Unified Energy-Based Diffusion Model

• The model unifies energy-based formulations with diffusion processes, interpreting reverse diffusion as a form of Langevin dynamics within a shared probabilistic system.
• It enables simultaneous training of diffusion samplers and energy-based models, bridging continuous and discrete settings for applications in image, motion, and molecular domains.
• By coupling energy gradients with denoising transitions, the approach addresses challenges in sample quality and temporal consistency while opening new avenues for architecture design.

A unified energy-based diffusion model denotes a class of generative formulations in which diffusion and energy-based modeling are treated as parts of the same probabilistic system rather than as separate modules. In this literature, unification can mean that reverse diffusion is interpreted as Langevin dynamics in an energy landscape, that an energy-based model (EBM) and a diffusion sampler are trained by a single objective, that a single energy function is defined over both data and diffusion time, or that denoising transitions themselves are parameterized as EBMs. The same idea also appears in discrete settings, where diffusion paths are cast as energy-minimizing trajectories, and in architectural settings, where message passing or attention is derived from diffusion-induced energy descent (Luo, 2022, Yoon et al., 2023, Zhang et al., 2023).

1. Conceptual genealogy

A major precursor is the variational–score unification of diffusion models. In the hierarchical-VAE view, a diffusion model is a Markovian latent hierarchy whose training reduces to predicting either the clean sample, the source noise, or the score of a noisified sample; Tweedie’s formula then makes explicit that score prediction is equivalent to learning gradients of log densities over noisy marginals (Luo, 2022). In that sense, diffusion already admits an implicit energy interpretation, since the score is the gradient of a time-dependent negative log-density.

A second line of work makes the EBM connection explicit at the objective level. Diffusion Contrastive Divergence (DCD) interprets classical Contrastive Divergence as one member of a larger family of KL-contractive divergences induced by diffusion processes, and shows that classical CD is recovered when the diffusion operator is the EBM-induced Langevin dynamics (Luo et al., 2023). Generalized Contrastive Divergence (GCD) goes further by replacing fixed MCMC with a trainable diffusion-style sampler and optimizing the pair as a minimax system, so that energy and sampler are no longer separate training components but a single coupled generative game (Yoon et al., 2023).

Taken together, these developments make “unified” a precise technical claim rather than a slogan. It refers either to a shared objective, a shared dynamics, or a shared probabilistic representation in which diffusion-time transport and energy-based density modeling are mathematically identified.

2. Canonical mathematical formulations

Several recurrent formulations now define the field.

Formulation	Core expression	Representative paper
Score as energy gradient	$p_\theta(z\mid c)\propto \exp(-E_\theta(z,c))$, with $$\nabla_z E_\theta(z_t,c)\approx \epsilon_\theta(z_t,t,c)$$	compositional latent diffusion (Zhang et al., 2024)
Joint EBM over diffusion time	$p_\theta(x,t)=\exp(-U_\theta(x,t))/Z(\theta)$	diffusion-assisted EBM (Zhang et al., 2023)
Per-step conditional EBM	$$p_\theta(x_{t-1}\mid x_t)=\dfrac{\exp(E_\theta(x_{t-1},t-1))\,q(x_t\mid x_{t-1})}{Z_{\theta,t}(x_t)}$$	adversarial stepwise EBM (Geng et al., 2024)
Joint energy–sampler game	$$\min_\theta \max_\phi \ \mathbb{E}_{p}[E_\theta(x)]-\mathbb{E}_{\pi_\phi}[E_\theta(x)]+\tau\mathcal{H}(\pi_\phi)$$	GCD (Yoon et al., 2023)

The first formulation identifies the denoising network with an energy gradient. In latent motion diffusion, for example, the reverse step

$$z_{t-1}=z_t-\epsilon_\theta(z_t,t,c)+\mathcal{N}(0,\tilde{\beta}_t I)$$

is compared directly to Langevin dynamics,

$$z_{t-1}=z_t-\eta \nabla_z E_\theta(z_t,c)+\mathcal{N}(0,\tilde{\beta}_t I),$$

yielding the interpretation that reverse diffusion is a noisy gradient descent trajectory on $E_\theta$ (Zhang et al., 2024). This formulation is especially natural when the energy is only available through its gradient, as in score-based models.

The second formulation absorbs diffusion time into the state space. Diffusion-assisted EBMs define a single normalized model over pairs $(x,t)$,

$$p_\theta(x,t)=\frac{\exp(-U_\theta(x,t))}{Z(\theta)},$$

so that clean data, partially diffused data, and near-Gaussian noise are all marginals or conditionals of one joint energy model (Zhang et al., 2023). The training signal is then maximum-likelihood-style on this joint density, with persistent sampling in the augmented space.

The third formulation places an EBM at each reverse denoising step. Instead of modeling the full marginal directly, one parameterizes

$$\nabla_z E_\theta(z_t,c)\approx \epsilon_\theta(z_t,t,c)$$0

as an energy-tilted version of the known forward kernel $$\nabla_z E_\theta(z_t,c)\approx \epsilon_\theta(z_t,t,c)$$1, which makes each reverse step an EBM-defined conditional while the full reverse chain remains a diffusion model (Geng et al., 2024). This decomposition is motivated by the claim that denoising conditionals are simpler than the full data marginal.

The fourth formulation unifies training rather than parameterization. GCD treats the EBM as a critic or reward and the diffusion model as a policy or sampler, with equilibrium at $$\nabla_z E_\theta(z_t,c)\approx \epsilon_\theta(z_t,t,c)$$2 when both model classes contain the data distribution (Yoon et al., 2023). A closely related divergence view appears in DCD, where one optimizes

$$\nabla_z E_\theta(z_t,c)\approx \epsilon_\theta(z_t,t,c)$$3

so that diffusion transport itself defines the contrastive divergence used to train the EBM (Luo et al., 2023).

3. Composition, attention, and path-space energy

Unified energy-based diffusion is not limited to unconditional density modeling. In compositional human motion generation, EnergyMoGen introduces two distinct but coupled energy spectra: a latent-aware EBM, where classifier-free guidance becomes a linear energy-gradient mixture in latent space, and a semantic-aware EBM, where cross-attention is interpreted as an energy over text embeddings and motion latents. Its Synergistic Energy Fusion combines latent-aware score, semantic-aware score, and single-text multi-concept score,

$$\nabla_z E_\theta(z_t,c)\approx \epsilon_\theta(z_t,t,c)$$4

with $$\nabla_z E_\theta(z_t,c)\approx \epsilon_\theta(z_t,t,c)$$5, thereby making compositional logic a score-level energy fusion problem inside one diffusion backbone (Zhang et al., 2024).

A different generalization appears in energy-constrained diffusion for neural message passing. Here diffusion operators on observed or latent structures are placed in one-to-one correspondence with energy functions implicitly descended by the diffusion process, and finite-difference iterations of the diffusion PDE induce the layers of MLPs, GNNs, and Transformers. DIFFormer is derived from this correspondence: its global attention is not introduced heuristically, but as the diffusivity inferred by an energy-constrained diffusion system (Wu et al., 2024). This suggests that unified energy-based diffusion can describe representation propagation as well as sample generation.

In discrete masked diffusion, the same theme appears at the level of path functionals. Masked diffusion models have been interpreted as minimizing three equivalent energies—weighted kinetic, weighted conditional kinetic, and weighted geodesic energy—with the optimal schedule satisfying

$$\nabla_z E_\theta(z_t,c)\approx \epsilon_\theta(z_t,t,c)$$6

This result turns mask scheduling into an energy minimization problem in discrete optimal transport and makes post-training schedule selection a low-dimensional geometric design problem rather than an ad hoc heuristic (Chen et al., 17 Sep 2025).

4. Representative instantiations across domains

Human motion: In motion generation, the unified view is instantiated in latent space. EnergyMoGen treats latent diffusion as an implicit EBM, adds a semantic-aware energy model derived from cross-attention, and uses adaptive gradient descent on text embeddings to resolve multi-concept semantics. The framework covers text-to-motion, compositional motion generation, and multi-concept motion generation, and the paper states that it outperforms existing state-of-the-art models on these tasks while also supporting compositional data augmentation (Zhang et al., 2024).

Image generation and density estimation: Two image-centric lines exemplify different unification strategies. Diffusion-assisted EBMs learn a joint EBM over diffusion time and use persistent training with MGMS, claiming that, for the first time for image data, persistently trained EBMs can simultaneously achieve long-run stability, post-training image generation, and superior out-of-distribution detection (Zhang et al., 2023). By contrast, adversarial diffusion EBMs embed an EBM into every denoising step, replace MCMC with a learned conditional generator, and use symmetric Jeffrey divergence plus a variational posterior to stabilize the generator–energy game while retaining useful density estimates (Geng et al., 2024).

Geometric and ODE formulations: A more structural unification appears in continuous-time models. “A Flexible Diffusion Model” parameterizes the forward SDE through a Riemannian metric $$\nabla_z E_\theta(z_t,c)\approx \epsilon_\theta(z_t,t,c)$$7 and an anti-symmetric matrix $$\nabla_z E_\theta(z_t,c)\approx \epsilon_\theta(z_t,t,c)$$8, yielding a family of forward diffusions with Gaussian stationary distribution that subsumes VP and includes critically-damped Langevin-style structure (Du et al., 2022). The ODE-style force-field framework instead constructs a deterministic probability flow from transport and continuity equations, interpreting generative trajectories as motion along force fields and Green’s functions in extended spacetime (Jin et al., 2023).

Molecular systems: In coarse-grained molecular modeling, unification is operationalized as consistency between two uses of the same model: iid equilibrium sampling by reverse diffusion and Langevin simulation using score-derived forces. The key observation is that standard diffusion models may sample equilibrium conformations well yet yield inconsistent molecular dynamics at small diffusion times. The proposed remedy is an energy-based diffusion model with a Fokker–Planck-derived regularization term enforcing temporal consistency of the energy $$\nabla_z E_\theta(z_t,c)\approx \epsilon_\theta(z_t,t,c)$$9, so that the same model acts as score network, Boltzmann emulator, and force field (Plainer et al., 20 Jun 2025).

Language: Text brings discrete-state variants to the foreground. Glauber-dynamics text diffusion uses a pretrained LLM as an energy function,

$p_\theta(x,t)=\exp(-U_\theta(x,t))/Z(\theta)$0

so that the stationary distribution of the discrete diffusion process is the LLM itself; generation is then iterative noising–denoising over token sequences rather than a single autoregressive pass (Kathuria et al., 5 May 2026). A separate decoding-oriented line defines invariant energy, independent energy, and unified energy for diffusion LLMs, with the unified energy

$p_\theta(x,t)=\exp(-U_\theta(x,t))/Z(\theta)$1

designed to correct dependency and invariance errors during parallel denoising (Yan et al., 8 Jun 2026).

5. Recurring technical issues and common misconceptions

One recurring misconception is that a unified model must expose an explicit scalar energy in closed form. In fact, several formulations define energy only up to an additive constant through its gradient or score. This is explicit in the latent-diffusion-as-EBM interpretation, where $p_\theta(x,t)=\exp(-U_\theta(x,t))/Z(\theta)$2 is operationalized through $p_\theta(x,t)=\exp(-U_\theta(x,t))/Z(\theta)$3, and in the general score-based perspective, where learning the score of a noisy marginal is already equivalent to learning a time-dependent negative log-density gradient (Zhang et al., 2024, Luo, 2022).

A second misconception is that unification implies a single sampling backend. The literature shows the opposite. GCD replaces classical MCMC with a learned diffusion sampler and makes this replacement part of the objective itself (Yoon et al., 2023). Diffusion-assisted EBMs retain persistent MCMC in the enlarged space $p_\theta(x,t)=\exp(-U_\theta(x,t))/Z(\theta)$4, using MGMS as the sampling mechanism required by maximum-likelihood-style training (Zhang et al., 2023). Adversarial stepwise EBMs avoid MCMC by introducing a variational generator trained against conditional energies through Jeffrey divergence (Geng et al., 2024). “Unified” therefore names a coupling between energy and diffusion, not a mandatory choice of sampler.

A third technical point is that sample quality and dynamical consistency need not coincide. In molecular modeling, standard diffusion training can yield correct-looking equilibrium samples while failing to satisfy the Fokker–Planck equation near $p_\theta(x,t)=\exp(-U_\theta(x,t))/Z(\theta)$5, which then produces inconsistent force fields for Langevin simulation. The energy-based correction is not merely cosmetic; it changes the temporal evolution of the learned log density so that diffusion sampling and molecular dynamics agree more closely (Plainer et al., 20 Jun 2025).

A fourth issue concerns schedules and structural degrees of freedom. In masked diffusion, the training objective can be invariant to the interpolation schedule while sampling performance changes substantially, and the schedule can be characterized as the minimizer of an explicit path energy (Chen et al., 17 Sep 2025). In diffusion-inspired message passing, the choice of coupling matrix $p_\theta(x,t)=\exp(-U_\theta(x,t))/Z(\theta)$6 is likewise not a cosmetic implementation detail but the object that determines which energy is descended by the layer update (Wu et al., 2024).

A fifth issue is specific to language. The gap between diffusion decoding and autoregressive decoding has been analyzed into capacity, dependency, and invariance factors. This framing makes clear that scaling model size addresses only one part of the problem; the remaining gap is structural and must be handled by energy corrections or by redesigned decoding planners (Yan et al., 8 Jun 2026).

6. Open problems and future directions

No single canonical unified energy-based diffusion model has emerged. Current directions include richer energy parameterizations, alternative diffusion architectures, continuous-time or path-integral samplers, and objective functions in which energy acts simultaneously as density model, critic, or reward. GCD explicitly points to different energy classes, different diffusion architectures, and trajectory-dependent rewards as natural extensions, while EnergyMoGen argues that its latent-and-semantic construction is not motion-specific and can transfer to other latent diffusion modalities (Yoon et al., 2023, Zhang et al., 2024).

A second frontier is physical consistency at scale. The molecular-dynamics line identifies Fokker–Planck consistency near $p_\theta(x,t)=\exp(-U_\theta(x,t))/Z(\theta)$7 as a concrete bottleneck and proposes fine-tuning existing diffusion models with FP regularization, with larger biomolecular systems and more complex solvent environments left for future work (Plainer et al., 20 Jun 2025). In image modeling, the adversarial per-step EBM line points toward combining energy-based reverse conditionals with analytic diffusion solvers and richer symmetric divergences, while the force-field and flexible-SDE lines suggest that forward-process design remains under-explored relative to denoiser design (Geng et al., 2024, Du et al., 2022, Jin et al., 2023).

A third frontier concerns structure learning and decoding. Energy-constrained diffusion on graphs and latent manifolds suggests continuous-depth message passing and broader architecture families beyond current Transformer and GNN variants (Wu et al., 2024). In discrete diffusion, schedule optimization via low-dimensional Beta-CDF parameterizations shows that post-training energy-aware tuning can materially improve few-step sampling (Chen et al., 17 Sep 2025). In diffusion LLMs, model-agnostic energy correction via AR proxies indicates that dependency- and invariance-aware decoding is a promising route for narrowing the gap to autoregressive generation without sacrificing parallelism (Yan et al., 8 Jun 2026).

Across these directions, the durable idea is not a single formula but a modeling principle: diffusion is most “unified” when its forward process, reverse sampler, architectural couplings, and objective can all be interpreted as operations on an energy landscape, whether that landscape is defined over data, latent variables, trajectories, diffusion time, or decoding paths.