after latent denoising.
Figure 4: Comparison of molecular geometry. From left to right: ground truth, reconstruction from raw codes, and reconstruction from denoised codes.
Figure 5: Cumulative distribution of field RMSD versus ground truth for NF and Diff on 1,000 molecules from QM9 and GEOM-drugs. Both fields closely match the ground truth.
Figure 6: Robustness of the learned neural field to isotropic Gaussian noise in the latent code space.
Appendix figures & tables8 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 7: Geometric intuition of gradient-field–guided molecule reconstruction. Left: An intuitive analogy between the predicted gradient vector field and a conceptual energy-like landscape. The scalar potential itself is not explicitly modeled and may not exist globally; the visualization serves only to illustrate the geometry induced by the learned vector field. Particles (red dots) follow local vector directions and move along curved trajectories toward stable equilibrium regions, which correspond to atomic centers. Right: Evolution of particle clustering during field-to-molecule reconstruction. Particles are initialized uniformly within a bounding box and iteratively updated by following the predicted vector field. Over successive iterations, particles converge toward stable equilibrium regions and form dense clusters. Final atomic coordinates are recovered by applying DBSCAN to the converged particle set, with different colors indicating different atom types.
QM9
GEOM-Drugs
Iterations niter
500
500
Step size η
0.1
0.1
Clustering radius ε
0.1
0.1
Minimum samples Nmin
3
3
Query points (C)
200
1000
Query points (H)
200
1000
Appendix
Table 4: Reconstruction parameters for query-point-based field-to-molecule conversion on different datasets.
Figure 8: Robustness of density-based atom extraction to post-processing parameters. Left: Reconstruction success rate and RMSD as functions of the DBSCAN clustering radius ε on the QM9 test set. Performance remains nearly unchanged across a wide range of ε , indicating that particles are already tightly localized around atomic centers after convergence. Right: Effect of the number of particle evolution iterations niter . Increasing niter improves reconstruction quality until convergence, after which gains saturate. Based on this analysis, we use ε=0.1 and niter=500 in all experiments.
Figure 9: One-dimensional illustration of different field definitions. Two atoms are placed on a line, and the induced field values along the spatial axis are visualized. Red markers indicate atomic positions.
Field Definition
Data Source
Success Rate (%)
RMSD (Å)
Gaussian
Ground Truth
100
–
Gaussian
Codes
70.67
0.106829
Tanh
Ground Truth
100
–
Tanh
Codes
94.13
0.094145
Gaussian-Clip
Ground Truth
100
–
Gaussian-Clip
Codes
96.75
0.055479
Appendix
Table 5: Quantitative comparison of different field definitions on the QM9 test set.
Figure 10: Ablation study of the Gaussian-Clip field parameters. We fix σsf and σmag to control the general shape of the Gaussian, and vary dclip to examine its effect on the gradient field. Small dclip values create steep local gradients that are hard to learn, while large dclip values suppress long-range gradients. The selected combination ( σsf=0.1,σmag=0.45,dclip=0.8 ) achieves a balance between smoothness and long-range interactions.
Figure 11: Failure cases without type-exclusive field components. Examples of reconstructed molecules exhibiting spurious atoms when type-exclusive field components are removed. Light green, blue, and red spheres denote F, N, and O atoms, respectively. Atoms circled by dashed lines correspond to spurious atoms that do not exist in the ground-truth molecules but are incorrectly generated during field-to-atom reconstruction.
Figure 12: EGNN radius ablation for latent diffusion. Left: Validation loss (MSE between clean and denoised latent codes) over training steps for different neighborhood radii. Intermediate radii yield the most stable optimization and lowest residual error. Right: Illustration of radius graphs and number of neighbors per anchor. Small radii produce sparse connectivity, limiting message passing, while overly large radii weaken locality bias. The optimal radius balances connectivity and local geometric structure.
Generating accurate 3D molecules requires jointly determining a variable number of atoms, their identities, and geometry across spatial scales: global shape emerges coarsely, whereas chemical bonds demand fine local precision. Existing atom-based diffusion and flow models operate largely at a single scale, while dense spatial fields become costly at high resolution. We introduce OctMole, a multiscale octree flow framework for 3D molecular generation. A hierarchical occupancy flow progressively resolves sparse molecular support from coarse to fine, and a joint atom flow assigns atom types and continuous in-cell offsets to occupied leaves. This decomposition combines efficient global structure with subcell geometric precision, while cell corruption improves robustness to generated support. On QM9 and GEOM-DRUG, OctMole achieves strong chemical, distributional, and geometric quality, with particularly clear gains over Uni-3DAR on GEOM-DRUG and substantial efficiency gains over dense 3D generation. As an additional capability, intermediate occupancy states and selected regions provide spatial conditioning without task-specific retraining, supporting multiscale shape conditioning, regional editing, linker generation, and shape interpolation. Code and pretrained checkpoints are available at https://github.com/octree-nn/octmole and https://huggingface.co/Disembo/octmole, respectively.
Yu-Tong Jin, Yi-Yang Tian, Qiwei Ye +1
Peking University · Beijing Academy of Artificial Intelligence
Three-dimensional (3D) molecule generation has been dominated by diffusion models, which achieve strong generation quality but typically require the molecular size to be specified a priori. Recent autoregressive approaches have substantially narrowed the performance gap while naturally supporting variable-length generation and conditioning on partial molecular context. However, balancing unconditional and context-conditioned generation remains challenging. We introduce KRONOS, a latent autoregressive diffusion framework that generates molecules in the latent space of a pre-trained autoencoder, jointly modeling molecular graph topology and geometry, while retaining the flexibility of autoregressive generation. We further introduce a mixed training strategy inspired by Fill-in-the Middle (FIM) paradigm, enabling both unconditional and fragment-conditioned molecular generation within a single left-to-right autoregressive model. Experiments on QM9 and GEOM-Drugs demonstrate that KRONOS achieves leading unconditional generation performance among autoregressive methods, while remaining competitive with diffusion models. Moreover, fragment-conditioned generation is achieved with negligible impact on unconditional generation performance, demonstrating that both generation paradigms can be supported within a single architecture.
Federico Ottomano, Gaopeng Ren, Yingzhen Li +2
Department of Chemistry Imperial College London · Department of Computing Imperial College London
Molecular size is intrinsically linked to composition, structure, and properties, but most 3D molecular generators treat size as a predefined input rather than a quantity to be generated. We introduce Equivariant-Free Transformer-Autoencoded Latent Flow Matching, a two-stage framework that samples a fixed-dimensional latent vector using flow matching and employs an autoregressive Transformer to determine molecular size, atom types, coordinates, and chemical attributes. Canonical atom ordering and rigid-pose alignment enable Transformers without equivariant layers, while decoded attributes guide bond reconstruction. On PCQM4Mv2, unconditional generation yields 87.9% unique and novel molecules passing sanitization and PoseBusters checks, exceeding baselines with lower end-to-end training and sampling time and higher end-to-end throughput. Across ten target HOMO-LUMO gaps, internal ranking retains 30% of the screened candidates and increases the density functional theory-verified hit rate within 0.1 eV of the target from 25.0% to 52.4%, while largely preserving novelty and diversity. These results demonstrate fixed-dimensional latent generation with autoregressive decoding as a practical approach to molecular design without prespecifying size.
Weichi Yao, Cameron Gruich, Bryan R. Goldsmith +1
Michigan Institute for Data & AI in Society · Department of Chemical Engineering, University of Michigan, Ann Arbor, USA · Department of Statistics, University of Michigan, Ann Arbor, USA