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Equivariant Efficient Joint Discrete and Continuous MeanFlow for Molecular Graph Generation

Unknown · 2026 · arxiv_cs

arXiv CS · Papers · License: Open Access · 2026

machine learning, deep learning, neural networks

arXiv:2604.08189v1 [cs.LG] 9 Apr 2026

Equivariant Efficient Joint Discrete and Continuous MeanFlow for Molecular Graph Generation Rongjian Xu1 , Teng Pang1 , Zhiqiang Dong1 , and Guoqiang Wu *1 *

Corresponding Author

School of Software, Shandong University, Jinan, Shandong, China , [email protected], [email protected], [email protected] *1 School of Software, Shandong University, Jinan, Shandong, China; Luke EI, Jinan, China , [email protected] April 10, 2026 Abstract 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.

Introduction

Graph generation has become a central problem in modern machine learning, with broad applications in chemistry, biology, material science, and network analysis [1, 2, 1

3, 4, 5]. Recent years have witnessed rapid progress in generative models for graphs, enabling the synthesis of complex relational structures with increasing fidelity and diversity. Graph-structured data is inherently discrete at its core: nodes and edges correspond to categorical entities and relational types, respectively. Accordingly, a prominent body of research centers on discrete graph generation, where the primary goal is to model probability distributions over valid node-edge configurations. Most existing methods leverage diffusion-based or flow-based frameworks—paradigms that enable principled likelihood-based training and improved scalability for large-scale graph dat. For example [6] proposed discrete-time discrete-state graph diffusion models. Subsequent works [7, 8] extended this line of research to continuous-time discrete-state graph diffusion. More recently, discrete flow-based graph generation models [9, 10] have emerged, which are specifically designed to mitigate the computational inefficiencies inherent to graph diffusion models. For many real-world applications, including molecular generation, graph structure alone is insufficient: incorporating 3D geometric information has been shown to significantly improve generation quality, validity, and downstream performance, and equivariant architectures can further ensure geometric consistency under rigid transformations. However, existing flow and diffusion models [11, 12, 13, 14] for discrete-continuous molecular generation often decouple topology from geometry, lacking cross-modal synchronized dynamics and inductive biases. This leads to physically inconsistent conformations and slow sampling, hindering their iterative generation process. To address these drawbacks, we propose Equivariant MeanFlow (EQUIMF), a unified generative framework with SE(3) equivariance, which couples the modeling of discrete structural components and continuous geometric properties by leveraging synchronized MeanFlow dynamics. Specifically, we propose a new discrete MeanFlow model for efficient generation over discrete graph structures, leveraging a new, simple yet effective model parameterization. Further, through a unified temporal alignment mechanism and mutual conditioning between structural and geometric representations, our approach enables the efficient generation of molecular structures in just a small number of steps. Our results on a series of benchmarks show that EQUIMF consistently improves generation quality, physical validity, and sampling efficiency over almost 2x faster than state-of-the-art (SOTA) flow-matching and diffusion models. We summarize our main contributions as follows: • New discrete MeanFlow. We propose a new discrete MeanFlow model for discrete domains, which achieves efficient few-step sampling through a simple yet effective model parameterization strategy. • Hybrid MeanFlow with new mutual conditioning. We propose a unified hybrid MeanFlow that jointly models discrete graph structures and continuous 3D geometries via a synchronized time-bridge and iterative mutual conditioning, supporting efficient sampling. • Equivariance-aware design and Empirical improvements. We design an SE(3)equivariant continuous head with theoretical symmetry guarantees, and our equiv-

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Figure 1: Architecture of Equivariant MeanFlow. This framework jointly models discrete graph e = (X, E)) and continuous 3D molecular geometries (R) via a shared SE(3)structures (G 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.

ariant MeanFlow achieves superior performance on molecular generation benchmarks over SOTA flow/diffusion-based baselines.

Preliminaries and Background

2.1

Problem Setup

Notations. In this paper, a plain lowercase letter (e.g., a) represents a scalar, a bold lowercase letter (e.g., a) denotes a vector, and a bold uppercase letter (e.g., A) denotes a matrix. For a vector a, its i-th entry is written as ai . Let [n] = {1, 2, . . . , n} be the index set of the first n integers. Problem Setup. We consider attributed (geometric) graphs with categorical node/edge types and 3D coordinates of all nodes. A graph with n nodes is denoted by G = (X, E, R), where X = x(i) i∈[n] ∈ [b]n , E = e(ij) i,j∈[n] ∈ [a + 1]n×n , and R = r(i) i∈[n] ∈ Rn×3 . 1 Here a and b denote the numbers of actual edge and node types, respectively. We treat the absence of an edge as a special edge type, so the total number of edge types is a + 1. The matrix R collects 3D coordinates of all nodes, where each row r(i) ∈ R3 represents the spatial position of node i. For simplicity, e = (X, E) for a (geometric) graph G. Given a training set D = {G(i) }N denote G i=1 1 Note that here we denote nodes as X instead of x to be more compatible with E and R, thereby helping easy readability.

i.i.d.

with N graphs, where each graph G(i) ∼ pdata , the goal is to learn a generative model (or generator) to match the (unknown) data distribution pdata over graphs.

2.2

Flow Matching For Graph Generation

Flow Matching (FM) [14, 15, 13] aims to learn how to transform a simple, easy-tosample noise distribution p0 = pϵ into a target data distribution p1 = pdata . It is originally proposed for the generative modeling of continuous domains (e.g., R) via an t Ordinary Differential Equation (ODE) dz dt = u(zt , t), t ∈ [0, 1]. For discrete random variables z ∈ [b] (e.g., graph node types), Discrete Flow Matching (DFM) [9, 10] provides an elegant framework by modeling the generation process as a Continuous-Time Markov Chain (CTMC), leading to the Kolmogorov equation, an ODE (a.k.a., probability flow): ∂t pt = K⊤ t pt , where t ∈ [0, 1], pt = [pt (z1 = 1, ..., p(z1 = b)], and Kt (·, ·) is the (instantaneous) transition rate matrix. In the noisy process, DFM defines a deterministic noising trajectory that starts from a data point z1 ∼ p1 and gradually interpolates to an initial noise distribution p0 (typically a uniform distribution over the discrete domain), given by pt|1 (zt |z1 ) = t δ(zt , z1 ) + (1 − t) p0 (zt ), where δ(zt , z1 ) is the Kronecker delta function (δ = 1 if zt = z1 and 0 otherwise). When t = 1, the distribution transforms to the original data point z1 ; as t decreases to 0, it smoothly evolves to the initial noise p0 . In the denoising process, to generate new samples, DFM considers a conditional transition rate matrix Kt|1 (·, ·|z1 ) to reverse the noising trajectory, which can be defined as: ReLU ∂t pt|1 (zt+dt |z1 ) − ∂t pt|1 (zt |z1 ) Kt|1 (zt , zt+dt |z1 ) = b · pt|1 (zt |z1 ) where ReLU(x) = max(0, x) and dt is next infinitesimal time. Then, it has Kt (zt , ·) = Ep1|t (z1 |zt ) Kt|1 (zt , ·|z1 ) . This rate matrix Kt governs the CTMC dynamics, which allows us to start from a sample drawn from p0 and evolve it to recover the data distribution pdata , where the transition probability between discrete states is given by: pt+dt|t (zt+dt |zt ) = δ(zt , zt+dt ) + Kt (zt , zt+dt )dt,

(1)

where dt is replaced with a finite time interval ∆ in practice. Finally, to (implicitly) model the instantaneous rate matrix Kt , DFM learns p1|t (z1 |zt ) which is parameterized by a neural network ϕ(zt , t). While DFM shows promise in molecular graph generation, it has two critical limitations. First, it (implicitly) models the instantaneous rate matrix, which requires numerous fine-grained time steps, leading to low sampling efficiency. Second, most DFMbased models only focus on discrete graph structures, ignoring continuous 3D geometric information and producing geometrically inconsistent molecules. These challenges motivate our unified generative framework, which accelerates sampling while jointly modeling discrete structure and continuous geometry. 4

2.3

Equivariance

Let G be a group acting on an input space X and an output space Y. For each g ∈ G, let Sg : X → X and Tg : Y → Y denote the corresponding linear representation operators. A function f : X → Y is said to be equivariant to the action of G if Tg f (x) = f Sg (x) , ∀g ∈ G, x ∈ X . If Tg is the identity mapping for all g, the function f is invariant to the group action. In molecular and geometric modeling, the relevant symmetry group is the Euclidean group SE(3), generated by translations, rotations, and reflections in R3 . An element g = (Q, a) ∈ SE(3) acts on a point r ∈ R3 as Sg (r) = Qr + a, where Q ∈ R3×3 is an orthogonal matrix (Q⊤ Q = I) and a ∈ R3 is a translation vector. For the function f that outputs geometric quantities in R3 , SE(3)-equivariance requires f (Qr + a) = Qf (r) + a,which is the modeling core. To model equivariant distributions over molecular graphs within our MeanFlow framework, we adopt widely-used E(3)-Equivariant Graph Neural Networks (EGNNs) [16], a type of graph neural network that satisfies the equivariance constraint as our backbone.

Method

In this section, we introduce the Equivariant MeanFlow (EQUIMF), a unified SE(3)equivariant generative framework that jointly models discrete and continuous domains via synchronized MeanFlow dynamics.

3.1

Noising and Denoising Process with Unified Time

We model the dynamics of graph Gt = (Xt , Et , Rt ) with a synchronized time trae t = (Xt , Et ), and continuous geometry Rt are jectory, where discrete structure G constructed by discrete and continuous MeanFlow respectively, which we will discuss in details in subsequent sections. Noising Process. For molecular graphs, a synchronized noising process jointly perturbs the discrete structure and continuous geometry, with et | G e 1 ) pt|1 (Rt | R1 ), pt|1 (Gt | G1 ) = pt|1 (G as follows. 1. Sample Time Pair. Uniformly sample a base time step t ∼ U(0, 1) and a small time interval ∆, such that the subsequent time step s = t + ∆ satisfies s ≤ 1. To guarantee this, we enforce ∆ ∈ [∆min , 1 − t], where ∆min is a hyperparameter controlling the minimum interval length. e t , we inject noise via 2. Inject Noise into Discrete Structure. For discrete structure G DFM trajectory, which interpolates between the data point and a noise distribution p0 : Y Y (i) (i) (ij) (ij) e t |G e 1) = , pt|1 (G pt|1 xt |x1 pt|1 et |e1 i<j

i∈[N ]

(i) (i) (i) (i) (i) where for each node i ∈ [N ], pt|1 xt |x1 = tδ xt , x1 + (1 − t)p0 xt , and (ij) (ij) (ij) pt|1 et |e1 is similarly defined for each edge et . 3. Inject Noise into Continuous Coordinates. For the continuous atomic coordinates Rt (or Rs ), we inject noise using a linear interpolation schedule: Y (i) (i) pt|1 (Rt |R1 ) = pt|1 rt |r1 , i∈[N ] (i)

(i)

where for each i ∈ [N ], rt = tr1 + (1 − t)ϵ, ϵ ∼ N (0, I). Similarly, we define the counterpart ps|1 (Rs |R1 ) of Rs . Denoising Process with New Conditional Generation. In the denoising (or sampling) process, we model the distribution p1|t (G1 |Gt ) according to the following formula: e 1 |Gt )p̂1|t (R1 |Gt ). p̂1|t (G1 |Gt ) := p̂1|t (G

(2)

Note that this is different from the following decomposition in [9]: e 1 |G e t )p̂1|t (R1 |Rt ), p̂1|t (G1 |Gt ) := p̂1|t (G

(3)

e and continuous one R independently, except with which denoises the discrete part G shared time t. In contrast, our modeling approach allows each part to evolve not only conditioned on its own information of the preceding time but also on that of the other part. Intuitively, this can enable us to better keep the overall harmony of (geometric) graphs, including discrete structure and continuous geometry, which is also verified by our experimental results (see Sec. 5). We parameterized the above distributions by neural networks, where θ1 denotes shared parameters for modeling both discrete and continuous parts, and θ2 denotes unique parameters for the continuous part, while θ2 denotes those of the discrete one, and θ = (θ1 , θ2 , θ3 ). We will detail it in the subsequent sections (see Fig. 1 for the overall architecture).

3.2

Equivariant MeanFlow for Continuous Geometry

We introduce a conditional generative model for continuous graph structures based on the continuous MeanFlow [17], which defines the model parameterization to encode an average velocity field, in contrast to the instantaneous velocity fields typically employed in flow-based generative models. This allows us to frame the generative process e t , Rt ), t) = ut (Gt , t). 2 as a trajectory governed by an ODE Ṙt = ut ((G MeanFlow learns an averaged velocity field based on the instantaneous velocity field, where the target average velocity ut,s (Gt , t, s) between the interval [t, s] is defined as: Z s 1 uβ (Gβ , β)dβ. ut,s (Gt , t, s) = s−t t 2 Notably, we vectorize the matrix R and here we can view it as a vector. t

1 ,θ2 Then, we train a parameterized neural network ûθt,s (Gt , t, ∆) to approximate the target average velocity field with s = t + ∆, where the training loss is Lcont (θ1 , θ2 ) = ER1 ,R0 ,t,s w(t, s)× 2 1 ,θ2 ûθt,s (Gt , t, ∆) − sg utgt (G , t, ∆) , (4) t t,s

where w(t, s) is a sampling weight, sg denotes a stop-gradient operation, and the target average velocity utgt t,s is utgt t,s (Gt , t, ∆) := ut (Gt , t) + ∆

1 ,θ2 1 ,θ2 ∇Rt ûθt,s (Gt , t, ∆) · ut (Gt , t) + ∂t ûθt,s (Gt , t, ∆) , which can be efficiently computed by the Jacobian-vector products (JVP). Based on the approximate average velocity, we can transport Rt to Rs by 1 ,θ2 Rs = Rt + ∆ × ûθt,s (Gt , t, ∆).

(5)

1 ,θ2 For convenience, we denote the induced corresponding distribution by ûθt,s (Gt , t, ∆) 1 ,θ2 as pθ1|t (R1 |Gt ). Equivariance. Our parameterized neural network is based on the EGNN [16], a type of graph neural network that satisfies the equivariance constraint as our backbone. Therefore, our model keeps the equivariance.

3.3

Discrete MeanFlow for Molecular Graph Structure

Here, we propose a new discrete MeanFlow model to generate discrete graph structures, where the core idea is to provide a new model parameterization for modeling the average transition rate matrix rather than the instantaneous one in discrete flow matching (DFM). For simplicity, we discuss only one dimension x(i) ∈ [b] of discrete node structures X and omit the similar edge ones E. Recall that in DFM, the temporal evolution of the marginal distribution over discrete (node) states is characterized by an ODE that governs the conservation law of proba(i)

(i) ⊤ (i)

(i)

bility mass: ∂t pt = Kt pt , where pt = [pt (x(i) = 1), . . . , pt (x(i) = b)], and (i) Kt ∈ Rb×b is the instantaneous transition rate matrix. In the practical denoising (or sampling) process, the transition probability between discrete states is given by Eq. (1) with the finite time interval ∆. New Parameterization. In DFM, it learns p1|t parameterized by a neural network (i)

ϕ(Gt , t), to (implicitly) model the instantaneous rate matrix Kt . However, this requires an extremely small time interval ∆, leading to a low convergence rate and sampling efficiency. Intuitively, a finite time interval can introduce an average transition

rate matrix as (i)

K̄t,s (·, ·) :=

1 ∆

Z s t

(i)

Kβ (·, ·)dβ.

If this average matrix could be modeled well, it is expected to use a larger time interval than that of the instantaneous one, thereby accelerating the sampling process. (i) Therefore, based on this insight, to model the average rate matrix K̄t,s , we learn p1|t parameterized by a neural network ϕ(Gt , t, ∆) with additional input ∆ compared with the one in DFM. Similarly to DFM, the training loss with a weighting λ is h 1 ,θ3 (X1 |Gt ) Ldisc (θ1 , θ3 ) = Et,p1 (G1 ),pt|1 (Gt |G1 ) − log pθ1|t i 1 ,θ3 − λ log pθ1|t (E1 |Gt ) , (6) 1 ,θ3 where pθ1|t (X1 |Gt ) =

θ1 ,θ3 i p1|t

Q (i) (ij) 1 ,θ3 1 ,θ3 x1 |Gt and pθ1|t (E1 |Gt ) = i̸=j pθ1|t e1 |Gt .

Algorithm 1 Training Algorithm of EQUIMF Require: Graph dataset D = {G1 , . . . , GN }. 1: Initialization: θ, ∆min 2: while θ not converged do 3: Sampling G1 ∼ D, t ∼ U(0, 1), ∆ ∼ U(∆min , 1 − t) e t ∼ pt|1 (G e t |G e 1) 4: Sampling G 5: Sampling Rt ∼ pt|1 (Rt |R1 ) 6: pθ1|t (·|Rt ) ← ϕθ1 ,θ2 (Gt , ∆) e t ) ← ϕθ ,θ (Gt , ∆) 7: pθ1|t (·|G 1 3 8: Lcont ← Eq. (4) 9: Ldisc ← Eq. (6) 10: Ljoint ← Eq. (7) 11: θ ← Optimize(Ljoint ) 12: end while 13: Return θ

▷ Noising ▷ Noising ▷ Denoising Prediction ▷ Denoising Prediction

Algorithm 2 Sampling Algorithm of EQUIMF Require: Trained models ϕθ1 , ϕθ2 , ϕθ3 . Ensure: Generated graph G1 = (X1 , E1 , R1 ) 1: Sampling noised data: X0 , E0 ∼ p0 ; R0 ∼ N (0, I); 2: t ← 0 3: for t = 0 to 1 − ∆ with step ∆ do 4: s←t+∆ 1 ,θ2 5: ûθt,s (Gt , t, ∆) ← ϕθ1 ,θ2 (Gt , ∆) θ e t ) ← ϕθ ,θ ({Gt , ∆) 6: p1|t (·|G 1 3 7: Rs ← Calculate with Eq. (5) e s ← Sample from pθ (G e s |G e t) 8: G s|t 9: t←s 10: end for 11: Return G1

3.4

▷ Prediction ▷ Prediction ▷ Denoising ▷ Denoising

Joint Training Objective with Mutual Conditioning

To enable tight coupling between discrete graph structure and continuous 3D geometry, we introduce a shared SE(3)-equivariant backbone encoder ϕθ1 that unifies the feature extraction and cross-modal information fusion for both generation heads (parameterized by θ2 and θ3 ). The joint training objective combines the task-specific losses of both heads, with weight hyperparameters λdisc and λcont > 0 to balance their contributions: Ljoint (θ) = λdisc Ldisc (θ1 , θ3 ) + λcont Lcont (θ1 , θ2 ), (7) where θ = (θ1 , θ2 , θ3 ). Time distortion. Besides, we adopt a time distortion strategy, similar to the one proposed in [10]. The key idea is to apply a non-uniform time step discretization during the sampling process, particularly emphasizing critical regions where fine-grained control is needed. The detailed implementation refers to Appendix D. Shared Representation Encoder. To enable tight coupling between discrete graph structure and continuous 3D geometry, we introduce a shared SE(3)-equivariant backbone encoder ϕθ1 that unifies the feature extraction and cross-modal information fusion for both generation heads. The encoder takes Gt and a time embedding τ = MLPt (t) as input and generates a unified representation ht = ϕθ1 (Gt , t), which is a condition for their individual generation process. This design enables the discrete graph and continuous geometry generation tasks to be mutually conditioned on the same information-rich latent representation, laying the foundation for joint generation. (See details in Appendix F). Overall, the training and sampling processes of our proposed EQUIMF are summarized in Algorithm 1 and 2.

3.5

Theoretical analysis

In this section, we analyze the equivariance property of our proposed equivariant Meanflow (EQUIMF) generative model, which is formally stated as follows, where the full 9

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. QM9

# Metrics

DRUG

Atom Sta (%) Mol Sta (%) Valid (%) Valid & Unique (%) Atom Sta (%) Valid (%) Data

99.0

95.2

97.7

86.5

99.9

ENF [18] G-Schnet [19] GDM [12] GDM-AUG [12] EDM [12] EDM-Bridge [20] EQUIFM [11]

85.0 95.7 97.0 97.6 98.7 98.8 98.9 ± 0.1

4.9 40.2 68.1 85.5 63.2 – 71.6 90.4 82.0 91.9 84.6 92.0∗ 88.0 ± 0.3 94.2 ± 0.2

39.4 80.3 – 89.5 90.7 90.7 93.2 ± 0.2

– – 75.0 77.7 81.3 82.4 84.2

– – 90.8 91.8 92.6 92.8∗ 98.9

EQUIMF (ours)

98.9 ± 0.1

93.0 ± 0.2 95.8 ± 0.4

95.0 ± 0.3

84.5

98.7

proof is in Appendix A. Proposition 1 (Equivariance of EQUIMF). Assume that (i) the nodes and edges features is SE(3)-invariant, (ii) the average velocity field of the continuous head is SE(3)equivariant, and (iii) the rate matrix of the discrete head is SE(3)-equivariant. Then, the whole generation process of our proposed EQUIMF is SE(3)-equivariant. We can prove that the assumptions of the above proposition hold in our proposed EQUIMF (see Appendix A), which indicates EQUIMF keeps the equivariance inductive bias.

Related Work

Graph Generative Models. Graph generative models aim to learn the distribution of complex graphs and enable sampling from this distribution, and have been widely applied to tasks such as molecular design [1]. From the perspective of generation paradigms, existing approaches can be broadly categorized into four classes: Autoregressive methods [5, 21, 22] generate graphs incrementally by treating them as sequences, but often face challenges in modeling node order and permutation invariance; VAE-based method [23, 24] reconstruct graph structures via latent variables, including both one-shot decoding and stepwise generation variants; GAN-based method [25] generate graphs or molecules through adversarial training; Normalizing flow methods [26, 27] characterize graph distributions via invertible transformations. Diffusion model methods [28, 29] that generate novel data samples from a given data distribution that employs two Markov chains. Discrete Diffusion and Flow Matching. In recent years, the diffusion/flow paradigm has emerged as one of the mainstream approaches for graph generation. Early works often relax the adjacency matrix to a continuous space to reuse continuous diffusion frameworks [28, 3]. But this weakens the discrete structural properties of graphs and introduces inappropriate noise injection mechanisms. In contrast, discrete diffusion 10

[6, 7, 8] directly defines transitions in the discrete state space, thus naturally preserving the discreteness of nodes/edges and demonstrating strong performance on various graph generation tasks. Closely related to discrete diffusion are discrete flow matching/discrete flow models [30, 10, 31] based on Continuous-Time Markov Chains that their core is to characterize instantaneous transition rates using a rate matrix, with the time evolution of marginal distributions connected by the Kolmogorov equation. Geometric Graph Generation Models. Recent advances move toward joint generative modeling of molecular topology and geometry. A prominent line of work adopts continuous diffusion [28, 12, 32] or score-based [33] models or flow-based models [11] in Euclidean space, where atomic coordinates are gradually denoised from isotropic Gaussian noise. TTo respect physical symmetries, these models commonly incorporate equivariant neural architectures tailored to SE(3) rigid transformations . [16, 34, 35, 36], such as equivariant message passing or tensor field networks, ensuring that predicted geometric updates transform consistently under rotations and translations. This design has led to substantial improvements in stability, sample validity, and physical plausibility for 3D molecule generation.

Experiments

In this section, we justify the advantages of the proposed equivariant meanflow with comprehensive experiments. The experimental setup is introduced in Section 5.1. Then we report and analyze the evaluation results for the 3d geometric graph generation in Section 5.2. We provide the performance of controllable molecule generation that targets predefined desired properties in Section ??. We provide detailed ablation studies in Sections 5.3 and 5.4 to further gain insight into the effects of different methods. At last, we demonstrate the high sampling efficiency in Section 5.5. Other experimental details are in Appendix C.

5.1

Setup

Evaluation Tasks. In this study, following the prior work [11, 12], we evaluate EQUIMF on several tasks related to 3D molecular graph generation. Specifically, we assess the model’s performance on the tasks of Molecular Modeling and Generation and Conditional Molecule Generation. Datasets. We use two commonly used datasets for our experiments. The QM9 dataset [37] is widely used for 3D molecular generation studies and includes 134k small organic molecules with information about various molecular properties. We use this dataset for both unconditional and conditional generation tasks. Specifically, for conditional tasks, we train models to predict chemical properties based on molecular graphs. We also evaluate EQUIMF on the GEOM-DRUG dataset [38], which is used for generating large molecular geometries. This dataset consists of large-scale molecular graphs with 3D atomic positions. It is a suitable testbed for our model’s capacity to generate molecules with realistic geometries.

5.2

Molecular Graph Generation

Evaluation Metrics. Following [11], to assess the model’s effectiveness, we evaluate the chemical viability of the molecules it generates—this metric reflects the model’s ability to capture inherent chemical principles from the training data. Subsequently, we gauge the quality of the predicted molecular graphs using two core stability metrics: atom stability and molecule stability. The atom stability metric calculates the fraction of atoms that exhibit the correct valence state, whereas the molecule stability metric measures the percentage of generated molecules where every atom meets stability requirements. In addition to these stability metrics, we further report validity and uniqueness: validity denotes the proportion of molecules deemed chemically valid by RDKit, and uniqueness is the percentage of distinct compounds among all generated samples. Baselines. Following the prior work [11, 12], we compare our proposed method with existing methods on molecular generation using the methods of Equivariant models [19] and Equivariant Normalizing Flows [16]. Besides that, we also compared with the equivariant graph diffusion models [39] and its non-equivariant variant and improved version. Finally, and most importantly, we compared it with the current SOTA method [11], which is an equivariant flow matching method with Hybrid probability transport for 3d molecule generation. Results. We quantitatively evaluate the performance of our method against state-ofthe-art baselines on both QM9 and DRUG datasets, with results summarized in Table 1. Following [11], we also get the above metrics from 10000 samples for each method. As shown in the table, on the QM9 dataset, our method demonstrates significant superiority across all key metrics. On the DRUG dataset, our method also delivers competitive performance. Overall, our method exhibits consistent and outstanding performance across both datasets. It not only guarantees the stability and validity of generated molecules but also enhances the diversity of outputs, thereby validating its effectiveness and competitiveness in 3D molecular generation tasks. Table 2: Mean Absolute Error for molecular property prediction. Property QM9* EDM† EQUIFM‡ EQUIMF (ours) †

Qmin Bohr2

∆e ∆v ϵLUMO eV meV eV

0.10 2.76 2.45 2.37

64 655 599 594

39 536 337 322

36 584 545 494

µ D

Cv cal mol K

0.043 0.00 1.11 1.101 1.112 1.038 1.080 1.011

[12], ‡ [11]

Baselines. We select the existing method [11] and [12] as our baselines. Results and Analyses. Table 2 reports the mean absolute error (MAE) between the target property values and the values predicted from generated molecules. Overall, our method achieves best performance across most properties, indicating improved controllability and reduced bias in conditional generation. This demonstrates our discrete meanflow and improving cross-domain conditioning can materially enhance the fidelity 12

of property-controlled generation.

5.3

Ablations on the Impacts of Equivariance

To evaluate the effect of equivariant inductive biases in our framework, we construct an ablation that differs only in whether the continuous geometric backbone and the shared backbone encoder keep equivariant coordinate updates. Specifically, we compare: (i) EQUIMF: our default model using an SE(3)-equivariant backbone (e.g., EGNN-style message passing) to predict the geometric MeanFlow field; (ii) NormalMF: a nonequivariant counterpart where the backbone is replaced by a standard graph MLP that takes the same inputs but does not guarantee equivariance (i.e., coordinates are updated without the SE(3)-equivariant constraint). We evaluate on QM9 using Atom Stable (%) and Mol Stable (%) following common practice. Table 3 shows that equivariant inductive biases consistently improve both stability metrics. These results validate that incorporating SE(3)-equivariant inductive bias into the continuous MeanFlow backbone is a key factor for stable and physically consistent molecule generation. Table 3: Ablation results of our models trained with/without equivariance on the QM9 dataset.

M ETHOD EQUIMF N ORMAL MF

5.4

ATOM S TABLE (%)

M OL S TABLE (%)

98.9 ± 0.1 97.3 ± 0.1

93.0 ± 0.2 88.1 ± 0.1

Ablations on the Impacts of Mutual Conditioning

To validate the necessity of bidirectional mutual conditioning between discrete topology and continuous geometry, we consider four different approaches to model the distribution p1|t (G1 |Gt ): P1 :

1 ,θ3 e 1 ,θ2 pθ1|t (G1 |Gt ) := pθ1|t (G1 |Gt )pθ1|t (R1 |Gt ),

P2 :

3 e 1 |G e t )pθ1 ,θ2 (R1 |Gt ), pθ1|t (G1 |Gt ) := pθ1|t (G 1|t

P3 :

1 ,θ3 e 2 pθ1|t (G1 |Gt ) := pθ1|t (G1 |Gt )pθ1|t (R1 |Rt ),

P4 :

3 e 1 |G e t )pθ2 (R1 |Rt ). pθ1|t (G1 |Gt ) := pθ1|t (G 1|t

As illustrated in Table 4, the results validate that mutual conditioning is critical for generating stable molecules: geometry-aware topological updates reduce chemically invalid edges, while structure-guided geometric updates prevent physically implausible conformations.

Table 4: Ablation results of meanflow models trained with/without mutual conditioning on the QM9 dataset.

5.5

M ETHOD

ATOM S TABLE (%)

M OL S TABLE (%)

P1 ( OURS ) P2 P3 P4

98.9 ± 0.1 85.1 ± 0.1 83.6 ± 0.1 33.7 ± 0.1

93.0 ± 0.2 79.4 ± 0.1 77.2 ± 0.1 28.5 ± 0.1

Sampling Efficiency

We evaluate the convergence efficiency and generation quality of our method by tracking the molecular stability during the sampling process, with EquiFM as the baseline. The variation curve of stability with sampling steps is shown in Figure 2. At the 0.95 stability threshold, our method requires only half the number of steps of the baseline, representing a nearly 2× improvement in efficiency. Besides, we get higher final stability.

Figure 2: Molecular Stability vs. Step Number

Conclusion and Discussion

We present EQUIMF, a unified SE(3)-equivariant generative framework that models discrete and continuous domains jointly via synchronized MeanFlow dynamics. By coupling discrete structural and continuous geometric modeling and establishing theoretical guarantees for equivariant graph distribution learning, EQUIMF outperforms existing flow-matching and diffusion models across benchmarks in generation quality, physical validity, and sampling efficiency. Besides, our proposed discrete MeanFlow can be used in other discrete domains, e.g., text generation. EQUIMF achieves highly efficient few-step sampling but does not fully utilize the core MeanFlow merit of single-step sampling. Its current design balances efficiency and quality through a few-step evolution, leaving one-step discrete-continuous generation unexplored.

Impact Statement “This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.”

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Formal Proof of Propositions

Let g = (Q, a) ∈ SE(3) where Q ∈ O(3) and a ∈ R3 . For atomic coordinates R = [r1 , . . . , rN ]⊤ ∈ RN ×3 , we define the rigid action g · R = RQ⊤ + 1a⊤ .

g · ri = Qri + a,

(8)

For discrete node/edge states (X, E), we treat them as SE(3)-invariant: g · (X, E, R) ≜ (X, E, g · R).

(9)

For any i ̸= j and any g = (Q, a) ∈ SE(3), ∥(g · ri ) − (g · rj )∥22 = ∥Q(ri − rj )∥22 = (ri − rj )⊤ Q⊤ Q(ri − rj ) = ∥ri − rj ∥22 . (10) Hence all pairwise squared distances are SE(3)-invariant.

A.1

Invariant Discrete Nodes and edges features

Proposition 2 (Invariant discrete node and edge features). Let Gt = (Xt , Et , Rt ) be a noisy molecular graph at time t. Assume node states Xt and edge states Et represent discrete types (e.g., atom/bond categories). Then for any g ∈ SE(3), (X′t , E′t ) = (Xt , Et )

G′t = g · Gt = (X′t , E′t , g · Rt ).

where

(11)

That is, discrete node/edge features are SE(3)-invariant. Proof. By definition, discrete node features Xt and edge features Et encode the intrinsic chemical properties. These properties are independent of the global coordinate frame of Rt , as they do not depend on the spatial position or orientation of the molecule. For any SE(3) transformation g = (Q, a) ∈ SE(3) (consisting of a rotation Q ∈ SO(3) and a translation a ∈ R3 ), the action of g only affects the 3D atomic coordinates Rt (transforming them to g · Rt = QRt + a). Since Xt and Et are independent of Rt , applying g does not alter the discrete types of nodes or edges. Thus: X′t = Xt ,

E′t = Et

This implies (Xt , Et ) = (X′t , E′t ), confirming that discrete node and edge features are SE(3)-invariant.

A.2

Equivariance of Average Velocity

Proposition 3 (Equivariance of average velocity). Let t < s and define the (groundtruth) average velocity field ut→s (Rt ) ≜

Rs − R t ∈ RN ×3 . s−t

(12)

Then for any g = (Q, a) ∈ SE(3), ut→s (g · Rt ) = ut→s (Rt ) Q⊤ ,

(13)

i.e., the average velocity is SE(3)-equivariant. Moreover, if the continuous MeanFlow head is implemented by an SE(3)-equivariant network (e.g., EGNN) producing ut→s = ϕθ2 (·), then ût→s (g · Gt ) = ût→s (Gt ) Q⊤ . (14) Proof. Using Eq. (8), we have g · Rs − g · Rt = (Rs Q⊤ + 1a⊤ ) − (Rt Q⊤ + 1a⊤ ) = (Rs − Rt )Q⊤ . Dividing by (s − t) yields ut→s (g · Rt ) =

g · Rs − g · R t (Rs − Rt )Q⊤ = = ut→s (Rt ) Q⊤ , s−t s−t

which proves Eq. (13). The learned equivariance Eq. (14) follows directly from the assumed SE(3)-equivariance of the backbone and the head (e.g., EGNN layers preserve the rigid-motion action while operating only on invariants such as Eq. (10)).

A.3

Invariance of rate matrices

Proposition 4 (Invariance of rate matrices). Let the discrete head parameterize node/edge E CTMC rate matrices K̄X t and K̄t conditioned on Gt = (Xt , Et , Rt ). Assume the rate predictors depend on coordinates only through SE(3)-invariant quantities (e.g., pairwise distances ∥rt,i − rt,j ∥22 , or other rigid invariants) and on (Xt , Et ) through their discrete values. Then for any g ∈ SE(3), X K̄X t (g · Gt ) = K̄t (Gt ),

E K̄E t (g · Gt ) = K̄t (Gt ).

(15)

Consequently, the induced discrete transition kernel pdisc θ (Xs , Es |Gt , t, ∆) is SE(3)invariant. Proof. Under g = (Q, a), the discrete inputs (Xt , Et ) are unchanged (Proposition 2). By Eq. (10), all pairwise squared distances (and any function thereof) are unchanged. Hence every scalar input used by the rate predictors is identical under (Xt , Et , Rt ) and (Xt , Et , g·Rt ), implying the predicted rate matrices are identical, i.e., Eq. (15). Since a CTMC transition kernel over discrete states is fully determined by its rate matrix (e.g., via matrix exponential or Euler discretization), the resulting conditional distribution over (Xs , Es ) is also unchanged under g, thus SE(3)-invariant.

A.4

Equivariance of EQUIMF: Proposition 1

Proposition 5 (Equivariance of EQUIMF). Consider one coupled MeanFlow step along a time bridge (t, s): (i) sample/update (Xs , Es ) from the discrete kernel deE termined by K̄X t , K̄t , and (ii) update coordinates by the average-velocity field Rs = Rt + (s − t) ut→s (Gt ). 21

(16)

Table 5: Graph generation performance on the synthetic datasets: Planar, Tree and SBM. Results are mean ± std over five runs (40 graphs/run). Model

Class

Planar

Tree

SBM

V.U.N. ↑

Ratio ↓

V.U.N. ↑

Ratio ↓

V.U.N. ↑

Ratio ↓

100

1.0

100

1.0

85.9

1.0

Train set

—

GraphRNN ( [21]) GRAN ( [40]) SPECTRE ( [41]) DiGress ( [6]) EDGE ( [32]) BwR (EDP-GNN) ( [42]) BiGG ( [43]) GraphGen ( [44]) HSpectre ( [45]) GruM ( [46]) CatFlow ( [47]) DisCo ( [7]) Cometh ( [8]) DeFoG (5% steps) DeFoG ( [10])

Autoregressive Autoregressive GAN Diffusion Diffusion Diffusion Autoregressive Autoregressive Diffusion Diffusion Flow Diffusion Diffusion Flow Flow

0.0 0.0 25.0 77.5 0.0 0.0 5.0 7.5 95.0 90.0 80.0 83.6±2.1 99.5±0.9 95.0±3.2 98.5±1.0

490.2 2.0 3.0 5.1 431.4 251.9 16.0 210.3 2.1 1.8 — — — 3.2±1.1 1.4±0.4

0.0 0.0 — 90.0 0.0 0.0 75.0 95.0 100.0 — — — — 73.5±9.0 96.5±2.6

607.0 607.0 — 1.6 850.7 11.4 5.2 33.2 4.0 — — — — 2.5±1.0 1.4±0.4

5.0 25.0 52.5 60.0 0.0 7.5 10.0 5.0 75.0 85.0 85.0 66.2±1.4 75.0±3.7 86.5±5.3 90.0±5.1

14.7 9.7 2.2 1.7 51.4 38.6 11.9 48.8 10.5 1.1 — — — 2.2±0.3 4.9±1.3

EQUIMF (Our method)

Flow

99.6±1.0

1.6±0.1

97.2±2.2

1.6±0.2

91.2±3.0

4.9±2.0

Assume Proposition 2, 3, and 4 hold. Then the overall one-step transition operator is SE(3)-equivariant: for any g ∈ SE(3), if G′t = g ·Gt , then the next state G′s produced by the same transition satisfies G′s = g · Gs . (17) Proof. (Discrete part). By Proposition 4, the rate matrices (hence the discrete kernel) are invariant under g; therefore sampling (Xs , Es ) from Gt or from G′t = g · Gt yields the same distribution. Since (X, E) are invariant variables, we have (X′s , E′s ) = (Xs , Es ). (Continuous part). Let g = (Q, a). Using Eq. (16) and Proposition 3, R′s = g · Rt + (s − t) ut→s (G′t ) = (Rt Q⊤ + 1a⊤ ) + (s − t) ut→s (Gt ) Q⊤ . Factorizing Q⊤ gives R′s = Rt + (s − t)ut→s (Gt ) Q⊤ + 1a⊤ = Rs Q⊤ + 1a⊤ = g · Rs . Combining with (X′s , E′s ) = (Xs , Es ) yields G′s = (X′s , E′s , R′s ) = g · Gs , proving Eq. (17).

Synthetic Graph Generation Performance

Setup. As we know, our method is a union of discrete and continuous domains; we test the discrete generation performance in this section. Following the [10], we evaluate on standard synthetic graph benchmarks that cover diverse structural patterns, including Planner, SBM [48], Tree datasets [49]. we test the common metrics that valid, unique, and novel. The baseline are all followed [10]. 22

Results and Analysis. The results are presented in Table 5, where performance is measured by VUN and structural Ratio metrics, averaged over five independent runs. The consistent performance gains across all three synthetic datasets confirm the effectiveness of our discrete graph geometry.

Sample Distortion

In our project, we adopt a time distortion strategy similar to the one proposed in [10].The core idea is to apply a non-uniform time step discretization during the sampling process, particularly emphasizing critical regions where fine-grained control is needed. This time distortion function address the issue where uniform time step discretization may not preserve essential properties of the graph during critical stages of sampling, such as when fine local variations are crucial for global graph characteristics like planarity or connectivity. In particular, we define a distortion function fdist (t) that increases the granularity of time steps during the most critical stages of the graph evolution, such that: t′ = fdist (t)

where

fdist (t) ∈ [0, 1],

is a strictly increasing function that stretches the final parts of the evolution process to capture more subtle structural changes in the graph. For instance, one such function we apply is: fdist (t) = 2t − t2 , which accelerates sampling during the initial phase and slows down the final parts to capture critical graph characteristics. This is similar to the polynomial distortion used in [10].The sample distortion strategy, designed to emphasize key transitions in graph dynamics, improves our model’s sensitivity to intricate local changes.

Experimental Details

For the dataset, we primarily use the publicly available dataset QM9 [37] and GEOMDRUG dataset [38]. Our experiments are implemented with a Pytorch based architecture, and the experimental environment is Ubuntu 22.04. For computational resources, we use an PRO6000 96GB GPU. Detailed hyperparameter settings are provided in Table.

Table 6: Hyperparameter Settings

Hyperparameter

Value

Batch Size Optimizer Learning Rate Hidden Layer Hidden Dimension Distortion function iteration λcont λdisc NFE

64 Adam 1 × 10−4 9 256 Ploydec 1000 0.2 0.8 50

Shared Representation Encoder

E.1

Shared Representation Encoder

The invariant structural feature hdisc t , which implicitly encodes geometric information from Rt via the shared encoder is fed to the discrete MeanFlow head to condition graph evolution on geometry. Meanwhile, the equivariant geometric feature hcont t , which implicitly encodes structural information from (Xt , Et ) — is passed to the continuous MeanFlow head to inform velocity field prediction with structural context. This design ensures that both generation heads operate on a unified, information-rich latent representation, laying the foundation for mutually conditioned joint generation.

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