Controlled Generation with Equivariant Variational Flow Matching
Authors: Floor Eijkelboom
Heiko Zimmermann
Sharvaree Vadgama
Erik J Bekkers
Max Welling
Christian A. Naesseth
Jan-Willem van de Meent
Venue: ICML 2025 poster (2025)
Keywords: Variational Flow Matching
Conditional Generation
Equivariance
Molecular Generation
Primary area: deep_learning->generative_models_and_autoencoders
TL;DR: We propose Variational Flow Matching for controlled generation, unifying conditional training and post hoc Bayesian inference with an equivariant formulation for molecular generation.

Abstract: We derive a controlled generation objective within the framework of Variational Flow Matching (VFM),
which casts flow matching as a variational inference problem.
We demonstrate that controlled generation can be implemented two ways: (1) by way of end-to-end training of conditional generative models, or (2) as a Bayesian inference problem, enabling post hoc control of unconditional models without retraining. 
Furthermore, we establish the conditions required for equivariant generation and provide an equivariant formulation of VFM tailored for molecular generation, ensuring invariance to rotations, translations, and permutations. 
We evaluate our approach on both uncontrolled and controlled molecular generation, achieving state-of-the-art performance on uncontrolled generation and outperforming state-of-the-art models in controlled generation, both with end-to-end training and in the Bayesian inference setting. 
This work strengthens the connection between flow-based generative modeling and Bayesian inference, offering a scalable and principled framework for constraint-driven and symmetry-aware generation.

Controlled Generation with Equivariant Variational Flow Matching
Floor Eijkelboom'
Heiko Zimmermann 2 Sharvaree Vadgama"
Erik J Bekkers"
Max Welling'
Christian A. Naesseth" Jan-Willem van de Meent"
Abstract
flows (CNFs) (Chen et al., 2018), models continuous-time transformation between distributions (Song et al., 2021), enabling direct sampling without Markov chain steps. Yet, CNFs have historically been hindered by their reliance on solving high-dimensional ordinary differential equations (ODEs), making both training and sampling computation- ally expensive (Ben-Hamu al., 2022; Rozen al., 2021; Grathwohl al., 2019).
We derive a controlled generation objective within the framework of Variational Flow Matching (VFM), which casts flow matching as a variational inference problem. We demonstrate that con- trolled generation can be implemented two ways: (1) by way of end-to-end training of conditional generative models, or (2) as a Bayesian inference problem, enabling post hoc control of uncondi- tional models without retraining. Furthermore, we establish the conditions required for equiv- ariant generation and provide equivariant for- mulation of VFM tailored for molecular genera- tion, ensuring invariance to rotations, translations, and permutations. We evaluate our approach on both uncontrolled and controlled molecular gen- eration, achieving state-of-the-ard performance on uncontrolled generation and outperforming state- of-the-art models in controlled generation, both with end-to-end training and in the Bayesian in- ference setting. This work strengthens the con- nection between flow-based generative modeling and Bayesian inference, offering a scalable and principled framework for constraint-driven and symmetry-avee awarr generation.
To address these challenges, Lipman et al. (2023) introduced Flow Matching (FM), a simulation-free method for training CNFs by regressing onto vector fields that define proba- bility paths between noise and data distributions. Unlike traditional CNF training, which requires maximum likeli- hood estimation through ODE solvers, FM directly learns vector fields through a per-sample objective, enabling scal- able training without numerical integration. FM general- izes beyond diffusion methods by accommodating arbitrary probability paths, including those based on optimal trans- port (Chen & Lipman, 2024; Klein et al., 2023), enabling faster training while maintaining expressiveness. Empiri- cally, FM has outperformed diffusion models in likelihood estimation and sample quality on datasets such as ImageNet (Wildberger et al., 2024; Dao et al., 2023; Kohler et al., 2023).
Variational Flow Matching (VFM) (Eijkelboom et al., 2024) frames flow matching as posterior inference in the distribu- tion over trajectories induced by the used interpolation. The key idea is approximate the posterior probability path, i.c. the probability distributions (for different points in time) over endpoints given the current point in space, using se- quence of variational distributions. For generation, VFM approximates the true vector field with the expected vector field based on the learned variational approximation. This approach achieves tate-of-the-aa results categorical data generation while maintaining the computational efficiency that makes FM practical. While VFM has demonstrated con- siderable success in categorical data generation and shows initial promising results on general geometries and molecu- lar generation tasks (Zaghen et al., 2025; Guzmán-Corden et al., 2025), its broader potential, both methodologically and in terms of real-world applications, has not been ex- plored. In particular, key distinguishing feature of VFM its ability to directly reason about (approximate) posterior
1. Introduction
Generative modeling has seen remarkable advances recent years, particularly in image generation (Ramesh et al., 2022; Rombach et al., 2022), where diffusion- approaches based on score matching (Vincent, 2011) have proven highly effective (Ho et al., 2020a; Song et al., 2020b). However, these methods rely on stochastic dynamics that require iter- ative denoising steps during sampling, leading to significant computational overhead (Song et al., 2020a; Zhang & Chen, 2022). An alternative approach, continuous normalizing
'qqual contribution "Bosch-Delte Lab 2 AMMab, Correspon- dence to: Floor Eijkelboom <f.ojjjelboo@@uxy @uvv.nl>.
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Controlled Generation with Equivariant Variational Flow Matching
probability paths, which makes it well suited to address two fundamental challenges in flow-matching- baate generative modeling: controlled generation and the incorporation of inductive biases such as symmetries.
equivariant formulation of VFM, ensuring that the gen- erative process respects symmetries such rotations, translations, and permutations critical for molecular modeling.
Controlled generation is a fundamental challenge in gener- ative modeling, requiring models to produce outputs that satisfy specific constraints while maintaining natural vari- ation in unconstrained aspects. This is particularly crucial for applications like molecular design, where the interplay between discrete (atom types, bonds) and continuous (spa- tial positions) properties typically necessitates combining multiple generative approaches. VFM's unified treatment of mixed modalities could potentially address this challenge, though developing effective control mechanisms remains an open problem. Additionally, many real-world applica- tions exhibit inherent symmetries for instance, molecular structures should remain valid under rotations, translations, and permutations. Incorporating such domain-specific con- straints into VFM would be essential for producing outputs with consistent structure and improved generalization. Thus, extending VFM to handle both controlled generation and symmetry constraints represents key direction for devel- oping more practical and reliable generative models.
Results for Molecular Generation: We validate our method on both unconditional and controlled molec- ular generation, achieving state-of-theaat results on unconditional tasks and significantly outperforming ex- isting models in conditional generation - both with and without end-to-end training. Notably, Bayesian infer- encematches or surpasses explicitly trained conditional models, demonstrating its flexibility and scalability.
These advances establish VFM as robust and efficient framework for constraint-driven generative modeling, with broad applications in molecule generation, e.g. material design, drug discovery, and beyond.
2. Background
2.1. Transport Dynamies for Generative Modeling
Generative modeling through transport dynamics is flex- ible framework for approximating complex distributions, such distributions over valid molecular structures, by trans- forming a simple distribution po (e.g., a standard Gaussian) into a target distribution P1. This transformation is typi- cally described by a deeendepnddett process, governed by an ordinary differential equation (ODE):
In this work, we address both of the aforementioned chal- lenges. We extend VFM to controlled generation, deriving principled formulation that enables generative models to sat- isfy explicit constraints. We show that this formulation nat- urally emerges from the connection between flow matching and variational inference (Eijkelboom et al., 2024), allow- ing conditional generation to be understood inference problem. Additionally, this perspective enables post-hoc control of pretrained generative models without requiring conditional training, providing cheap and flexible alternative to standard end-to-end approaches.
Bayesian
dt (1())) = (x(()x(x))) with (o((r) = x,
(I)
where u [0, 1] x RDD RR. a velocity field that guides the evolution over time. The task to approximate d, us- ing parameterized model of (z:), such neural network. While ODEs are invertible (under Lipschitz continuity of u&) com-
Furthermore, we show how we can design variational ap- proximations that are group-equivariant with respect to its expectation, which ensures that the generative dynamics respect key invariances. We demonstrate the utility of these advancements on problems in molecular generation, a field that, as of late, has received great attention and as a result many advances of the state-of-the ofte-oo-thheart.
defining a likelihood through a putation - solving ODEs during training is computationally expensive.
Flow Matching addresses this limitation by directly learning the time-dependent vector field u, on [0, 1] through:
Our key contributions are:
(PM( = Ee,x [1 [
[())
(2)
Controlled Generation as Variational Inference: We derive a controlled generation objective within VFM and show that it can be used in two ways: 1) for end- to-end training of conditional generative models, or 2) Bayesian inference problem, enabling post hoc control of unconditional models without retraining.
While on is intractable, can make assumption on the velocity field uz(z x1) for generative process that is con- ditioned on a specific endpoint æ1 (e.g., a target molecule). The marginal field (x) can then be expressed as an ex- pected value with respect to the posterior probability path Pt (x) x)), which defines the distribution over possible end points for interpolations intersect x at time
Equivariant Formulation: We establish the condi- generation and provide an
tions required for g
u = E Epe(os (x) [ue( x:
(3)
2

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Controlled Generation with Equivariant Variational Flow Matching
enabling efficient estimation through conditional samples. key insight is that minimizing the loss for the conditional velocity field yields the same gradient as minimizing it for the marginal field, leading to the conditional flow matching loss:
3. Controlled and Equivariant VFM
Controlled generation crucial for practical applications in generative modeling, In this section, we 1) extend VFM to controlled generation by deriving a unified objective for both end-to-end training and control using post-hoc Bayesian inference and 2) develop fully equivariant framework that ensures invariance to key symmetries.
(0) == 2 [luua( -z) (z)])
(4)
2.2. Variational Flow Matching
3.1. Controlled Variational Flow Matching
Variational Flow Matching (VFM) extends Flow Matching by introducing a variational perspective. Instead of directly regressing to the true vector field, VFM parameterizes it through a variational distribution of [x1 x):
Controlled generation extends generative modeling by guid- ing the generative process to satisfy constraints imposed by conditioning on additional information y. In Flow Match- ing, the primary goal is to learn a sequence of distributions (Pr)ocr<< that evolve from a source distribution Po to a tar- get distribution P1. the context of controlled generation, introduce controlled velocity field u. y) and corre- sponding controlled probability density path (- y). which dictates how the distribution evolves in response to To distinguish this from standard conditional modeling, we ex- plicitly use the terms controlled velocity field and controlled probability (density) path to emphasize the dependence on the control variable y.
of (z) := of (m) [ur( x1)]
(5)
This reformulation transforms the problem into one of vari- ational inference, minimizing the Kullback-Leible diver- gence between the true posterior pr(21 a)) and its varia- tional approximation (x(1 z)::
(0) == E, [KL (21 2) de [2),
(6)
- [log d (x1 1r]] const.
(7)
The main observation make is that, given datapoint ø1, the intermediate latent variables (re)o<tt<1 are conditionally independent of y,
When the conditional velocity field is linear æ1 (e.g. straight-line interpolation, diffusion), this objective sim- plifies matching only the posterior mean, then
Pe(x F1, 2) p x1).
(11)
E (p) (1 [x))] = (x [ep(es)))(<1)))
This makes intuitive sense: once the final sample Ij is known, it encapsulates all the information about the control variable y, rendering the intermediate states Dy unneces- sary for inferring y- Or, equivalently, given ø1, the control variable y does inform our noising process. We can therefore express the probability path conditioned y as the marginal,
Moreover, the latter expectation only depends element-wise on the marginal expectation, implying we can use a fully factorized variational form without loss of generality, which results in the simplified mean-field objective:
D CME-VEM(O) -E log
[
of (af ( m))
Pr(x w) / (2 an (w) da
-
Σ log of (24 xx)
(10)
= f pz(2 (x)e (x) I y) dr,
Notice that this reduces learning single high-dimensional distribution into learning D univariate distributions, much simpler task.
To arrive at formulation similar to standard VPM we have to find the velocity field that generates this probability path. We establish the corresponding controlled velocity field in Proposition 3.1.
VFM's flexibility terms of the choice of variational distri- bution qe makes it particularly well-suited for molecular gen- eration tasks. For instance, using categorical factors enables modeling of discrete molecular features like atom types and bond orders, which can be combined with Gaussian factors to represent continuous atomic coordinates. This unified treatment of discrete and continuous variables, combined with the efficiency inherited from Flow Matching, makes VFM attractive for mixed-nn moddlitty tasks.
Proposition 3.1. The controlled (marginal) velocity field
ue(x y) = Epp (a [ue [x))];
generates the controlled probability path pe(x v)-
Proof. See Appendix A.I.
3

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This decomposition highlights two key components: (1) the base generative model Pe(21 | æ). which is learned using VFM, and (2) the task-specific likelihood p(y x.),, which enforces the desired constraints. That is, the unnormalised density p(x1) x pe( x) x))(y) | m1) can be interpreted as Controlled Generation with Equivariant Variational Flow Matching
Algorithm 1 Sampling Controlled VFM
reweighted version of pe(22 I x), where p(y x1) acts task-specifie weight. Approximating the influence of p(y 21) near the mean of Pz (r), we treat the prior as the dominant component.
Require: ply x1), m8
2o ~ Po
for k= = 0, 0,... K do
t 2+ k/K
We can view the (approximate) solution to the task-specific generation problem as a fixed-point equation. The mode of the unnormalized density 5(x1) satisfies the fixed-point equation:
$1.0- = (Ixx
for 0,... , do
I1,5+1 t ~1,1 1 + En V.V, 21 ply (1,s))
end for
01 (11.5- - F-t) 21)/(1 - t)
Vx log x) + Vz, logply x1) = 0.
(14)
TR+11 f Tt + (1/T)Ut
end for Return 2T
Using this formulation we propose an iterative method to approximate the mean of p(x1) using fixed-point iteration. Assuming (21 x)) is Gaussian with mean (e(x) and co-
variance Σ,, the mean can iteratively be refined as follows:
In short, we can generate samples from (z y) by in- tegrating the ODE defined by the controlled velocity field ux(x y). If we can approximate pr(x1 | 2, y), then this allows us to compute (x y) in terms of Ep.(2: (a11 (x.w)) [21] in a manner that analogous to the uncontrolled case in (8), as long ut (x x1) is linear in ø1- Note that similar result was found in Zheng et al. (2023).
11 (k-1) = me(x) + V, logp(y (
(15)
where the initialization is =/60 Me(x). See provided pseudo code for details.
This iterative process serves as a practical method to approx- imate the mean of p(x1) without directly normalizing the density. The step size in each iteration is governed by the covariance E,, and convergence depends on properties such as the smoothness and log-concavity of log p(y 21). This underscores the modularity: the pretrained generative model Pt (x x)) serves reusable prior, while task-specific con- straints are incorporated through p(y .11). By iteratively refining the samples, we enable controlled generation tai- lored to specific tasks without requiring retraining of the generative backbone. The fixed-point approach provides an efficient and adaptable solution for incorporating task- specific constraints into generative modeling. Note that our aim is not full Bayesian treatment, but to show that min- imal inference-tine updates - without retraining and with negligible overhead - enable effective post-hoc control. It hence serves as a proof of concept for integrating traditional inference techniques with SOTA generative models without the need of extra training. Note that work by e.g. Park et al. (2024) achieves similar slightly goal through the Tweedie approximation, though said approximation provides ar- guably noisier estimate of the gradient and would be more difficult to learn
A straightforward way to approximate (x æ. y) model it directly by means of a variational distribution of (#1 | a, y) that additionally conditions on the contol y. This leads to the Controlled Variational Flow Matching objective:
[()) - , [log de (x11 xr, [x, const.
(12)
As before, when (1 21) is linear in æ1, it suffices to employ a mean-field variational distribution. In practice, this allows for an efficient and scalable implementation of controlled generation, where the model only needs to learn how the expectation of 21 influences the velocity field, rather than explicitly modeling the full posterior. This sim- plification makes Controlled VFM computationally feasible, preserving its ability to handle complex constraints while maintaining the efficiency advantages of Flow Matching.
3.2. Controlled Generation as Inference
Rather than directly training a conditional model Pe(11 y) from scratch, we can leverage the structure of VFM and formulate controlled generation as a Bayesian inference problem. Specificallyy, we observe that the target posterior distribution can be written as
By explicitly separating the generative model from the task- specific component, this formulation introduces a modular and flexible approach to controlled generation. Pretrained VFM models can be reused across multiple tasks by sim- ply adapting replacing the classifier p(y (x1). avoiding the need to retrain the generative model entirely. This is particularly well-suited for scenarios where tasks or control variables y frequently change. Moreover, unlike standard classifier -uaffine- methods in diffusion (e.g. Chung et al. (2022)), which require a time-dependent classifier pe x))
xr,y) v) x pe(21 p(y 21)
(13)
VFM
Classifier
4

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Controlled Generation with Equivariant Variational Flow Matching
classification in VFM is performed on data pairs (x1, y)- This allows direct usage of off-the-shelf classifiers re- ward functions, which are typically trained on æ1 rather than intermediate states z, enabling controlled generation without approximations or modifications to the underlying flow matching framework.
conditional velocity field is bi-equivariant under G, and 3) the variational posterior is equivariant under G with respect to its expected value. Then the marginal path pc(x) is invariant under G.
Proof. See Appendix A.2.
This modularity is particularly valuable in applications like molecular generation, where a pretrained model (x | x)) can remain fixed while p(y) (z1) is tailored to predict spe- cific molecular properties. Specifically, it opens up the door to using recent complex graph models that are c.g. based on geometry (Zhdanov et al., 2024), topology (Bodnar et al., 2021a;b), or both (Eijkelboom et al., 2023; Kovač et al., 2024; Battiloro et al., 2024; Liu et al., 2024). Similarly, in image generation, the same generative backbone can be paired with new classifiers to enforce style content- specific constraints, facilitating efficient domain adaptation. From a practical standpoint, this framework also opens the door to a variety of inference tools for controlled generation. For instance, energy-based approaches such as Langevin dynamies can be applied iteratively to guide the genera- tion process by exploiting the gradient of the unnormalized posterior.
When G acts linearly, the optimal transport conditional ve- locity naturally satisfies the bi-equivariance property, as:
9 9. - g 1
I1 1-1
(18)
This is particularly relevant for groups such as S,, (the sym- metric group) and SO(n) (the special orthogonal group), both of which act linearly on R". Consequently, under appropriate choices of prior distribution and model archi- tecture, G- GFV ensures that the generative dynamics invariant under S, and SO(n) when employing the optimal transport conditional velocity field, arguably the two most relevant groups in real-life applications.
4. VFM for Molecular Generation
A Unified Perspective on Molecular Generation Tasks. Molecular generation can be divided into three distinct tasks, each addressing different aspects of molecular generation!:
3.3. Group Equivariant Variational Flow Matching
Equivariance describes the property that a model's output transforms in the same way as the input under symme- try, such as a rotation or translation, leading to consistent preservation of the underlying structure. In (Variational) Flow Matching, we aim for all pr to be invariant, ensuring the generative process respects the symmetries of the target distribution. This is crucial for modeling real-world data with inherent symmetries, improving both efficiency and alignment with the data's ' structure.
Discrete-Molecuie Generation focuses on molecular graphs (A, E), i.e. only considers atom types and bond types. Both are treated as categorical variables, with models capturing relationships between atoms and bonds without considering spatial positions.
Generation focuses on atomic positions R to define the molecular geometry. Features like atom and bond types are inferred later using external tools, making this approach ideal for applications where spatial configuration is key, such as molecular docking. Sometimes, the atom type is included as continuous variable in this formulation well.
Continuous-Molecule
Informally, since we only access the mean of the posterior distribution during generation, this can be achieved by de- signing the model qf (z1 Ir) to be group-equivariant with respect to the expectation of the distribution. Specifically, for all g E G in a given group G, we require that:
of (x))g=))) [x1] = og (ma (m) [x1].
(16)
Joint-Molecular Generation is the most comprehen- sive setting, which considers both discrete and con- tinuous features, typically of the form (R, A, C, E), representing positions (continuous), atom types, for- mal charges, and edge types (discrete), capturing the full complexity of molecular data.
In addition, we must ensure that the prior distribution ad- heres to the same symmetry constraints; otherwise, the sym- metry cannot be preserved. Moreover, the conditional ve- locity field used to compute the marginal field must satisfy bi-equivariance, expressed as:
we g æ1) = g ue(x æ1).
(17)
Variational Flow Matching ( (VFM) provides a unified ap- proach that can be applied to any combination of discrete
This leads the following proposition:
Heere, we ignore string representations such as SMILES, as these are primarily studied in the context of auto-regressive model- ing.
Propusition 3.2. Let G be an arbitrary group, and suppose that 1) the prior distribution invariant under G, 2) the
5

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Controlled Generation with Equivariant Variational Flow Matching
Table Results discrete molecular generation QM9 and Zinc250k. Architecture used
VFM is the one used in Digress.
QM9
ZINC250k
Valid
Unique
FCD 4
Valid
Unique
FCD 4
MoFlow
91.36
98.65
4.467
63.11
99.99
20.931
EDP-GNN
47.52
99.25
2.680
82.97
99.79
16.737
GraphEBM
8.22
97.90
6,143
5.29
98.79
35.471
GDSS
95.72
98.46
2,900
97.01
99.64
14,656
Digress
99.00
96.20
Dirichlet FM
99.10
98.15
0.888
97.52
99.20
14.222
Discrete FM
98.01
99.90
0.991
96.21
100.00
14.252
VFM
99.84
99.92
0.471
98.80
99.99
13.805
and continuous molecular features. This makes applicable to all three molecular generation tasks, unlike approaches that require separate techniques for different modalities.
Our focus is not on introducing new architectures but on demonstrating how VFM enables a unified generative frame- work across discrete and continuous data types while also providing scalable approach to controlled generation. Through these experiments, we assess both VFM's suitabil- ity as general-purpose approach to flow-based modeling and its advantages for property-targett generation.
The variational distribution in VFM is defined using a net- work that outputs variational parameters for each feature in the model. For continuous features such as atomic posi- tions R, the network can define a Gaussian variational form, whereas for discrete features, like atomic and bond types (A, E), it can output categorical form. This data-driven posterior formulation results in flow-matching objective that matches the specific structure of the data, eliminating the need for separate sampling strategies, auxiliary networks, or cantrtte-ooninuuuu hybridization techniques. As a result, VFM provides a fully general and efficient framework for molecular generation, allowing any combination of discrete and continuous molecular features to be modeled without modifying the underlying generative process.
4.1. Unconditional Generation
Discrete-Molecular Generation. As part of this work, we have reproduced the discrete molecular generation re- sults presented Eijkelboom et al. (2024) and compared them against discrete flow matching (which was published after VFM) that is the strongest existing discrete flow model at this moment. Our results in Table confirm strong performance across both QM9 and ZINC250K datasets, and we observe that VFM outperforms discrete flow matching, though we do not claim this as a contribution of this work.
Experiments. We evaluate VFM through two sets of ex- periments designed to assess both its general generative performance and its controlled generation capabilities.
Continuous-Molecular Generation. We evaluate G- VFM on the QM9 dataset using established metrics in- cluding negative log-likelihood (NLL), atom stability, and molecule stability. Our model achieves competitive results (- 120.7 NLL, 98.7% atom stability, 82.0% molecule stability), matching or exceeding its flow-based approach. These re- sults suggest that variational treatment of flow matching enhances model performance.
First, we investigate how the transition from Flow Matching to Variational Flow Matching impacts geometric-informet generative performance. To this end, we take state-of-the FM models and retrain them using VFM rather than FM, allowing for direct comparison. We evaluate this across discrete, continuous, and joint molecular generation tasks to determine whether VFM maintains or improves perfor- mance while preserving the benefits of the FM framework.
While recent work like PONITA (Bekkers et al., 2024), which leverages molecular symmetries through weight- sharing and a specialized Sphere2 architecture, has pushed the sfate-of-the-ard further (-137.4 NLL, 98.9% atom stability, 87.8% molecule stability), our focus was not on achieving absolute best performance but rather on explor- ing the potential of variational approaches within the flow matching framework. We deliberately built upon existing flow Row-matching architectures to isolate the impact of our methodological contributions. The strong performance of G- G-VFM, despite using a simpler architecture, suggests that
Second, we assess the effectiveness of our controlled gen- eration formulations by comparing against state-of-the-art conditional generative models. We evaluate VFM in two settings: (1) end-to-end training with property constraints and (2) post-hoe conditioning via variational inference on a pretrained unconditional model. This allows us to test whether our approach can match or exceed dedicated con- ditional models while benefiting from the flcxibility and efficiency of inference-band based control.
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Controlled Generation with Equivariant Variational Flow Matching
stability drop for tasks below 80%.
Table Results continuous molecular generation on QM9.
Architecture used is the one in Klein et al. (2023).
4.2. Conditional Generation
Models
NLL v
Atom stability t Mol stability t
In the conditional generation experiments, we compare both end-to-end VFM and variational inference VFM (VI-VFM) with of-te-of-the-art conditional models QM9 dataset. For end-to-end VFM, we train the model with property supervision, while for VI-VFM we use a pre-trained uncon- ditional model and frame the property control as Bayesian inference using a property classifier pvy x1). We target key molecular properties including polarizability (a), or- bital energies (EhOMO. fLUMO) and their gap (Aæ), dipole moment (14), and heat capacity (C,). Property classifiers trained following (Hoogeboom et al., 2022a).
E-NF
-59.7
85.0
4.9
G-Schnet
N.A.
95.7
68.1
GDM
-94.7
97.0
63.2
GDM-aug
-92.5
97.6
71.6
EDM
-11.77
98.7
82.0
EFM
-11.77
98.7
82.0
G-VFM
-120.7
98.7
82.0
our variational approach provides a promising direction for improving flow-matching methods while maintaining their computational advantages. Incorporating PONITA's archi- tectural innovations into the VFM framework represents an exciting direction for future work.
For evaluation, we measure the Mean Absolute Error (MAE) between target and predicted property values from the clas- sifier on generated molecules, as well as standard molecular quality metrics: atom stability (percentage of atoms with valid valency), molecule stability (percentage of molecules with all atoms stable), and validity (ability to be parsed by RDKit (Landrum, 2016)).
Joint-Molecular Generation. We evaluate G- G-V's's ca- pabilitiees for joint molecular generation on the QM9 and GEOM-Drugs datasets, focusing on key metrics includ- ing atom stability, molecular stability, validity, uniqueness, and the Gensen-Shannor divergence of energy distributions (IS(E)). On QM9, G-VFM achieves strong results (99.6% atom stability, 99.5% molecular stability, 98.9% validity) that are comparable to SemlaFlow, while maintaining the same computational efficiency advantage of requiring only 100 function evaluations (NFE) compared to diffusion- bassd approaches that need 500-1000 NFEs. On GEOM-Drugs, G-VFM achieves competitive performance (99.8% atom stability, 96.5% molecular stability, 95.3% validity) closely matching SemlaFlow (99.8%, 97.7%, and 95.2% respec- tively), though with a slightly higher IS(E) (0.18 vs 0.14). Note that only MiDi, SemlaFlow, and EQGAT-diff generate everything jointly, whereas the other methods use external tools to infer e.g. bond information, which could affect the relatively strong performance of these models.
Results and Discussion. Our end-to-end VFM achieves 2.05 MAE on polarizability prediction, outperforming pre- vious methods like EDM (2.76), EQUIFM (2.41), and GE- OLDM (2.37). The VI-VFM approach achieves competitive results with 2.25 MAE using only an unconditional model. By combining both approaches training end-to-end and then applying variational inference steering we achieve our best results of 1.98 MAE.
These results demonstrate the effectiveness of our varia- tional inference framework for controlled generation. The strong performance of VI-VFM (2.25 MAE) compared to specialized conditional models like EDM (2.76 MAE) sug- gests a promising new direction for post-hoc conditional generation. While D-Flow achieves better performance (1.39 MAE), requires significantly more computational resources during inference. D-Flow searches over many can- didate initializations (high NFE), where our sampler needs one forward pass plus short fixed-poin point calibration (low NFE, no gradients), i.c. where D-Flow needs O(k x N) NFEs, we need only O(N), where & denotes the number of 'starting points' considered by D-Flow and N denotes the number of integration steps. That is, our approach of- fers an effective compromise, with the flexibility choose between pure VI for simplicity the combined approach for enhanced performance. Most importantly, the success of VI-VFM demonstrates that pre-trained unconditional mod- els can be effectively repurposed for controlled generation through variational inference, opening new possibilities for adapting existing models to conditional generation tasks.
Using the same underlying architecture as SemlaFlow but replacing its discrete flow matching approach with VFM allows us to handle both discrete and continuous variables in a unified framework. This unified treatment not only simplifies implementation by eliminating separate sampling procedures for different variable types but also maintains strong performance across datasets of varying complexity. The ability of G-VFM to match SemlaFlow's 's results while reducing architectural complexity demonstrates the poten- tial of variational approaches in flow matching, particularly for challenging mixed-modality tasks like molecular gener- ation.
Though not explicitly provided in the table, we want to emphasize that the performance of G-VEM drops when equivariance is not enforced, e.g. atomic and molecular
7

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Controlled Generation with Equivariant Variational Flow Matching
Table Results joint molecular generation on QM9 and GEOM-Drugs Architecture used for G- G-VFM one in SemlaFlow.
Dataset
Model
Atom Stab
Mol Stab f
Valid
Unique
JS(E)
NFE 4
EDM
98.7
82.0
91.9
98.9
0.12
1000
GCDM
98.7
85.7
94.8
98.4
0.10
1000)
MUDiff
98.8
89.9
95.3
99.1
0.09
1000
QM9
GFMDiff
98.9
87.7
96.3
98.8
0.09
500
EquiFM
98.9
88.3
94.7
98.7
0.08
210
SemlaFlow
99.9
99.6
99.4
97.9
0.08
100
G-VFM
99.6
99.5
98.9
97.5
0.09
100
MiDi
99.8
91.6
77.8
100.0
0.23
500
GEOM
EQGAT-diff
99,8
93.4
94.6
100.0
0.11
500
SemlaFlow
99.8
97.7
95.2
100.0
0.14
100
G-VFM
99.8
96.5
95.3
100.0
0.18
100
Table 4. Quantitative evaluation of conditional molecule generation. Values reported in the table are MAE (over 10K samples) for molecule property predictions (lower is better).
Property
Ac meV
HOMMO meV
ÉLUMO meV
Ce
Units
Bohr
D
mei K
QM9
0.10
64
39
36
0.043
0.040
EDM
2.76
655
356
584
1.111
1.101
EQUIFM
2,41
591
337
530
1.106
1.033
GEOLDM
2.37
587
340
522
1.108
1.025
D-Flow
1.39
344
182
330
0.300
0.784
G-VFM (End-to-End)
2.05
512
298
445
0.923
0.901
G-VFM (Bayesian Inference)
2.25
534
312
468
0.978
0.956
G-VFM (Both)
1.98
498
289
432
0.901
0.889
5. Related Work
neural networks for molecular generation. Recent work (Bekkers et al., 2024; Vadgama et al., 2025) leverages pre- conditioned diffusion (Karras et al., 2022) for faster sam- pling. These approaches maintain E(3)/EE(3) symmetries through equivariant graph neural networks (Satorras et al., 2021; Gasteiger et al., 2022: Bekkers et al., 2024), handling continuous atomic positions while encoding discrete proper- ties like atom types one-hot vectors. Finally, Cornet et al. (2024) aims similarly at improved conditional generation but achieves this through making the noising process learnable and not compared against as the conducted experiments different from the setup considered in this work.
Diffusion and Flow-based Methods for Discrete Data. Recent advances in diffusion models have enabled vari- ous approaches for discrete data generation. (Luo et al., 2021) introduced GraphDF, using discrete latent variables for molecular graphs, while GDSS (Jo et al., 2022) devel- oped score-based approach using stochastic differential equations. DiGress (Vignac al., 2023a) employed graph transformer architecture for progressive molecular modifica- tion through edge and node operations. Recent flow-based methods include Discrete Flow Matching (Campbell al., 2024), which uses continuous-timg Markov Chains for flex- ible sampling of both discrete and continuous data, and the Dirichlet Flow framework (Stark et al., 2024), which models conditional probability paths via Dirichlet distributions.
Joint Modeling of Discrete and Continuous Data. Sev- eral recent approaches tackle unified representation of dis- crete and continuous features. MiDi (Vignac et al., 2023b) generates both molecular graphs and 3D atomic arrange- ments, while JODO (Huang et al., 2023) produces com- plete molecules with atom types, charges, bonds, and 3D coordinates. EQGAT-Diff (Le et al., 2023) extends joint modeling to include hybridization states, improving sample validity. FlowMol (Dunn & Koes, 2024) represents categor- ical variables in continuous space using scaled conditional
Croup-Equivariatt Diffusion for Continuous Data. In- corporating data symmetries into generative models has proven crucial for improving performance (Hoogeboom et al., 2022b; Gebauer et al., 2020; Bekkers et al., 2024). EDMs (Hoogeboom et al., 2022b) combined score-basee based diffusion (Ho al., 2020b) with group-equivariant graph
8

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Controlled Generation with Equivariant Variational Flow Matching
flow matching. Theoretical advances include discrete state space flows via Continuous Time Markov Chains (Camp- bell et al., 2024; Gat et al., 2024) and Fisher Flow Matching (Davis et al., 2024), which develops Riemannian flows on probability simplexes for joint modeling.
tein design, where complex structural constraints must be satisfied while maintaining biological feasibility. These advances could solidify VFM's role as comnerttnn of generative modeling for complex physical systems.
This project was support by the Bosch Center for Artificial Intelligence. JWvdM additionally ac- knowledges support from the European Union Horizon Framework Programme (Grant agreement ID: 101110122200 120237).
Acknowledgments
6. Conclusion
This paper builds on the Variational Flow Matching frame- work to developed methods for controlled generation. To do so, we define controlled velocity fields in terms of their cor- responding controlled probability density paths, along with variational methods for approximating these paths, offer- ing a principled approach to generating data under specific constraints. We demonstrate how conditional generation can be formulated as a Bayesian inverse problem, bridging generative modeling with classical probabilistic reasoning. Furthermore, we propose a fully equivariant formulation that ensures generative processes respect key symmetries such as rotations, translations, and permutations, which are particularly relevant for molecular and material design. These advances achieve state-of-the-an results on bench- mark datasets and provide a unified and efficient framework for discrete, continuous, and joint molecular generation tasks.
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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Controlled Generation with Equivariant Variational Flow Matching
A. Proofs
A.1. Controlled Variational Flow Matching
Proposition A.1. The controlled (marginal) velocity field
ue(x [ v) == (e)la))) (u(a Ix)),,
generates the controlled probability path pr v)-
Proof. By the continuity equation, we know that uz(2 y) = (x)lee))))) [ue xi)) generates pr y)) if
dt Pi v) div (pe(x) (8) (z)la)))))) [ue( [1)))..
(19)
First, we realized the time derivative of the controlled probability path indeed can be written as
d de pc v) = ( (
= = a(<1 v) dx = - div (ue( mi)pe(x a²)) q( v) dæ1
( (m)) g(ar lv)
(20)
(21)
Then, by Fubini's thoerem and direct manipulation, we observe that indeed
-f div (ue(æ 21)po(> xx1) q(<1 v) dæ = div (aeszz x1)pe(<) x1)g(<1 v) dz
div Pe(x v) / velx x palar x (n(g 1) dxy
pr( v)
div (Pe(2) (z) Be (u( mi)))
and thus, that (xxx) v) E (2)(1))) [ue(x] x1)] generates pr(x v), which is what we wanted to show.
A.2. Equivariance
Proposition A.2. Let G be arbitrary group, and suppose that the prior distribution is invariant under G, 2) the conditional velocity field is bi-equivariant under G. and 3) the variational posterior is equivariant under G with respect to its expected value. Then the marginal path pr(x)) is invariant under G.
Proof. Clearly, if our parameter model 0 is equivariant and ue(x x1) is bi-equivariant, we have that
0g x) velg ( =)) ue(g g p (z)) ux(x [x(x))) = 1.gg-o(ee (z)).
(22)
Per definition, point from our generated distribution of (x:) is obtain through solving an ODE, i.c.
d==t (el(e))))) [we(xr sx)] dé,
20
(23)
1=
for some Zo in Pa- Hence, define bijection (under Lipschitz continuity of of) &vr 20 x, such that
= Eir (xr) = + f. (es [Mr x1)] dt.
(24)
Note that if of is equivariant, have that (v 20) g & (20) directly. Since po(20) = po(g 20) per assumption, have that of (x: of (g 2),, which is what we wanted to show.
13


Reviews:
[{"review_id": "zghAhs7xHD", "reviewer": "Reviewer_k6L2", "summary": "The core contributions of the paper are twofold.\n\n**[1.Inference time control of VFM]**\nThe authors show that a conditional VFM distribution can be factorized into unconditional VFM part and the classifier part. Based on this factorization, the authors propose an iterative approximate solution to $\\underset{x_1}{\\mathrm{argmax}}\\\\, p_t(x_1|x,y)$ that can be solved during the inference time.\n\n**[2.Achieving equivariance in VFM]**\nThe authors show that equivariance of VFM can be achieved by using an equivariant variational distribution, when the conditional vector field is equivariant.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the clear summary and thoughtful comments. We appreciate the recognition of our core contributions, as well as the constructive suggestions that helped us clarify and strengthen the presentation. Below, we address the reviewer’s main points.\n\n**1. Inference-Time Control: Assumptions and Approximation Accuracy**\n\nWe agree that our formulation suggests simplifying assumptions (eg, log-concavity of the classifier). These are now stated explicitly and discussed in the revised manuscript.\n\nWhile VFM connected variational inference to unconditional flow matching, our goal is to extend this to conditional generation and introduce an equivariant formulation. We show that one can perform variational inference on $p_t(x_1 \\mid x, y)$, and more importantly, **use this perspective to rethink conditioning in generative models**. A simple proof-of-concept demonstrates this idea.\n\nOur aim is not a full Bayesian treatment, but to show that minimal inference-time updates—without retraining and with negligible overhead—enable effective post-hoc control. Using only the posterior mean is sufficient: in endpoint-based models (eg, flows, diffusions), the expected conditional velocity depends only on this value (see VFM). This eliminates the need for higher-order terms, keeping the method scalable while still improving empirical performance. As such, issues like log-concavity are less critical for our setting—we propose a simple integration of inference with generative modeling and focus not on recovering the full posterior, but on obtaining a reliable estimate of the posterior mean.\n\n\n\nBroadly, this work opens a direction within VFM: enabling modular inference-time control by integrating classical inference tools with learned approximations. We clarified this vision in the revision, and hope it inspires further work in this efficient design space.\n\n\n**2. Clarification of Notation and Presentation**\n\nWe thank the reviewer for catching unclear notation and formatting issues. We confirm that $\\mu_t(x)$ denotes the mean of $q_t(x_1 \\mid x)$ and have made this explicit. We revised the explanation around Eq. 15 to clarify its relation to VFM: the fixed-point update approximates the mean of $p(x_1 \\mid x, y) \\propto p(x_1 \\mid x)p(y \\mid x_1)$.\n\nWe also corrected the bolding in Table 3 and improved formatting throughout.\n\n\n**3. Modeling Details of $q_t(x_1 \\mid x)$**\n\nYes, we use a mean-field Gaussian for continuous variables and a categorical distribution for discrete ones, as in the original VFM. We added a brief clarification in the revised text.\n\nOnce again, we thank the reviewer for the insightful feedback, which helped refine both the clarity and precision of our work.", "claims_and_evidence": "## 1.Inference time control of VFM\n\nFor the inference-time control, the authors propose to optimize $\\underset{x_1}{\\mathrm{argmax}}\\\\, p_t(x_1|x,y)$ by iteratively solving $x_1$ that satisfies the fixed-point equation $\\nabla_{x_1} \\log p_t(x_1|x) + \\nabla_{x_1} \\log p_t(y|x_1)=0$. Since computing the score function $\\nabla_{x_1} \\log p_t(x_1|x)$ is intractable, the authors propose to approximate the score function with only the first moment $\\mu_t$, resulting in equation (15).\n\nHowever, several strong assumptions are not sufficiently justified. Although the proposed approximate guidance appears to offer some empirical benefits, it lacks thorough theoretical or experimental validation beyond incremental metric improvement (e.g., adding the approximate guidance increases metric XX by YY%). \n\nFor example, the authors assume log-concavity of the classifier $\\log p(y|x_1)$. However, most of the deep classifiers are not log-concave. Hence, as mentioned in the paper, the convergence is not guaranteed, which is in contrast to typical classifier guidance for SDE-based models. Another strong assumption is the approximation of $p_t(x_1|x)$ with a Gaussian centered at the mean of the variational distribution. Although the update rule derived from this approximation is simply a VFM + classifier gradient, which seems reasonable, it is unclear whether this provides a good Bayesian approximation.\n\n\n## 2. Achieving equivariance in VFM.\nThe authors show that VFM can be made equivariant by utilizing an equivariant variational distribution instead of directly modeling the equivariance to the marginal vector field network. This is indeed a significant benefit of the VFM approach, as it allows one to flexibly put inductive bias to the distribution instead of the marginal vector field.", "methods_and_evaluation_criteria": "By utilizing VFM framework, the authors shows that it is possible to simplify the implementation complexity of an equivariant flow matching model for a mixed discrete-continuous molecular generation tasks. In particular, the proposed method, G-VFM, matches the performance of SemlaFlow with a simpler training pipeline. The authors have also demonstrated the benefit of the proposed inference-time controlled generation.", "theoretical_claims": "As mentioned earlier, the authors used several strong assumptions to derive Equation 15. This makes its Bayesian correctness questionable. However, the final update rule looks reasonable, regardless of its theoretical accuracy. Other proofs look correct.", "experimental_designs_or_analyses": "As mentioned before, the paper could be improved with an in-depth qualitative/quantitative analysis on the accuracy of the Bayesian posterior approximation for inference-time controlled generation.", "relation_to_broader_scientific_literature": "The flexibility of the proposed equivariant VFM approach could be beneficial for other areas such as 3D vision or robotics.", "essential_references_not_discussed": "Relevant works are appropriately referenced.", "other_strengths_and_weaknesses": "There are several unreference varibles in the paper. For example, it is unclear where $\\mu_t(x)$ comes from. I presume this is the mean of the variational distribution $q_t(x_1|x)$. The presentation of the paper could be improved if the authors explicitly state that $\\mu_t(x)$ is approximated with that of $q_t(x_1|x)$. Also, it would be helpful for the readers to understand equation 15 if the authors explain how equation 15 is related to VFM.", "other_comments_or_suggestions": "There is a typo in Table 3. The Mol Stab of G-VFM (=99.5) is highligted in bold, although SemlaFlow has higher score of 99.6.", "questions_for_authors": "At first glance, it is unclear how exactly $q_t(x_1|x)$ is modeled. I presume that $q_t(x_1|x)$ is simply a mean field Gaussian for continuous variables and a categorical distribution for discrete variables as in the original VFM paper. Is this correct?", "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": "TOVT4Lb2AI", "reviewer": "Reviewer_TqUv", "summary": "The paper focuses on extending the recently proposed Variational Flow Matching (NeurIPS 2024) approach for conditional generation and for incorporating inductive biases as symmetries. They derive two different ways for controlled generation, the first one is similar to conditional diffusion models, with the difference that in this case, they learn a conditional vector field and one that resembles classifier guidance, where the conditioning is post-hoc. They then focus on the problem of generating molecules where the symmetries of the true underlying distribution that has to be learned present specific symmetries. They consider three different molecular generation tasks, discrete, continuous, and joint generation tasks: in the first one they just focus on atom types and bond types, in the second one they focus on atom positions and sometimes atom type, and in the third one they focus on everything plus formal charges. Results show that the variational formulation of flow matching is beneficial.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer TqUv,\n\nThank you for the detailed and thoughtful review and for engaging deeply with both the theoretical and empirical aspects of our work. Below we respond to the key points and how they will be addressed in the revised manuscript.\n\n**1. Unified Objective and Variational Distributions**\n\nWe fully agree: while VFM provides a unified objective, it still requires variational distributions per modality. However, our contribution is to show that these choices can be **modular and plugged directly into the same objective, avoiding the need to redesign the loss or architecture per data type—a common challenge in standard FM pipelines**. Though not appropriately explored here, this even naturally extends to other distributions (eg, Poisson for neural data) by learning the conditional rate and optimizing the NLL. We clarified this core motivation in the revision.\n\n**2. Fairness of Table 3 Comparisons**\n\nWhile our method is flexible enough to generate any subset of molecular features (e.g., just positions or positions plus atom types), in all experiments, we matched the generation scope of each baseline for a fair comparison. For example, if a baseline did not model bond structures or formal charges, we excluded those as well. We now clearly state this in the text and annotate Table 3 to indicate which components each model generates.\n\n**3. Simplifying Assumptions and Classifier Guidance**\n\nWe now explicitly state and discuss simplifying assumptions (eg, classifier log-concavity) in the revised manuscript.\n\nThis work extends VFM with an inference-based view of conditional generation. We perform variational inference on $p_t(x_1 \\mid x, y)$ in a simple proof-of-concept setup that yields meaningful improvements. Due to the linearity of the conditional velocity (as in flows/diffusions), matching only the posterior mean suffices (see VFM), and **a full Bayesian treatment is unnecessary**. Our main contribution is showing that this inference view enables scalable post-hoc control by combining classical inference tools with learned approximations. For more detail, we refer to our response to reviewer k6L2. This also addresses the reviewer's question about the normalization constant in Eq. 13—your interpretation is correct. \n\n\nFurthermore, while related to classifier guidance (CG/CFG) in diffusion, our approach differs significantly in cost and flexibility. CG typically requires backpropagating through a denoised prediction at each timestep, which can be expensive and must be trained conditionally. In contrast, **our fixed-point method operates post-hoc at the endpoint, with no joint training**. Finally - even though is it true that true the Tweedie transform a similar effect can be obtained, the approach would arguably be rather noisy and therefore hard to learn. We now clarify this key distinction in the paper.\n\n\n**4. Author questions / Minor comments:**\n\n- **Missing citations:** We agree and will cite [4], [5], and other related work on conditional generation and classifier guidance. These citations have been added in Section 3.1. \n- **Missing Experimental Details (Table 2):** We have included full model architecture and training details in the appendix, and clarified in Table 2 whether atom type modeling is included per experiment.\n- **Table Formatting and Metric Presentation:** We corrected the inconsistent bolding and formatting in Tables 1–3. Bolding is now applied uniformly (best per column, including ties), and we added global method rankings to aid interpretability across datasets.\n-  **Table 1 Repetition:** We included Table 1 to demonstrate that G-VFM reproduces the results of standard VFM models as a consistency check. That said, we agree that this space could be better used to highlight new contributions, so we have moved Table 1 to the appendix.\n- **Typo:** We have corrected “Fischer flow” to “Fisher flow” (line 422).\n\nWe again thank the reviewer for their constructive feedback. Your comments helped significantly improve the clarity, fairness, and completeness of the revised manuscript.", "claims_and_evidence": "The main claim in the papers is that Variational Flow matching is beneficial on top of Flow matching especially for conditional generations and when the true distribution presents symmetries. They considered mostly the problem of molecular generation and the results show that the variational formulation is helpful.\n\nThe authors mention several times that \"VFM provides a unified approach that can be applied to any combination of discrete and continuous molecular features\", while I agree that the benefit is that the loss is the same, one still has to choose a different type of variational distribution. I found it a bit unfair with respect to diffusion models for example as, in that case, one has to choose a specific stochastic process instead of a different variational distribution. But this is a minor detail and the authors might also disagree with me.", "methods_and_evaluation_criteria": "Yes, the benchmark datasets are the usual one that people uses for evaluating generative models for molecular generation, and also the metrics considered are correct. I have a few comments about the tables that I leave in the experimental analysis sections.", "theoretical_claims": "They have two main propositions in the paper. The first one is about showing that the conditional on some observations vector field generates the conditional probability path. This was also derived in [1], but [1] did not provide a full proof. The second theoretical claim shows that the marginal path is invariant under G if we have the following elements: a prior that is invariant under G, a conditional velocity field that is bi-equivariant, and a variational posterior that is equivariant. \n\n**References**\n\n[1] Zheng, Q., Le, M., Shaul, N., Lipman, Y., Grover, A., & Chen, R. T. (2023). Guided flows for generative modeling and decision making. _arXiv preprint arXiv:2311.13443_.", "experimental_designs_or_analyses": "The design of the experimental design is valid. I just have a few comments regarding the analysis: \n- the use of bold numbers is a bit misleading. For example, in Table 1 the authors bold both $100.00$ and $99.99$, but not if the difference is $0.02$ in the unique results column. In Table 2, EDM and EFM perform exactly the same in terms of atom stability and molecule stability, but their results are not bolded. In Table 3, Semla Flow is performing better on QM9 in terms of Mol. stability but the result of G-VFM is bolded. Also, EquiFM results on QM9 for JS(E) should be bold as they perform the same as SemlaFlow.\n- I think that the results in Table 3 do not present an apple-to-apple comparison as some of the methods like EDM, GCDM, and EquiFM do not generate bond information, which has been shown to be helpful for getting higher validity. Therefore, I will invite the authors to make it clear in the main text what the different models are generating, e.g. if they are generating bonds and charges. That's the main reason for the huge difference in terms of metrics on QM9 in Table 3. \n- I don't really see why having Table 1 in the main text as results and comparison were presented already in the Variational Flow Matching paper.\n- It would be nice if the authors presented the full details of model parameters and training details in the appendix. Also, the paper will benefit if the results in Table 2 authors specify if they are modeling atom types or not.", "relation_to_broader_scientific_literature": "The paper places itself in the flow matching landscape, building on top of Variational Flow Matching approach [1]. They proposed techniques specific to this approach for conditional generation similar to what people usually do when they train conditional diffusion models [2] or perform post-hoc conditioning using classifier guidance [for example 3]. However, I feel that the method section does not cite any relevant references for the entire Section 2 and 3, which I invite the author to add. For example, the main part section 3.1, and the proposition were also derived in [4], which the authors do not cite. As the main application is molecular generation, I think that the paper presents methods relevant to people working in that field. Although [5] is not evaluating log-likelihood it is a related work for Table 2.\n\n**References**\n\n[1] Eijkelboom, F., Bartosh, G., Andersson Naesseth, C., Welling, M., & van de Meent, J. W. (2024). Variational flow matching for graph generation. _Advances in Neural Information Processing Systems_, _37_, 11735-11764.\n\n[2]Ho, J., Jain, A., & Abbeel, P. (2020). Denoising diffusion probabilistic models. _Advances in neural information processing systems_, _33_, 6840-6851.\n\n[3]Chung, H., Kim, J., Mccann, M. T., Klasky, M. L., & Ye, J. C. (2022). Diffusion posterior sampling for general noisy inverse problems. _arXiv preprint arXiv:2209.14687_.\n\n[4] Zheng, Q., Le, M., Shaul, N., Lipman, Y., Grover, A., & Chen, R. T. (2023). Guided flows for generative modeling and decision making. _arXiv preprint arXiv:2311.13443_.\n\n[5] Cornet, F., Bartosh, G., Schmidt, M., & Andersson Naesseth, C. (2024). Equivariant neural diffusion for molecule generation. Advances in Neural Information Processing Systems, 37, 49429-49460.", "essential_references_not_discussed": "See above", "other_strengths_and_weaknesses": "The paper is nicely written and it tackles an important task in deep generative models which is conditional generation and application to model distribution that has symmetries, which is of interest to the community.\nI would like just to add one more comment on something that the authors mention in the paper on line 181 (right column): \"unlike standard classifier-guidance methods in diffusion, which require a time-dependent classifier $p_t(y |x)$, classification in VFM is performed on data pairs $(x_1,y)$\". I think this is not entirely correct as in diffusion one can also use a classifier trained on the clean samples, but at each denoising step, one has to first apply Tweedie to get the approximate clean sample.", "other_comments_or_suggestions": "- Line 422: Fischer flow instead of Fisher flow", "questions_for_authors": "I have just one question for the authors, if I am not completely wrong, it seems that equation 13 can be written as:\n$$\np_t(x_1|x,y) = \\frac{p_t(x_1|x)p_t(y|x_1)}{p_t(y|x)} \\approx p_t(x_1|x)p_t(y|x_1)\n$$\nwhere at the denominator the time-dependent classifier appears. Is it something that can be used to improve the guidance? It might be a completely useless question, but since I try to derive Eq. 13 I was curious to ask if that can be used.", "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": "o573g28P2X", "reviewer": "Reviewer_6iCA", "summary": "The paper proposes two novel methods within the variational flow matching (VFM) framework for generative modeling. The first is controlled generation, which enables conditional generation using unconditional generative models without requiring retraining (though it is not necessarily limited to this scenario). The second method introduces equivariant generative models, which are well-suited for molecular generation tasks, where invariance to rotations, translations, and permutations is beneficial.\n\n### **Background on flow matching and variational flow matching**\nIn the standard flow matching (FM) framework, models directly parameterize the velocity field $u_t(x)$, i.e., the expected value of the endpoint-conditional velocity field $u_t(x|x_1)$ over the posterior distribution (i.e., the distribution of the endpoint given the current position).\n\nIn contrast, the variational flow matching (VFM) framework does not directly model the velocity field. Instead, it models the posterior distribution and takes the expectation later. Thus, the model is trained by minimizing the forward KL divergence between the ground truth and the model posterior. This training process simplifies into matching posterior means when the endpoint-conditional velocity field is linear, such as in conditional optimal transport (condOT). Consequently, an unimodal distribution, such as a Gaussian, can be used for continuous random variables.\n\nMoreover, under the linearity assumption on the endpoint-conditional velocity field, computing the velocity field is further simplified—it can be done by parameterizing the mean of a Gaussian, for example.\n\n### **Controlled generation**\nFor a controlled generation, the paper highlights a key observation: the endpoint-conditional velocity field should remain unchanged regardless of additional conditions since the endpoint already contains information about those conditions. Leveraging this insight, the authors show that the velocity field of a conditional generative model corresponds to the expected value of the same endpoint-conditional velocity field as in the unconditional model. However, in the conditional case, the expectation is taken with respect to the conditional posterior of the endpoint (rather than the unconditional posterior), given both the additional condition and the current position.\n\nThe paper then shows that the conditional posterior of the endpoint can be rewritten in terms of the unconditional posterior and a classifier (Equation 13) using Bayes' theorem up to a normalizing constant. This formulation enables reusing the unconditional model without retraining. However, taking an expectation with respect to an unnormalized distribution is typically intractable, making direct computation infeasible.\n\nTo address this, the authors leverage the linearity assumption on the endpoint-conditional velocity field. Since VFM only requires an unimodal distribution, they propose estimating the mean of the conditional posterior by solving a fixed-point equation (Equation 14) iteratively. This iterative process enables controlled generation without sampling from unnormalized distributions.\n\n### **Equivariant generative models**\nIn addition to the controlled generation, the paper introduces a method for equivariant generative modeling under the VFM framework. A key requirement for this approach is that the expected endpoint-conditional velocity field (computed over the posterior distribution of the endpoint given the current position) must be equivariant under group actions. The paper establishes that this can be achieved if:\nThe prior distribution is group-equivariant.\n\nThe endpoint-conditional velocity field is bi-equivariant with respect to both the endpoint and the current position.\nThe expected value of the model's posterior, which is a function of the current position, is group-equivariant with respect to its input (i.e., the current position).\n\nThe authors emphasize that when using a conditional optimal transport (condOT) map, the second condition is automatically satisfied if the relevant groups act linearly on the domains of interest, such as SO(n), and this would be the case for molecular generation, which are the main application of interest of the paper.\n\n### **Experimental results**\nThe paper demonstrates the effectiveness of the proposed methods on several benchmark molecular generation datasets, including QM9, ZINC250k, and GEOM.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer 6iCA,\n\nWe sincerely thank the reviewer for their comprehensive assessment of our paper. \n\nWe appreciate your recognition that these contributions are \"conceptually insightful and practically impactful\" and that our work addresses an important gap in flow matching frameworks. We are glad to hear the reviewer is on the same page about the interesting avenues to explore regarding integrating 'classical' inference techniques with SOTA generative modeling.", "claims_and_evidence": "In my understanding, the paper's contributions are clear. The proposed methods--controlled generation and equivariant generative modeling under the variational flow matching (VFM) framework--are both conceptually insightful and practically impactful.\n\nThese contributions are particularly significant for several reasons. First, the controlled generation approach introduces an efficient mechanism for performing conditional generation without retraining (though it is not necessarily limited to this scenario, as discussed in the paper). Lacking an efficient controlled generation method for the flow matching framework has led many practitioners in various applications to prefer denoising diffusion. The proposed controlled generation is particularly valuable in scenarios where retraining is computationally expensive or infeasible.\n\nSecond, the introduction of equivariant generative models under the VFM framework is a notable theoretical contribution, as it ensures that the generative process respects fundamental symmetry constraints. This is crucial for applications such as molecular generation, where invariance to rotations, translations, and permutations is essential for generating physically meaningful structures.\n\nFurthermore, the paper provides sufficient mathematical formulation and justification to support the proposed methods concisely, allowing readers to understand them easily. The experimental results on benchmark molecular datasets further validate the effectiveness of these approaches, demonstrating their potential real-world impact.\n\nOverall, I find that the paper not only addresses a well-motivated problem but also offers valuable solutions that could inspire future research directions in both generative modeling and scientific machine learning.", "methods_and_evaluation_criteria": "N/A", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "N/A", "relation_to_broader_scientific_literature": "N/A", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "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": "JXN1GiJWuB", "reviewer": "Reviewer_uXdp", "summary": "In this paper, the authors present a controlled generation objective in the framework of Variational Flow Matching (VFM), as well as an equivariant formulation of VFM which has applications in 3D molecule generation. For controlled generation, the authors demonstrate that both end-to-end constrained training and post-hoc modification of pretrained models is possible under their methodology. Results are demonstrated on several unconditional and conditional molecular generation tasks, including the QM9, GEOM-Drugs, and Zinc250k datasets.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer uXdp,\n\nThank you for the detailed and thoughtful review. We appreciate your close reading and thank you for your constructive suggestions. Below we address the main points raised in your review and clarify several aspects that were not sufficiently emphasized in the original submission.\n\n**1. Contribution in Unconditional Generation (Table 2)**\n\nWe agree that the motivation behind the unconditional generation section was not clearly explained. Our goal with Table 2 was not to introduce a new architecture, but to show that strong performance can be achieved via a simple, unified formulation under the VFM framework. By linearly interpolating between noise and data—whether continuous or discrete—and selecting an appropriate variational distribution (eg, Gaussian or categorical), one can train a single model with a shared objective across modalities. This avoids the need for specialized architectures or training regimes typically required in standard FM for mixed data types, and even generalizes to other distributions (eg, Poisson for neural activity). We have clarified this conceptual simplicity in the revision and added citations for baselines directly in the tables.\n\n\n**2. Comparison to D-Flow and Efficiency Tradeoff**\n\nWhile D-Flow performs better in Table 4, it does so by evaluating the forward process over many candidate initializations ($n \\times K$ evaluations, eg, $10 \\times 100$). In contrast, our method uses a single forward pass and a short fixed-point calibration for conditional guidance—requiring only $K$ forward evaluations and no additional gradient computations. We prioritize simplicity and scalability, whereas D-Flow trades compute for precision. Both are valid design choices, and we now highlight this tradeoff explicitly in the paper, including a discussion of guidance cost and number of function evaluations (NFE).\n\n\n**3. Equivariance Evaluation and Practical Utility**\n\nYou are correct that we do not isolate the empirical effect of equivariance. While the theoretical formulation is a key contribution, we agree that deeper empirical validation would strengthen the work. We now clarify which models enforce which equivariant properties and will provide an ablation in the final version. \n\n\n**4. Generality Beyond Molecular Generation**\n\nWhile molecular generation is a natural testbed due to its symmetry constraints, the proposed methods are general. VFM provides a flexible framework for combining learned dynamics with structural inductive biases. We now emphasize this more clearly in the discussion and are actively exploring applications beyond molecules, including images (to be included in the final version, at least as a proof of concept).\n\n\n**5. Comparison to CG / CFG Methods**\n\nYou raise a valid point. Our method assumes access to a classifier over $x_1$ rather than a time-dependent $p_t(y \\mid x_t)$. While related, these regimes differ in flexibility and cost. CG/CFG typically requires backpropagating through a denoised prediction at every timestep, which is expensive and needs conditional training. Our fixed-point procedure operates post-hoc on the endpoint and does not require joint training. We now clarify this key distinction in the manuscript.\n\n\n**6. Theoretical Novelty of Equivariance Result**\n\nWhile standard flow matching methods can exhibit equivariant behavior under certain conditions, our contribution is to show that equivariance can be guaranteed through the variational distribution—without requiring the velocity field itself to be equivariant. This decoupling is novel and enables greater modularity: symmetry constraints can be enforced directly through the variational family, independent of the learned dynamics. This allows for clean, flexible design of symmetry-preserving generative models and simplifies implementation in practice. We now clarify this point more explicitly in the final version.\n\nOnce again, we thank the reviewer for their careful and constructive feedback. Your comments helped us significantly improve the clarity, presentation, and positioning of our contributions.", "claims_and_evidence": "1. In Table 2, the authors present results on continuous molecular generation with G-VFM as “our variational treatment of flow matching”. However, as I understood, this Unconditional Generation section is simply reproducing/re-implementing the previously described VFM framework on these specific datasets. The authors state this explicitly for Table 1 (discrete generation), but not for Tables 2 and 3. Either the specific contributions from this work for unconditional generation should be made more clear, or this section should be moved to the Appendix, because it is not in service to the story/main contributions of the paper, which I interpreted to be controlled generation and equivariance.\n\n2. D-Flow seems to considerably outperform the proposed G-VFM methods for controlled generation in Table 4. The authors claim that this comes at the expense of increased generation cost for D-Flow, but I didn’t see a specific quantification of this cost (I may have missed it). NFE should be added to Table 4, unless I am missing something. Without this info, it is hard to make a judgment about whether the substantially worse results of G-VFM relative to D-Flow are justified by computational acceleration.", "methods_and_evaluation_criteria": "1. Given the exclusive focus on molecule generation and a methodological section devoted to proving equivariance properties of VFM, I did not find any investigation/mention of equivariance in the experiments/evaluation (for instance, a direct ablation/comparison between VFM with the equivariance properties satisfied vs not satisfied). This begs the question: if you can enforce equivariant/invariant generative dynamics, so what? Is it actually practically useful for, e.g., producing more physical realistic samples and/or achieving better conditioning scores? While the authors perhaps treated this as self-evident, there is growing evidence in adjacent fields that equivariance can be learned from data without explicit enforcement, and that this is actually the preferred approach as models/data scales [1]. I would like to see a more thorough investigation of this.\n\n\n2. Related to #1, given the lack of focus on equivariance in the results, I would also expect much broader evaluations/demonstration of the proposed method on different modalities in which equivariance is not required such as controlled image generation (with target properties like compressibility, prompt-image alignment, aesthetic quality, etc.) See [2] for details. Without this, I feel that the current scope of evaluation of the paper is too narrow to be of broad interest.\n\n3. Given the explicit mentioning of alternative conditioning strategies like classifier guidance (CG), I would have expected a direct comparison to CG and CFG to empirically demonstrate the benefits.\n\n[1] Qu, Eric, and Aditi Krishnapriyan. \"The importance of being scalable: Improving the speed and accuracy of neural network interatomic potentials across chemical domains.\" Advances in Neural Information Processing Systems 37 (2024): 139030-139053.\n\n[2] Black, Kevin, et al. \"Training diffusion models with reinforcement learning.\" arXiv preprint arXiv:2305.13301 (2023).", "theoretical_claims": "The primary theoretical claim is that given appropriate choices of the prior, conditional velocity field, and variational posterior, the generative dynamics of VFM are invariant under transformations in SO(n). I did not carefully review the proof. Since I am not very familiar with the literature in this area, can the authors comment on how different/surprising this result is compared to (what I assume are known equivariance properties) of regular FM?", "experimental_designs_or_analyses": "See above.", "relation_to_broader_scientific_literature": "This work builds upon Variational Flow Matching, first introduced in [1].\n\n[1] Eijkelboom, Floor, et al. \"Variational flow matching for graph generation.\" Advances in Neural Information Processing Systems 37 (2024): 11735-11764.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "The paper is well written and easy to follow", "other_comments_or_suggestions": "Include citations for baselines directly in the Tables.", "questions_for_authors": "How does the performance of the proposed post-hoc modification approach change as a function of how out-of-distribution the target property is for the generative model and/or classifier? \n\nIs classification always used as guidance even if the target is continuous-valued (e.g by binning into discrete classes). Or is the approach equally valid for regression as well?", "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:

Meta-review: This paper extends Variational Flow Matching (VFM) for conditional generation and introduces methods for inference-time control and symmetry-aware modeling. 

Pros:

+ Introduces an inference-time controllable approach for conditional generation using VFM, offering flexibility similar to classifier guidance in diffusion models.

+ Provides a method to achieve equivariance in generative models through variational distribution design.

Cons:

+ Relies on strong assumptions (e.g., log-concavity and Gaussian approximation) that lack rigorous theoretical or experimental justification.

+ Experimental comparisons are sometimes unclear due to differences in generated features across models.

+ Some notational and presentation issues (e.g., undefined variables and unclear derivation of equations) reduce clarity and accessibility.
Decision: Accept (poster)