Learning Energy-Based Generative Models via Potential Flow: A Variational Principle Approach to Probability Density Homotopy Matching
Authors: Junn Yong Loo
Leong Fang Yu
Michelle Adeline
Julia K. Lau
Hwa Hui Tew
Arghya Pal
VISHNU MONN BASKARAN
Chee-Ming Ting
Raphael CW Phan
Venue: Accepted by TMLR ()

Abstract: Energy-based models (EBMs) are a powerful class of probabilistic generative models due to their flexibility and interpretability. However, relationships between potential flows and explicit EBMs remain underexplored, while contrastive divergence training via implicit Markov chain Monte Carlo (MCMC) sampling is often unstable and expensive in high-dimensional settings. In this paper, we propose Variational Potential (VAPO) Flow Bayes, a new energy-based generative framework that eliminates the need for implicit MCMC sampling and does not rely on auxiliary networks or cooperative training. VAPO learns an energy-parameterized potential flow by constructing a flow-driven density homotopy that is matched to the data distribution through a variational loss minimizing the Kullback-Leibler divergence between the flow-driven and marginal homotopies. This principled formulation enables robust and efficient generative modeling while preserving the interpretability of EBMs. Experimental results on image generation, interpolation, out-of-distribution detection, and compositional generation confirm the effectiveness of VAPO, showing that our method performs competitively with existing approaches in terms of sample quality and versatility across diverse generative modeling tasks.

Published in Transactions on Machine Learning Research (05/2025)
Learning Energy-Based Generative Models via Potential Flow: A Variational Principle Approach to Probability Den- sity Homotopy Matching
Junn Yong Loo
won.junnyonn@@mnnss.ccmm edu
Monash University Malaysia
Fang Yu Leong
tenng.Jangguu@mmnnas..cc
Monash University Malaysia
Michelle Adeline
maade0005@@studert..mmoaaaa edu
Monash University Malaysia
Julia Kaiwen Lau
fulia.lavu@monnash..cl
Monash University Malaysia
Hwa Hui Tew Monash University Malaysia
lwaa. www.tew@monash. edu
Arghya Pal Monash University Malaysia
Vishmu Monn Baskaran
Disheu..mosn@mmonaat..cc
Monash University Malaysia
Chec-Ming Ting Monash University Malaysia
chng.
Raphaël C.-W. Phan
edu
Monash University Malaysia
Reviewed on OpenReview: https: openreview. net/ forum? id=oc7poEYOFK
Abstract
Energy-based models (EBMs) are a powerful class of probabilistic generative models due to their flexibility and interpretability. However, relationships between potential flows and explicit EBMs remain underexplored, while contrastive divergence training via im- plicit Markov chain Monte Carlo (MCMC) sampling is often unstable and expensive in high-dimensional settings. In this paper, we propose Variational Potential (VAPO) Flow Bayes, a new energy-based generative framework that eliminates the need for implicit MCMC sampling and does not rely on auxiliary networks or cooperative training. VAPO learns an euergy-parameterized potential flow by constructing a flow-driven density homo- topy that matched to the data distribution through a variational loss minimizing the Kullback-Teibler divergence between the flow-driven and marginal homotopies. This prin- cipled formulation enables robust and efficient generative modeling while preserving the interpretability of EBMs. Experimental results on image generation, interpolation, out- of of-distribution detection, and compositional generation confirm the effectiveness of VAPO, showing that our method performs competitively with existing approaches in terms sam- ple quality and versatility across diverse generative modeling tasks. The code is available at tttps://githe com/1jun0004/VAPP

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Published in Transactions on Machine Learning Research (05/2025)
1 Introduction
Energy-based models (EBMs) have emerged a flexible and expressive class of probabilistic generative models (Nijkamp et al., 2019; Du & Mordatch, 2019; Grathwohl et al., 2020b; Gao et al., 2020; Du et al., 2021; Gao et al., 2021; Grathwohl et al., 2020a; Yang et al., 2023; Zhu et al., 2024). By assigning potential energy that correlates with the unnormalized data likelihood (Song & Kingma, 2021), EBMs offer a structured energy landscape for probability density estimation, providing several notable advantages. First, EBMs are interpretable, as the underlying energy function can be visualized in terms of energy surfaces. Second, they are highly expressive and do not impose strong architectural constraints (Bond-Taylor et al., 2022), enabling them to capture complex data distributions. Third, EBMs exhibit inherent robustness to Out-of- Distribution (OOD) inputs, given that regions with low likelihood are naturally penalized (Du & Mordatch, 2019; Grathwohl et al., 2020a). Building on their origins in Boltzmann machines (Hinton, 2002), EBMs also share conceptual ties with statistical physics, allowing practitioners to adapt physical insights and tools for model design and analysis (Feinauer & Lucibello, 2021). They have demonstrated promising performance in various applications beyond image modeling, including text generation (Deng et al., 2020), robot learning (Du et al., 2020), point cloud synthesis (Xie et al., 2021a), trajectory prediction (Pang et al., 2021; Wang et al., 2023), molecular design (Liu et al., 2021), and anomaly detection (Yoon et al., 2023).
Despite these advantages, training deep EBMs often relies on implicit Markov Chain Monte Carlo (MCMC) sampling for contrastive divergence. In high-dimensional settings, MCMC suffers from poor mode mixing and slow mixing (Du & Mordatch, 2019; Nijkamp et al., 2019; Gao et al., 2020; Grathwohl et al., 2020a; Nijkamp et al., 2022; Bond-Taylor et al., 2022), yielding biased estimates that may optimize unintended objectives (Grathwohl et al., 2020b). Truncated chains, in particular, can lead models to learn an implicit sampler rather than a true density, which prevents valid stcady-state convergence and inflates computational overhead. As a result, the generated samples can deviate significantly from the target distribution (Grathwohl et al., 2020b). To mitigate these issues, some works propose auxiliary cooperative strategies that learn complementary models to either avoid MCMC via variational inference (Xiao et al., 2021a) or combine short- run MCMC refinements with learned generator distributions (Xie et al., 2020; Grathwohl et al., 2021; Hill et al., 2022). Nevertheless, these approaches could complicate model architectures and training procedures.
In parallel, flow-based models have advanced generative modeling by leveraging continuous normalizing flows and optimal transport techniques to surpass diffusion models in sample quality and efficiency (Kim et al., 2021; Song et al., 2021). Notable examples include Flow Matching (Lipman et al., 2023), which models diffcomorphic mappings between noise and data; Rectified Flow (Liu et al., 2023b), which optimizes sampling paths; Stochastic Interpolants (Albergo & Vanden-Eijnder 2023; Rezende & Mohamed, 2015; Chen et al., 2018), which incorporate stochastic processes into flows for complex data geometries; Schrödinger Bridge Matching (Shi et al., 2023), which intogrates regulai-ee optimal transport with diffusion; and Poisson Flow Generative Model (PFGM) (Xu et al., 2022), which introduces an augmented space governed by the Poisson equation. However, these methods do not directly parameterize probability density and lack the theoretical advantages of EBMs, such as generating conservative vector fields aligned with log- likelhhooo gradients (Salimans & Ho, 2021).
Recent approaches, such as Action Matching (Neklyudov et al., 2023), explicitly model the energy (ac- tion) to generate data-recovery vector fields, thus providing a structured approach to learning conservative dynamics. Meanwhile, Diffusion Recovery Likelihood (DRL) (Gao et al., 2021) and Denoising Diffusion Ad- versarial FBMs (DDAEBM) (Geng et al., 2024) refine conditional EBMs by improving sampling efficiency and training stability through diffusion-based probability paths. However, a direct connection between energy- parrme-prreeeeeeeeed flow models and explicit (marginal) EBMs remains unexplored, limiting the applica- tion of based techniques for learning EBMs. Furthermore, existing generative models have yet to adopt variational formulations, such as the Deep Ritz method, to align the evolution of density paths.
To address the computational challenges of existing energy-based methods, we propose Variational Potential (VAPO) Flow Bayes, a novel energy-based generative framework grounded in variational principles that climinates the need for auxiliary models and implicit MCMC sampling. VAPO employs the Deep Ritz method to learn an energy-paran parameteriede potential flow, ensuring alignment between the flow-driven density homotopy and the data-recovery likelihood homotopy. To address the intractability of homotopy matching,
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formulate variational loss function that minimizes the Kullback-Leibler (KL) divergence between these density homotopies. Additionally, we validate the learned potential energy an effective parameterixation of the stationary Boltzmann energy. Through empirical validations, we benchmark VAPO against state-of- the-art generative models, showcasing its competitive performance in Fréchct Inception Distance (FID) for image generation and excellent OOD detection with high Area Under the Receiver Operating Characteristic Curve (AUROC) scores across multiple datascts.
2
Background and Related Works
In this section, we provide overview of EBMs, particle flow, and the Deep Ritz method, collectively forming the cornerstone of the proposed VAPO framework.
2.1 Energy-based Models (EBMs)
Denote 0 Ω C IR as the training data, EBMs approximate the data likelihood Pdata (3) via defining a Boltzmann distribution
pB(x)
(1)
where 2B is the Boltzmann energy parameterized via neural networks and Z e((z) dæ is the nor- malizing constant. Given that this partition function is analytically intractable for high-dimensional data, EBMs perform the Maximum Likelihood Estimation (MLE) by minimizing the negative log-likelihood loss (0) = - Pa (x) [log pl (+) = pa [ (T)] pa) [log Z]. The gradient of this MLE loss with respect to model parameters is approximated via the contrastive divergence (Hinton, 2002) loss VoLMLE = (F) (nls(>) (x)( [V) Nonetheless, EBMs are computationally intensive due the implicit MCMC generating procedure required for generating negative samples æ ~ pB(x) implicitly during training.
2.2 Particle Flow
Particle flow, introduced by Daum & Huang (2007), is a class of nonlinear Bayesian filtering (sequential inference) methods designed to approximate the posterior distribution p(æ, Ze()) of the sampling process given observations. While closely related to normalizing flows (Rezende Mohamed, 2015) and neural ordinary differential equations (ODEs) (Chen al., 2018), these frameworks do not explicitly accommodate Bayes update. Instead, particle flow achieves Bayes update p(r, Fgg)) p(æ, [ee)) P(Ez #e,)<)) by transporting the prior samples I2 p(z, T<t) through ODE de v(z,t) parameterized by a velocity field o(z,)) t), over pscudo-time E [0, 1]. The velocity field is designed such that the sample density follows log-hon tog-homotopy that induces the Bayes update. Despite its effectiveness in time-series inference (Pal et al., 2021b; Chen et al., 2019b; Yang et al., 2014) and its robustness against the curse of dimensionality (Surace et al., 2019), particle flow, particularly potential flow where the velocity field v(r, () = D(x, 1) is the gradient of potential energy, remains largely unexplored in energy-based generative modeling.
2.3 Deep Ritz Method
The Deep Ritz method is deep learning-based variational numerical approach, originally proposed by E & Yu (2018), for solving scalar elliptic partial differential equations (PDEs) in high dimensions. Consider the following Poisson's equation, fundamental to many physical models:
A [(x),
x E Ω
(2)
u(x) = 0, x E a
where A is the Laplace operator, and 0 denotes the boundary of Q. For a Sobolev function u E H.( (definition in Proposition 2) and square-integrable P T22, the variational principle ensures that weak
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solution of the Euler-Lagrange boundary value equation (2) is equivalent to the variational problem of minimizing the Dirichlet energy (Mäller & Zeinhofer, 2019), as follows:
= arg min (2
dr + 11 "Ioo o(x)2 dx
(3)
where v denotes the Del operator (gradient). In particular, the Deep Ritz method parameterizes the trial function using neural networks and performs the optimization (3) via stochastic gradient descent. To enforce the Dirichlet boundary condition, the second component of the Dirichlet energy (3), weighted by a positive constant must be evaluated on the boundary 0Q. This necessitates acquiring additional boundary samples x E ON during neural network training, thereby introducing extra computational overhead. The Deep Ritz method is predominantly applied for finite element analysis (Liu al., 2023a) due its versatility and effectiveness in handling high-dimensional PDE systems. In (Olmez et al., 2020), the Deep Ritz method is employed to solve the density- weighted density-weighted Poisson equation arising from the feedback particle filter (Yang et al., 2013). However, its application in generative modeling remains unexplored.
3 Variational Potential (VAPO) Flow Bayes
In this section, we introduce VAPO, novel energy-based generative modeling framework inspired by par- ticle flow and the Deep Ritz method. VAPO encompasses four key elements: constructing a Bayesian marginal homotopy between the Gaussian prior and data likelihood (Section 3.1), designing a potential flow that aligns the flow-driven homotopy with the marginal homotopy (Section 3.2), formulating a variational loss function using the Deep Ritz method (Section 3.4), and establishing connections between homotopy matching, diffusion, and EBMs (Section 3.3).
3.1 Interpolating Between Prior and Data Likelihood: Log-Homotops Bayesian Transport
Let x E S denote the training data, with likelihood Pdata( (T), and let samples. First, we define Gaussian prior (=x) _N(0, I) and Gaussian conditional data likelihood p( x) = ,(T;,, both with isotropic covariances. This data likelihood satisfies the state space model x == H ve, where is the standard Gaussian noisc. The standard deviation v is usually set to be small so that x closely resembles z. The aim of flow-based generative modeling is to learn a density homolopy (path) interpolating between the prior and the data likelihood for generative modeling. On that account, consider the following conditional (data-conditioned) probability density log- rog-hoootopy p: S22 [0, R:
represent the generative
p( a,,)) In ch(z)2,tt dx
(4)
where h: 522 [0, 1] - IR is a log-linear function:
h(w 2zt) a(t) log + B(t) log [x)
(5)
where a : [0. + [0, and 3 : [0.1] + [0, 1] are both monotonically increasing functions parameterized
by time t. The following proposition shows that this hog-hooyt transformation results a Gaussian perturbation kernel.
Proposition 1. Consider a Gaussian prior q(x)= - N(;;; II and a conditional data likelihood p( ar)
N(T,,, The log-homolopy transport (4) corresponds to a Gaussian perturbation kernel p(x: T ((x; (1))?,, characterized by the time-varying mean and standard deviation:
((()) w²) a() v² )
u2
(tt) sigmoid log
o(t) - B(t) M(t)
(6)
where sigmoid(z) = Ite=s denotes the logistic (sigmoid) function.
Proof. Refer to Appendix C.I.
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Hence, the density homotopy equation represents tempered Bayesian transport mapping from the Gaus- sian prior ((x) to the posterior kernel
p( x) q(a) p 2,1)= In ch(x)aa1) dx Jn p(z x) g(z)()))))))) dr
= p(æ 2)
(7)
which the maximum a posteriori estimation centered on discrete data samples. To approximate the intractable data likelihood, we can then consider the following marginal probability density homotopy:
p(a, / Pitata (T) pl 8. dz,
(8)
where it remains that p(x, 0) q(x), and have p(z, Sq (z p(z z) p(z). Therefore, this marginal homotopy defines data-recovery path interpolation between the Gaussian prior ((x) and the approximate data likelihood p(z). In particular, p(z) represents Bayesian approximation of the true data likelihood, by convolving the discrete data likelihood (Tata()) with the posterior distribution p(r z). Nevertheless, the marginalization in (8) is intractable, thereby prechuding a closed-form solution for the marginal homotopy. To overcome this challenge, we propose potential flow-driven density homotopy, whose time evolution is aligned with this delavrec marginal homotopy.
3.2 Modeling Potential Flow a Data- Recoveyy Homotopy Landscape
Our goal to model potential flow whose density evolution aligns with the marginal homotopy, thereby directing samples toward the data likelihood. We begin by deriving the time evolution of the marginal homotopy in the following proposition.
Proposition 2. Consider the conditional homotopy p( t) in (4) with Gaussian conditional data likeli- hood p(ū .r) (i; v²1). Then, the time evolution (derivalive) of the marginal homolopy p(x, 1) is given by the following partial differential equation (PDE):
0p(x,t) t) Øt
1
E (F) [p(r I,t) ((1,,,,))-7x,,
(9)
where 7y denotes the innovation term
(1,,,1)= a(t)
}(t) u2
wa ||:|| +
|| - -
(10)
Here, d(l) and (()) denole the lime-derivatives, and 7(x, T. 1) =
denotes the expectation.
Proof. Refer to Appendix C.2.
A potential flow involves subjecting the prior samples to an energy-generated velocity field, where their trajectories (z(t)) satisfy the following ODE:
dx(t) V (z,
(1I)
dl
where Φ 2 [0, + R is a scalar potential energy, and V denotes the Del operator (gradient) with respect to the data samples a(t). The vector field V e Ω represents the divergence (irrotational) component in the Helmholtz decomposition. By incorporating this potential flow, the flow-driven density homolopy ps (xx I) evolves via the continuity equation (Gardiner, 2009):
Ops t) = V (pa(z,ttV
(12)
Øt
which corresponds to the transport equation for modeling fluid advection. Our aim is model the potential energy such that the evolution of the prior density under the potential flow emulates the evolution of the
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marginal homotopy. In other words, seek to achieve homotopy matching, Pe = p. by aligning their respective time evolutions as described in (9) and (12). This leads to the following PDE, which takes the form of a wensity- Poisson equation:
- 5 V. (po (z) (2 - T Pa (8) (ptx x. (1(,,,,0)5((,,.00) 2.0]
(13)
However, this Poisson equation remains intractable due to the lack of a closed-form expression for pg. To overcome this limitation, we substitute the intractable pa with the target marginal homotopy p, enabling direct sampling and variational principle approach. In the following proposition, we demonstrate that the revised Poisson's 's equation minimizes the KL divergence between the flow-driven and conditional homotopies, yielding statistically optimal homotopy matching.
Proposition 3. Consider a potential flow of the form (11) and given that & E Hg(S),), where Hi denotes the (Sobolev) space of n-limes differenliable functions that are compaclly supported, and square-integrable with respect to marginal homotopy p(x,t). Solving for the potential energy ((x) that satisfies the following density--weighted Poisson's equation:
- E [p(x I a,t) (7(2, 2,),
V..
(14)
is then equivalent to minimizing the KL divergence DEL (z, between the flow-driven homotopy and the condilional homolopy-
Proof. Refer to Appendix C.3.
Therefore, solving this wensity- Poisson's 's equation corresponds to performing homotopy matching P4 = p. In the following section, we demonstrate that this homotopy matching gives rise to a Boltzmann energy expressed in terms of the potential energy t when the marginal homotopy ` reaches its stationary equilibrium, thereby establishing a connection between our proposed potential flow framework and EBMs.
3.3 Connections to Diffusion Process and Energy-Basen Modeling
In this section, we clarify the relationship between diffussio models and flow matching within the homotopy matching framework. Building on this insight, we establish a link between our proposed potential flow framework and energy-based modeling.
First, we present results from diffusion models. It has been outlined in Song et al. (2021) that the condi- tional densily homotopy, represented by the Gaussian perturbation kernel p(x (2,)== = -Nx;; ((()),((())))) characterizes a diffusion process governed by the following stochastic differential equation (SDE):
dz(() - (1))(()) di g(() (I
(15)
where (t) E R" denote the standard Wiener process. Note that the time parameterization with respect to here is the reverse of the conventional parameterization used in diffusion models, where the diffusion process transitions from x11) ~ Pdata (ø) to z00) ~ q(x) = (O w2T) as defined in Scction 3.1. In addition, the tire-varying drift f [0, IR and diffusion g [0, 1] lR coelficients are shown by Karras et al. (2022)
to be given by
f(1) = i(()
s(t) = - 20(1) (5(1)+ + s(1))())
(16)
m(t)²
where ji(t) and (1) denote the time-derivatives. Tt has also been shown in Song et al. (2021) that the following deterministic probability flow ODE:
dx(l) = xt)) 9())² v_ log p(z,((zt) t)
(17)
dt
results in the same marginal probability homotopy p(z, t) as the forward-time diffusion SDE (16). Subse- quently, we highlight the link between the diffusion process and the vector field modeled in flow matching.
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Proposition 4. The conditional vector field in flow matching (Lipman et al., 2023), given by
dx(t) - ol [2,) =
(18)
dt
with standard Gaussian noise T), satisfies the conditional probability flow ODE governing the diffusion process conditioned on boundary condition x(1) Paata (T). It follows that the marginal vector field, given by the law of iterated expectation (tower property) BUUX = x] = =EE X = x, X æ|:
dx(t) v(z,t) Exp (u(2 1x,t x]] = / o(x 2,t) p(2: H. Pdata( di
(19)
dt
p(a, t)
also satisfies the marginal probability flow ODE (17).
Proof. Refer to Appendix C.4.
Building on this result, we establish a connection between the proposed potential flow framework and EBMs. The following proposition demonstrates that homotopy matching, e.g., pa = p leads to an energy- parameterized Boltzmann equílibrium.
Proposition 5. Given that the flow-driven homotopy po(x,)) t) matches the rec-rre marginal homotopy p(x,t), they exhibit the some Fokker Planck dynamics. As the time-varying marginal density p(x, t) converges to its stationary equilibrium Poc(x), i.e., when 0p(2.t)
dr - O, the Fokker-Planck dynamics reach the Boltzmann distribution (1), where the Boltzmann eneryy (3()) defined as follows:
(Φ(x)= th ox)) + soo ||0:1/
(20)
where &x (x), Soo, and 9so denole the steady-state potential energy, drift, and diffusion coelficients, respec- tively, associated with the stationary equilibrium.
Proof. Refer to Appendix C.5.
On that note, we uncover the connection between the proposed VAPO framework and EBMs, demonstrating the validity of the potential energy a parameterization of Boltzmann energy. This holds provided that p
converges to its stationary equilibrium and po learns to match these convergent dynamics. In the following section, we introduce a variational principle approach to solving the weishty- Poisson equation (14), thereby addressing the intractable homotopy matching problem.
3.4 Variational Potential Energy Loss Formulation: Deep Ritz Method
Solving the weighty Poisson's 's equation (14) is particularly challenging in high-dimensional settings. Numerical approximation struggles to scale with higher dimensionality, as selecting suitable basis functions, such as in the Galerkin approximation, becomes increasingly complex (Yang et al., 2016). Similarly, a diffu- sion map-based algorithın demands an exponentially growing number of particles to ensure error convergence (Taghvaci et al., 2020). To address these challenges, we propose a variational loss function using the Deep Ritz method. This approach casts Poisson's equation as variational problem compatible with stochastic gradient descent. Consequently, the proposed approach solves Eq. (14), effectively aligning the flow-driven homotopy with the marginal homotopy. Directly solving Poisson's 's equation (14) is challenging. Therefore, we first consider the following weak formulation:
Sa2 Epsnnt()) [p(x: E,t) (,(2, F,t) ((x,E,t)] v dr (V. V. (pla.)) v dr
(21)
This PDE must hold for all differentiable trial functions w. In the following proposition, we introduce a variational loss function that is equivalent to solving this weak formulation of the weighte Poisson's 's equation.

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Proposition 6. The variational problem of minimizing the following loss functional:
t( Cov p(anla,t) (ana)))) (((,)t) (0, 2. + Ep(x,t) ([,,
(22)
with respect to the polential energy D, is equivalent to solving the weak formalation (21) of the density- weighted Poisson's equation (14). Here, ||- denotes the Euclidean norm, and Cov denotes the covariance. Furthermore, the variational problem (22) admits a unique solution & E Hb(St; p) if the marginal homotopy p satisfy the Poincaré inequality:
E Eptx.r)
|| (xx, 1)] 2 Ep(r,))
[[)(
(23)
for some positive scalar constant 1j > 0 (spectral gup).
Proof. Refer to Appendix C.6.
The integration by parts in (66) and (87) require that the marginal density p(x) vanishes on the boundary an of some open, bounded domain Ω C R", so that the boundary integral Jan p(x) (b de = 0 holds. In standard implementations, although the training data x are typically normalized to lie within [-1,1]", we may define the perturbed samples as x E $2 C R", where Ω is chosen to contain the support of the data distribution. Accordingly, the open bounded domain Ω can be defined sufficiently large that the conditional homotopy p( I,t) approaches zero at the boundary An. Since p(xx zz,) is a Gaussian perturbation kernel, it decays exponentially and is effectively negligible near the boundary, thereby satisfying the required condition. As a result, the marginal distribution p(z, t) also vanishes at an, ensuring the validity of the integration by parts required to formulate both Proposition 3 Proposition 6.
Remark 1.
Overall, Propositions 3 and 6 recast the intractable problem of minimizing the KT. divergence between the flow-driven homotopy and the marginal homotopy as equivalent variational problem of solving the loss function (22). By optimizing the potential energy with respect to this loss and transporting the prior samples through the ODE (11), the prior particles evolve along a trajectory that aligns with the marginal homotopy. In particular, the covariance loss here plays an important role by ensuring that the normalized innovation (residual sum of squares) is inversely proportional to the potential energy. As a result, the energy-generater velocity field V, th consistently points in the direction of greatest potential ascent, thereby driving the flow of prior particles towards high likelihood regions. Given that homotopy matching is performed over the entire time horizon, apply stochastic integration to the loss function over time, where t ~ [1, tend) is drawn from the uniform distribution.
3.5 Training Implementation
In our implementation, we adopt the Optimal Transport Flow Matching (OT-FM) framework (Lipman et al., 2023) for training, where it corresponds to the SDE parameterization f(t) - g(t) /2(1-0) as outline in (Kingma & Gao, 2023), or equivalently att) derived in (5). To establish a Boltzmann equilibrium, we further require dt.
n² w² 1-t' B(t) = (1-1)= for the log-homotopy transformation
ap(x,)) + 0 so that the time- varying marginal density p(x, t) converges to the stationary Boltzmann distribution. However, the Gaussian employed by the flow- probability paths is defined
perturbation kernel p(x t) = N (u((t),,ot)) only over finite time interval & mmxx] Additionally, the marginal homotopy p(x, is not guaranteed to reach equilibrium within this prescribed time window.
To resolve these limitations of the flow-based probability paths, we explicitly enforee stationarity in our training implementation, by imposing a steady-state equilibrium (x)(x) = p(x, / 2 bmax) beyond some cutoff time fmax < tend close to the terminal time. This steady-state equilibrium (z) = p2(x) thus corresponds to the stationary Boltzmann distribution, parameterized by the energy function derived in (20) with fox (tmax) and J = (tmaxx). Given that steady-state equilibrium is enforced via Pee (z) = p(a, t 2 (maxx pdata(()) (T), the stationary Boltzmann distribution approximates the true data likelihood by design.
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Algorithm 1 VAPO Training
input: Initial model parameters 0, mean and standard deviation scheduling functions jut), a (t), cutoff time fmax: terminal time lend, decay exponent n; spectral gap constant 7), and batch size B.
repeat
Sample observed data Pdata (z), ~ ((,, fend), and e; ~ -V
Set 4 = min(t;, fmax) and sample ~ p(z ã, t,) via reparametcrization p(t; (t4) 4
Compute gradient V (0r 44) w.r.l. I3 via backpropagation
Calculate VAPO loss π B L(Dg, t;), backpropagate and update model parameters 0 until convergence
Finally, our VAPO loss function is implemented as follows:
L CVAO (T) = =| fond
1( di Re((o.mm)) [[ [)]
(24)
where
(r, v(x a,
C(#,t) p(zl,,)) (T)
(z, w(t) w(() (2,,
V (x, 0 |o 2,00
(25)
+ + p(la(a,)) (F)
[| ( || + (I,
Here, we incorporate an additional cosine distance between the potential gradient V and the conditional vector field in (18) to the loss function. While this cosine distance does not influence the learning of the potential energy's magnitude (magnitude learning is entirely supervised by the covariance loss), enforces directional alignment between the gradient and the vector field. To further enforce convergence Loward steady-state potential (x),,, additionally encourage quasi-static dynamics by minimizing the Euclidean norm of the time derivative
alongside the gradient norm during training. Also, weighting w(() = t)* with decay exponent k applied to the innovation term to balance the covariance loss across time stabilize training.
00 ae
Considering that the marginal homotopy may not satisfy the Poincaré inequality (23), we include the right- hand side of this inequality in the loss function to enforce the uniqueness of the minimizer. To empirically validate the existence of a positive Poincaré constant 1). Figure 11 plots the ratio between the mean gradient norm [[IVV, x |12] and the mean energy norm [[] over training iterations on CIFAR-10, without applying the additional Poincaré regularization loss. It shows that the ratio bounded below by n = 6.81 x 10-5, thereby confirming the existence of a positive Poincaré constant during training.
Nonetheless, our experiments indicate that the existence and magnitude of such an unenforced Poincaré constant vary across different neural architectures. For completeness, we incorporate the Poincaré regular- ization with small 7) for both the WideResNet and U-Net models, which we fine-tune during training for optimal results. The cutoff time fmax. terminal time fend, decay exponent and spectral gap constant nj are hyperparameters to be determined during training. Algorithm 1 summarizes the training procedure of our proposed VAPO framework.
4 Experiments
In this section, we validate the energy-based generative modeling capabilities of VAPO across several key tasks. Section 4.1 explores 2D density estimation. Section 4.2 presents the unconditional generation and spherical interpolation results on CIFAR-10 and CelebA. Section 4.4 evaluates mode coverage and model gencralization through energy histograms of train and test data and the nearest neighbors of generated samples. Section 4.5 examines unsupervised OOD detection performance on various datasets. Section 4.6 verifies the convergence of long-run ODE samples to a Boltzmann equilibrium. Additional results on ablation study and computational efficiency are provided in Appendix A. Additional discussions of the results are also provided in Appendix B. Finally, implementation details, including architecture, training, numerical solvers, datasets, and FID evaluation, are provided in Appendix D.

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T:0255
T-0.30
T+1.25
T-1.0
T=1.25
1-1.0
T-1.75
T=200
Figure 1: 2D potential flow. Top: Sample trajectories from the Gaussian prior noise distribution (black) to the target 2-Moons distribution (blue), driven by the potential energy ((x, t) and sampled using ODE solver. Bottom: Time evolution of the learned potential energy landscape (xr,t).
4.1 Density Estimation on 2D Data
To verify the convergence properties of the potential energy and to assess the validity of the Boltzmann energy (20), conduct density estimation 2D synthetic datasets. Specifically, we learn a potential flow that transforms unimodal Gaussian prior distribution into 2-Moons target distribution. Figure 1 shows the sample trajectories driven by the potential flow dz(t) = v. 2, t) dt, obtained via the deterministic Euler solver. Figure 2 presents the sample trajectories and density estimation of the Boltzmann distribution x e e(((x), obtained via the Stochastic Gradient Langevin Dynamics (SGLD). Notably, both the potential energy (xx) and the Boltzmann energy (x) exhibit stable convergence toward their steady-state equílibrium. Furthermore, the results indicate that the estimated Boltzmann density closely aligns with the ground- truth 2-Moons distribution. These results highlight the effectiveness of our variational principle approach in learning the Boltzmanu stationary distribution through homotopy matching against the stationary-enforceed marginal Poo (z).
Nonetheless, a standard formulation of the forward-time SDE (15), or equivalently, the marginal probability flow ODE (17), is valid only in the case unimodal Gaussian prior, e-g., ((x) = = x, = 0) = NO0 wfI), as discussed Kingma & Gao (2023). This assumption underpins the consistency of the Fokker-Planck dynamics with the continuous-time diffusion framework, ensuring the validity of the stationary Boltzmann energy in the limit. We acknowledge this limitation of current framework. As direction for future work, we propose extending the forward-time SDE or ODE formulation of continuous-tine diffusion to be admissible for more general prior distributions, such as mixtures of Gaussians or learned priors, to accommodate multi-modal data while maintaining consistency with our proposed energy-based framework.
4.2 Unconditional Image Generation
For image generation, we consider three VAPO model variants: an autonomous (independent of time) energy model ((x) parameterized by Zagoruyko & Komodakis (2016), and time-varying energy model (xx t) parameterized by U-Nct (Ronneberger et al., 2015). Figure shows the uncurated and unconditional
Figure 2: 2D Boltzmann density estimation. Top: Sample trajectories from the Gaussian prior noise distri- bution (black) to the target 2-Moons distribution (blue), driven by the Boltzmann energy and sampled via SGLD. Middle: Visualization of the log-density estimation (up to an additive constant) log (z()) = Φ parameterized by Boltzmann energy.
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Figure 3: Uncurated and unconditional samples generated for CIFAR-10 (left) and CelebA (right).
Male and Young
Male and Smile
Young and Smile
Figure 4: Compositional and conditional CelebA samples generated based on three attribute pairs.
image samples generated using the time-varying energy model CIFAR-10 32 x 32 and CelebA 64 x 64. The generated samples are of decent quality and resemble the original datasets, despite not having the highest fidelity as achieved by state-of-the-art models. Table 1 summarizes the quantitative evaluations of our framework in terms of FTD (Heusel et al., 2017) scores on the CIFAR-10. In partieular, the VAPO models achieved FID scores competitive to existing generative models. Figures and show additional uncurated samples of unconditional image generation on CIFAR-10 and CclcbA, respectively.
4.3 Image Interpolation and Compositional Generation
To achieve smooth and semantically coherent image interpolation, we perform spherical interpolation bet wetween two Gaussian noises and subsequently apply ODE sampling to the interpolated noises. Figures and 10 show additional interpolation results on CIFAR-10 and CclobA, respectively. For compositional sample generation, we first train a class-conditioned energy model ((x, c), and then sample by averaging the conditional energies across selected classes. Figure presents compositional generation results conditioned on composite CelebA
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Sample
Nearest Neighbours
Sample
Nearest Neighbours
Figure 5: Generated CIFAR-10 samples and their five nearest neighbors in train set based on pixel distance.
attributes, specifically (Male, Young), (Male, Smile), and (Young, Smile). However, certain compositional samples show limited variation across attribute pairs, suggesting that incorporating composition weights could improve attribute-specific conditioning.
4.4 Model Generalization and Mode Evaluation
To evaluate the generalization capability of the VAPO model, Figure 5 presents the nearest neighbors of the generated samples in the CIFAR-10 training set. The results show that nearest neighbors differ significantly from the generated samples, suggesting that our model does not overfit the training data and generalizes well to the underlying data distribution. To validate the mode coverage and over-fitting ability, Figure 6 presents a histogram of the CIFAR-10 training and test datasets across the potential energy estimated by VAPO. The histogram shows that the learned energy model assigns similar energy values to images from both sets. This indicates that the VAPO model generalizes well to unseen test data while maintaining broad mode coverage of the training distribution.
0.012
Dataset
Test
0.01
Train
Probabiity
0.004
0.002
1000
1100
1200
1300
1400
Energy
Figure 6: Histogram of the CIFAR-10 training and test datascts across potential energy.
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Table 1: FID scores on unconditional CIEAR-10 image generation.
Energy-based Models
FID
Other Likellhood-based Models
FID J
EBM-IG (Du & Mordatch, 2019)
38.2
ResidualF low (Chen et al., 2019a)
47.4
EBM-FCE (Gao et al., 2020)
37.3
Glow (Kingnn & Dhariwal, 2018)
46.0
CoopVAEBM, (Xie et al., 2021b)
36.2
DC-VAE (Parmar et al., 2021)
17.9
CoopNets (Xie et al., 2020)
33.6
GAN-based Models
Divergence Triangle (Han al., 2019)
30.1
SN-GAN (Miyato al., 2018)
21.7
VERA (Crathwohl et al., 2021)
27.5
SNGAN-DDLS Che et al. (2020)
15.4
EBM-CD (Du et al., 2021)
25.1
BigGAN (Brock al., 2019)
14.8
GEBM (Arbel al., 2021)
19.3
Score-based and Diffusion Models
HAT-EBM (Hill et al., 2022)
19.3
NCSN-v2 (Song Ermon, 2020)
10.9
CF-EBM (Zhao et al., 2020)
16.7
DDPM Distil (Luhman et al., 2021)
9.36
CoopFlow (Xie al., 2022)
15.8
DDPM (Ho et al., 2020)
3.17
VAEBM (Xiao et al., 2021a)
12.2
NCSN+ (Song et al., 2021)
2.20
DRL (Gao et al., 2021)
9.58
Flow-based Models
CLEL (Lee et al., 2022)
8.61
Action Matching (Neklyudov et al., 2023)
10.0
DDAEBM (Geng et al., 2024)
4.82
Flow Matching (Lipman et al., 2023)
6.35
CDRL (Zhu et al., 2024)
3.68
Rectified Flow (Liu et al., 2023b)
4.85
VAPO (Autonomous)
14.5
DSBM (Shi al., 2023)
4.51
VAPO (Time-varying)
6.72
PFGM (Xu al., 2022)
2.35
Table 2: AUROC scores f for OOD detection on several datasets.
Models
CIFAR-10 Interpolation
CIFAR-100
CelebA
SVHN
PixeICNN (Salimans et al., 2017)
0.71
0.63
0.32
GLOW (Kingma Dhariwal, 2018)
0.51
0.55
0.57
0.24
NVAE (Vahdat Kautz, 2020)
0.64
0.56
0.68
0.42
EBM-IG (Du & Mordatch, 2019)
0.70
0.50
0.70
0.63
VAEBM (Xiao et ul., 2021a)
0.70
0.62
0.77
0.83
CLEL (Lee et al., 2022)
0.72
0.72
0.77
0.98
DRL (Gao et al., 2021)
0.44
0.64
0.88
CDRL (Zhu et al., 2024)
0.75
0.78
0.84
0.82
VAPO (Ours)
0.78
0.67
0.84
0.61
4.5
But-of-Distribunn Detection
Given that the potential flow corresponds to a stationary Boltzmann distribution, the Boltzmann energy ØB from (20) can be used to distinguish between in-distribution and OOD samples based on their assigned energy values. Specifically, the polential energy model trained on the CIFAR-10 training set assigns energy values to both in-distribution samples (CIFAR-10 test set) and OOD samples from various other image datascts. We evaluate OOD dotection performance using the AUROC metric, where a higher score reflects better model's efficacy in accurately assigning lower energy values to OOD samples. Table compares the AUROC scores of VAPO with those of various likelihood-based and F.BMs. The results show that model performs exceptionally well on interpolated CIFAR-10 and CelebA 32 x 32, while performing moderately CIFAR-100 and SVHN.
4.6 Long-Run Steady-Sta State Equilibrium
Figure 12 illustrates long-run ODE sampling over an extended time horizon t e [0, 20] using the autonomous energy model parametorized by Ned. Additionally, Figure 13 illustrates long-run ODE sampling using the time-varying energy model parameterized by U-Net. The results indicate similar deterioration in image quality extended time periods, albeit to greater extent compared to the autonomous model.
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Figure 14 plots the mean gradient norm E [|VV and the mean energy norm [[ neither of which exhibit convergence. These results are consistent with those observed in EBMs trained using non-convergent short-run MCMC (Agoritsas et al., 2023; Nijkamp et al., 2020). This issue arises from the inherent difficulty neural network models face in learning complex energy landscapes in high-dimensional spaces. Regions that remain unseen during training can correspond to poorly modeled areas of the energy landscape, often resulting in the emergence of sharp local minima. Consequently, ODE-basod sampling may become trapped in these local minima, leading mode collapse and poor mixing, which manifest visual artifacts such excessive saturation and loss of background details.
To resolve these issues, we replace the deterministic ODE solver with the conventional SGLD sampler for image generation, enabling sampling from the Boltzmann energy via Tz+1 = It + A. (= + v where e ~ (0, X²I) denotes isotropic Gaussian noise with temperature scale (standard deviation), and A, is the step size. The injected stochasticity from the diffusive noise in SGLD facilitates escape from local minima and enhances mixing efficiency during sampling. As shown in Figures 15 and 17, SGLD mitigates mode collapse and the long-run image samples converge well to the stationary equilibrium. Furthermore, Figures 16 and 18 demonstrate that the gradient norm converges to zero, while the energy norm asymp- totically stabilizes, indicating stcady-state thermalization. These SDE-based sampling results confirm that equilibrium convergence is achievable with a stochastic sampler. Nonetheless, the temperature scale À must be carefully tuned to balance convergence speed and sample quality. Moreover, our experiments show that ODE-based sampling consistently yields better FID scores, potentially due to the deterministic nature of the proposed potential flow and the straightness of the linearly interpolated OT-FM trajectories, which contribute to sharper and more consistent sample generation.
5 Conclusion
We propose VAPO, a novel energy-based potential flow framework designed to reduce the computational cost and instability typically associated with EBM training. Empirical results demonstrate that VAPO out- performs several existing EBMs in unconditional image generation and achieves competitive performance in OOD detection, highlighting its versatility across diverse generative modeling tasks. Despite these promising results, future work will aim to refine the training strategy to improve scalability to higher-resolution images and other data modalities, while addressing the limitations outlined in this work. Additionally, exploring generative models that, inherently incorporate Neumann boundary conditions into the design of their blur- ring perturbation kernels (Rissanen et al., 2023; Hoogeboom & Salimans, 2023; Daras et al., 2023) presents a promising direction for improving energy landscape modeling and enhancing sample diversity without incurring the computational burden of long-run MCMC sampling.
Broader Impact Statement
Generative models represent a rapidly growing field of study with overarching implications in science and socicty. Our work proposes a new generative model designed for efficient data generation and OOD detection, with potential applications ions in fields such as medical imaging, entertainment, and content creation. However, as with any powerful technology, generative models come with substantial risks, including the potential misuse in creating deepfakes or misleading content that could undermine social security and trust. Given this dual-use nature, it is essential to implement safeguards, such as classifier based guidance, to prevent the generation of biased or harmful content. Morcover, generative models are vulnerable to backdoor adversarial attacks and can inadvertently amplify biases present in the training data, reinforcing social inequalities. Although our work uses standard datasets, it is important to address how such biases are handled. We are actively exploring methods to identify and mitigate biases during both the training and generation phases. This includes employing fairness-aware training algorithms and evaluating the model's output for biased palterns. Furthermore, while this work demonstrates the potential benefits of generative models, the ethical concerns surrounding their deployment must be considered. Addressing these issues will require ongoing collaboration to develop frameworks for responsible use, including transparency, model interpretability, and robust safeguards against malicious applications. By proactively engaging with these ethical concerns, the broader community can contribute to the responsible advancement. of generative modeling technologies.
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Published in Transactions on Machine Learning Research (05/2025)
Yang Zhao, Jianwen Xie, and Ping Li. Learning energy-based generative models via coarsc-to-fine expanding and sampling. In International Conference on Learning Represenlations, 2020.
Yaxuan Zhu, Jianwen Xie, Ying Nian Wu, and Ruiqi Gao. Learning energy-based models by cooperative diffusion recovery likelihood. In The Twelfth International Conference on Learning Represendations, 2024.
20


Reviews:
[{"review_id": "6uVeRZndl2", "reviewer": "Reviewer_q12m", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n* The paper presents a novel framework for using potential energy functions to define data-generating gradient flows. The paper brings together a variety of ideas from EBMs, diffusion models, flow matching, and deep Ritz learning into an interesting and novel paradigm.\n* Several non-trivial propositions are presented to describe the density homotopy evolution, define a Poisson equation that can minimize KL divergence with a learned model, define the Boltzmann steady-state distribution of the learned model, and to define a loss that can solve the Poisson equation. The appendix provides detailed derivations of each proposition.\n* The results show strong generative performance of the proposed method among recent EBMs and strong OOD performance from the Boltzmann density.\n\nWeaknesses:\n* To my understanding, the density defined by the Boltzmann energy $\\Phi_B$ does not necessarily follow the approximate data density $\\bar{\\rho} (x)$. Is this correct? In this case, the proposed method does not accomplish the typical EBM goal of learning a Boltzmann energy that can approximate the data density, but rather learns a potential flow that can guide samples from the noise density to the data density in finite time. It might not be entirely accurate to categorize this proposed method as an EBM in the sense used in many prior works.\n* Despite the strong performance of the method compared to other EBMs, the results still lag behind diffusion and flow matching models.\n* The density estimation application in Figure 1 doesn't seem very convincing, because the learned densities in the top row do not seem to match the expected 2-moons distribution.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work explores the problem of image generation with a time-dependent potential energy function whose gradient dynamics guide samples from a noise prior distribution to realistic image samples. The paper begins by defining a conditional density homotopy and an associated marginal density homotopy, where the marginal density homotopy at t = 0 follows the prior distribtution and the homotopy at t = 1 approximately follows the distribution of observed data. The timesteps 0 < t < 1 provide a path to guide samples from noise to data during the data generation process. The next section finds a formula for the evolution of the density homotopy over time and defines a Poisson equation that gives a condition under which the KL divergence between the homotopy from a learned potential energy flow and the target data homotopy is minimized. The steady-state behavior of such a learned homotopy is shown to follow a Boltzmann distribution with a certain form. Finally, a practical loss function for solving the Poisson equation and learning the potential energy flow is presented. Experiments are conducted for learning densities in low dimensions, generating CIFAR-10 and Celeb-A images, and the OOD performance of the Boltzmann energy.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "It would be helpful to include a section that describes the benefits of the proposed potential energy flow compared to the typical diffusion or flow matching models where the flow vector field is defined directly by the network output rather than the gradient of a scalar valued network.", "broader_impact_concerns": "Broader impacts were not discussed, although this does not impact my assessment of the paper.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "lSG0PnRUQt", "reviewer": "Reviewer_ruyf", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n- Novel Framework \\\nThe paper innovatively tackles longstanding challenges in training energy-based models (EBMs) by eliminating the reliance on computationally intensive MCMC sampling, which typically suffers from issues like poor mode mixing and slow convergence. Instead, it introduces a variational approach that combines the Deep Ritz method with density homotopies to enable efficient and principled training.\n\n- Justification \\\nThe introduction effectively motivates the proposed method by: 1. Clearly outlining the shortcomings of current EBM training techniques. 2. Discussing alternative generative modeling approaches (like flow-based models) and their limitations. 3. Positioning VPFB as a framework that retains the theoretical benefits of EBMs.\n\n- Theoretical rigor \\\nThe paper demonstrates strong theoretical contributions by: 1. Constructs a clear Bayesian marginal homotopy between a simple Gaussian prior and the data likelihood. 2. Formulating a potential flow governed by a conservative vector field, ensuring that the energy landscape is both well-defined and interpretable. 3. Employing the Deep Ritz method to derive a variational loss that minimizes the KL divergence.\n\n**Weakness**\n- Computational Cost Analysis Missing \\\nWhile the paper claims that VPFB reduces the computational burden by avoiding MCMC sampling, it does not provide a detailed comparison of training costs relative to traditional MCMC-based EBMs or state-of-the-art diffusion/flow-based models. The claim of avoiding computationally heavy MCMC sampling is not fully substantiated without detailed efficiency metrics.\n\n- Performance Gap \\\nAlthough the empirical results are competitive, there is a notable performance gap compared to state-of-the-art generative models, including leading EBM methods like DDAEBM. This discrepancy raises concerns about the practical applicability of VPFB. A thorough analysis is needed to explain the performance gap.\nMoreover, the number of function evaluations (NFEs) are not compared with baselines, which is critical for fair comparison.\n\n- Limited Ablation Studies \\\nThe paper does not include detailed ablation studies to isolate the contributions of individual components (e.g., potential flow alignment and the variational loss formulation). This omission makes it challenging to evaluate the specific impact and necessity of each component on the overall performance.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The work introduces Variational Potential Flow Bayes (VPFB), which tackles well-known challenges in training energy-based models (EBMs), such as the reliance on MCMC sampling with its inherent issues like poor mode mixing and slow convergence. By employing the Deep Ritz method and aligning density homotopies via a variational loss that minimizes KL divergence, the approach is innovative in that it seeks to remove the need for auxiliary models and implicit MCMC entirely. This direct connection between energy-parameterized flow models and explicit marginal EBMs seems to fill a gap in current research.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "**Include Detailed Computational Cost Analysis** \\\nProposed Adjustment: Provide a thorough comparison of training efficiency, including metrics such as the number of function evaluations (NFEs), training runtime, and resource usage, against MCMC-based EBMs and state-of-the-art diffusion/flow-based models. \\\nImpact: Critical – This is essential to substantiate the claim that VPFB avoids the computational overhead associated with MCMC sampling.\n\n**Conduct Comprehensive Ablation Studies** \\\nProposed Adjustment: Isolate and quantify the contributions of individual components (e.g., potential flow alignment, variational loss formulation via the Deep Ritz method) to assess their individual impact on performance. \\\nImpact: Critical – These studies are necessary to clearly demonstrate that each component meaningfully contributes to the overall effectiveness of the approach.\n\n**Analyze and Address the Performance Gap** \\\nProposed Adjustment: Provide a detailed analysis of the performance gap compared to state-of-the-art generative models (e.g., DDAEBM), including potential explanations and strategies for improvement. \\\nImpact: Strengthening – While not critical, this analysis would enhance the clarity of the contributions and provide valuable insights for further improvement.\n\n**Expand the Experimental Scope** \\\nProposed Adjustment: Evaluate the method on additional datasets or different data modalities (e.g., higher resolution images or non-image data) to better demonstrate the generality of the approach. \\\nImpact: Strengthening – Broader experiments would increase the work’s impact and validate its applicability beyond the current scope.", "broader_impact_concerns": "While the paper primarily focuses on a methodological contribution, there are several ethical implications that should be considered:\n\n**Dual-Use Concerns** \\\nAdvances in generative modeling, including improved image generation quality, can be misused to create deepfakes or misleading content. It is important to acknowledge this dual-use nature and discuss potential safeguards.\n\n**Data Bias and Fairness** \\\nAlthough the work uses standard datasets, any generative model can inadvertently amplify biases present in the training data. It would be beneficial to discuss how the approach handles such biases and what steps could be taken to mitigate fairness concerns.\n\nIncluding a Broader Impact Statement that addresses these points would strengthen the submission by proactively considering potential ethical issues associated with generative modeling advancements.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "fdl4dlWcng", "reviewer": "Reviewer_baM4", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n   The integration of potential flow with variational principles is interesting, offering a fresh perspective on EBM training, addressing longstanding issues with MCMC sampling. Extensive experiments validate the framework across diverse tasks. \n   Weaknesses:\n   - In Equation (67), the assumption that $\\(\\bar{\\rho}(x)\\)$ vanishes at the boundary $\\(\\partial\\Omega\\)$ (i.e., compact support) may not hold for real-world data distributions such as natural images.\n   - Proposition 6 postulates the existence of $\\(\\eta > 0\\)$ such that ${E}_{\\bar{\\rho}}|\\nabla \\Phi|^2 >=\\eta {E}{\\bar{\\rho}} \\Phi^2$, but lacks empirical validation or theoretical justification for this Poincaré-type inequality assumption.\n   - The contradiction between the theoretical steady-state convergence (assumed when $\\(t_{\\text{max}} = 0.9\\)$) and the empirical observation in Figure 6 showing degraded sampling quality with increasing $\\(t\\)$ requires clarification.\n   - Ambiguous notation: The manuscript frequently conflates $\\(\\rho_{\\Phi}\\)$ (flow-driven density) with $\\(\\bar{\\rho}\\)$ (target marginal density), necessitating clearer differentiation.\n   - While the introduction positions VPFB as addressing MCMC's computational bottlenecks, the experimental section lacks detailed comparative analysis with baseline MCMC methods to substantiate computational efficiency claims. Since numerical SDE solver might be another computationally costly method.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces Variational Potential Flow Bayes (VPFB), a novel framework for training energy-based models (EBMs) that circumvents the computational challenges of traditional contrastive divergence training with implicit MCMC sampling. VPFB constructs a flow-driven density homotopy aligned with the data distribution through a variational loss that minimizes the KL divergence between flow-driven and marginal homotopies. Experiments on image generation, interpolation, out-of-distribution (OOD) detection, and compositional generation demonstrate competitive performance (e.g., FID of 7.08 on CIFAR-10 and strong AUROC scores for OOD detection).", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Provide additional theoretical proofs or experimental validation for key assumptions (e.g., boundary behavior of data distributions, validity of Poincaré-type inequalities)\n   - Clarify the contradiction between experimental observations in Figure 6 (degraded sampling quality with increasing t) and the theoretical steady-state convergence assumption (when t_max = 0.9)\n   - Include comprehensive computational metrics (total parameter count, wall-clock time, inference latency) to quantitatively demonstrate VPFB's computational efficiency advantages over conventional MCMC methods\n   - Conduct thorough notation review to ensure consistent symbol usage. Typo: \"VAPO\" near Equation 24.", "broader_impact_concerns": "The paper's potential for misuse in generating harmful content is a concern.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper offers a novel and theoretically strong framework for training EBMs without MCMC. Despite some empirical limitations, all reviewers agreed that the method demonstrates competitive performance compared to existing EBMs and presents a promising direction for future research.
Decision: Accept (poster)