Equivariant Efficient Joint Discrete and Continuous MeanFlow for Molecular Graph Generation
Authors: Rongjian Xu, Teng Pang, Zhiqiang Dong, Guoqiang Wu
Organizations: School of Software, Shandong University, Jinan, Shandong, China · School of Software, Shandong University, Jinan, Shandong, China; Luke EI, Jinan, China
Graph-structured data jointly contain discrete topology and continuous geometry, which poses fundamental challenges for generative modeling due to heterogeneous distributions, incompatible noise dynamics, and the need for equivariant inductive biases. Existing flow-matching approaches for graph generation typically decouple structure from geometry, lack synchronized cross-domain dynamics, and rely on iterative sampling, often resulting in physically inconsistent molecular conformations and slow sampling. To address these limitations, we propose Equivariant MeanFlow (EQUIMF), a unified SE(3)-equivariant generative framework that jointly models discrete and continuous components through synchronized MeanFlow dynamics. EQUIMF introduces a unified time bridge and average-velocity updates with mutual conditioning between structure and geometry, enabling efficient few-step generation while preserving physical consistency. Moreover, we develop a novel discrete MeanFlow formulation with a simple yet effective parameterization to support efficient generation over discrete graph structures. Extensive experiments demonstrate that EQUIMF consistently outperforms prior diffusion and flow-matching methods in generation quality, physical validity, and sampling efficiency.
Figures & tables
Figure 1: Architecture of Equivariant MeanFlow. This framework jointly models discrete graph structures ( G=(X,E) ) and continuous 3D molecular geometries ( R ) via a shared SE(3)-equivariant encoder ϕθ1 . The encoder’s outputs drive a discrete MeanFlow head ϕθ3 and a continuous one ϕθ2 , which are optimized via a joint loss for mutually conditioned generation.
# Metrics
QM9
DRUG
Atom Sta (%)
Mol Sta (%)
Valid (%)
Valid & Unique (%)
Atom Sta (%)
Valid (%)
Data
99.0
95.2
97.7
97.7
86.5
99.9
ENF [ 18 ]
85.0
4.9
40.2
39.4
–
–
G-Schnet [ 19 ]
95.7
68.1
85.5
80.3
–
–
GDM [ 12 ]
97.0
63.2
–
–
75.0
90.8
GDM-AUG [ 12 ]
97.6
71.6
90.4
89.5
77.7
91.8
Table 1: Results of atom stability, molecule stability, validity, and validity × uniqueness. A higher number indicates a better generation quality. The results marked with an asterisk were obtained from our own tests.
Property
Qmin
Δe
Δv
ϵLUMO
μ
Cv
Bohr 2
eV
meV
eV
D
mol Kcal
QM9*
0.10
64
39
36
0.043
0.00
EDM †
2.76
655
536
584
1.11
1.101
EQUIFM ‡
2.45
599
337
545
1.112
1.038
EQUIMF (ours)
2.37
594
322
494
1.080
1.011
Table 2: Mean Absolute Error for molecular property prediction.
Method
Atom Stable (%)
Mol Stable (%)
EQUIMF
98.9±0.1
93.0±0.2
NormalMF
97.3±0.1
88.1±0.1
Table 3: Ablation results of our models trained with/without equivariance on the QM9 dataset.
Method
Atom Stable (%)
Mol Stable (%)
P1 (ours)
98.9±0.1
93.0±0.2
P2
85.1±0.1
79.4±0.1
P3
83.6±0.1
77.2±0.1
P4
33.7±0.1
28.5±0.1
Table 4: Ablation results of meanflow models trained with/without mutual conditioning on the QM9 dataset.
Figure 2: Molecular Stability vs. Step Number
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Model
Class
Planar
Tree
SBM
V.U.N. ↑
Ratio ↓
V.U.N. ↑
Ratio ↓
V.U.N. ↑
Ratio ↓
Train set
—
100
1.0
100
1.0
85.9
1.0
GraphRNN ( [ 21 ] )
Autoregressive
0.0
490.2
0.0
607.0
5.0
14.7
GRAN ( [ 40 ] )
Autoregressive
0.0
2.0
0.0
607.0
25.0
9.7
SPECTRE ( [ 41 ] )
GAN
25.0
3.0
—
—
52.5
2.2
DiGress ( [ 6 ] )
Diffusion
77.5
5.1
90.0
1.6
60.0
1.7
Appendix
Table 5: Graph generation performance on the synthetic datasets: Planar, Tree and SBM. Results are mean ± std over five runs (40 graphs/run).