VecMol: Vector-Field Representations for 3D Molecule Generation

Yuchen Hua; Xingang Peng; Jianzhu Ma; Muhan Zhang

stat.MLMar 13, 2026

VecMol: Vector-Field Representations for 3D Molecule Generation

Authors: Yuchen Hua, Xingang Peng, Jianzhu Ma, Muhan Zhang

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 ·

Abstract

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 $E_{\phi}$ into a grid-based latent field $\mathbf{z}\in\mathbb{R}^{L^{3}\times d}$ , where each spatial location stores a local latent code. Given a set of spatial query points $Q=\{\mathbf{q}_{i}\}_{i=1}^{m}\in\mathbb{R}^{m\times 3}$ , a neural field decoder $D_{\psi}$ maps the latent field to a continuous molecular vector field $\mathbf{V}=D_{\psi}(Q,\mathbf{z})$ . Bottom: Latent field diffusion and molecular generation. A denoising diffusion probabilistic model is trained in the latent field space. Starting from Gaussian noise $\mathbf{z}_{T}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$ , the diffusion model progressively denoises the latent variables to obtain a sampled latent field $\mathbf{z}_{0}$ . 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 $\%_{\uparrow}$	atom $\%_{\uparrow}$	$\%_{\uparrow}$	$\%_{\uparrow}$	${W_{1}}_{\downarrow}$	$\mathrm{TV}_{\downarrow}$	$\mathrm{TV}_{\downarrow}$	len ${W_{1}}_{\downarrow}$	ang ${W_{1}}_{\downarrow}$
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.

\uparrow

\downarrow

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 $\%_{\uparrow}$	atom $\%_{\uparrow}$	$\%_{\uparrow}$	$\%_{\uparrow}$	${W_{1}}_{\downarrow}$	$\mathrm{TV}_{\downarrow}$	$\mathrm{TV}_{\downarrow}$	len ${W_{1}}_{\downarrow}$	ang ${W_{1}}_{\downarrow}$
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.

\uparrow

\downarrow

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 $\%_{\uparrow}$	energy ↓	$\mathrm{TV}_{\downarrow}$	$\mathrm{TV}_{\downarrow}$	$\uparrow$	$\uparrow$	$\uparrow$
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.

\uparrow

\downarrow

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 $n_{\text{iter}}$	500	500
Step size $\eta$	0.1	0.1
Clustering radius $\varepsilon$	0.1	0.1
Minimum samples $N_{\min}$	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

\varepsilon

on the QM9 test set. Performance remains nearly unchanged across a wide range of

\varepsilon

, indicating that particles are already tightly localized around atomic centers after convergence. Right: Effect of the number of particle evolution iterations

n_{\text{iter}}

. Increasing

n_{\text{iter}}

improves reconstruction quality until convergence, after which gains saturate. Based on this analysis, we use

\varepsilon=0.1

and

n_{\text{iter}}=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

\sigma_{\text{sf}}

and

\sigma_{\text{mag}}

to control the general shape of the Gaussian, and vary

d_{\text{clip}}

to examine its effect on the gradient field. Small

d_{\text{clip}}

values create steep local gradients that are hard to learn, while large

d_{\text{clip}}

values suppress long-range gradients. The selected combination (

\sigma_{\text{sf}}=0.1,\sigma_{\text{mag}}=0.45,d_{\text{clip}}=0.8

) achieves a balance between smoothness and long-range interactions.

Panel of Figure 11 — 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.

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OctMole: Multiscale 3D Molecular Generation with Hierarchical Spatial Flows

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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.

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