Forward-Only Diffusion Generative Model

• Forward-only diffusion generative models are defined by a single forward transport mechanism that directly maps a source distribution to the target data, eliminating the standard noise reversal process.
• They employ methods such as diffusion bridge mixtures and mean-reverting SDEs to construct sample trajectories that align with the data distribution, showing practical benefits in tasks like image restoration.
• The models illustrate a taxonomy of generative approaches, ranging from strictly forward-time generators to forward-centric and hybrid reverse-dependent constructions, each with unique theoretical and empirical implications.

A forward-only diffusion generative model is a diffusion-inspired generative construction in which sampling is performed by a single forward-time transport from a source distribution to a target data distribution, rather than by the standard two-part recipe of forward noising and reverse denoising. In contemporary literature, however, the phrase sits inside a broader and often ambiguous family of “forward-centric” ideas. Standard diffusion tutorials and PDE analyses define diffusion generation as forward corruption toward a simple law, typically Gaussian, followed by a learned reverse-time process that reconstructs data; under that formulation, generation is not forward-only (Le, 2024, Cao et al., 28 Jan 2025). By contrast, a smaller set of works constructs genuinely forward-time generators, either by mixing diffusion bridges into a forward stochastic transport or by designing a mean-reverting SDE whose drift and diffusion both collapse at the clean target (Peluchetti, 2023, Luo et al., 22 May 2025). Around these strict constructions lies a larger literature on learnable forward processes, structured noising, forward-model supervision, and forward-free refinement, all of which modify the role of the forward process without necessarily removing reverse-time inference.

1. Standard diffusion and the obstacle to forward-only generation

The classical obstacle is explicit in continuous-time Ornstein–Uhlenbeck diffusion. In the tutorial formulation, the forward process is

$$\mathrm{d}X_t = -X_t\,\mathrm{d}t + \sqrt{2}\,\mathrm{d}B_t,\qquad X(0)=X_0,$$

with closed form

$$X_t = \gamma_t X_0 + \beta_t Z,\qquad Z\sim\mathcal N(0,I),$$

so that

$$q(x_t\mid x_0)=\mathcal N(\gamma_t x_0,\beta_t^2 I).$$

Because $\gamma_t\to 0$ and $\beta_t\to 1$, the forward process destroys information about $X_0$ and converges to a standard normal; starting from arbitrary noise and applying the same forward dynamics only yields more noise-like states, not samples from the data distribution (Le, 2024).

The PDE view sharpens why the missing piece cannot be supplied by naive time inversion. For the OU forward law $\rho(x,t)$,

$$\partial_t \rho = \nabla\cdot(x\rho) + \Delta \rho,$$

the formally reversed density path satisfies an anti-diffusive, ill-posed equation. Stabilization introduces a score-dependent drift, yielding the reverse SDE

$$d{X}_s = {X}_s\,d s + 2\,\nabla \ln \rho\left ({X}_s, T_*-s\right)\,d s + \sqrt{2}\,d B_s.$$

Generation therefore depends on $\nabla \log \rho(x,t)$, or on a learned surrogate for it, and not on the forward semigroup alone (Cao et al., 28 Jan 2025).

In discrete formulations the same dependence appears through the reverse conditional mean. The reverse Gaussian kernel depends on the unknown clean datum $$X_t = \gamma_t X_0 + \beta_t Z,\qquad Z\sim\mathcal N(0,I),$$0, so practical generation introduces a neural network that predicts $$X_t = \gamma_t X_0 + \beta_t Z,\qquad Z\sim\mathcal N(0,I),$$1, $$X_t = \gamma_t X_0 + \beta_t Z,\qquad Z\sim\mathcal N(0,I),$$2, or $$X_t = \gamma_t X_0 + \beta_t Z,\qquad Z\sim\mathcal N(0,I),$$3. The standard denoising objective is then

$$X_t = \gamma_t X_0 + \beta_t Z,\qquad Z\sim\mathcal N(0,I),$$4

and sampling starts from $$X_t = \gamma_t X_0 + \beta_t Z,\qquad Z\sim\mathcal N(0,I),$$5 and iterates learned reverse transitions until $$X_t = \gamma_t X_0 + \beta_t Z,\qquad Z\sim\mathcal N(0,I),$$6 is obtained (Le, 2024). In this canonical sense, diffusion generation is inherently reverse-time and iterative.

2. Genuine forward-time constructions

Two recent lines satisfy the strict criterion of forward-only sampling in markedly different ways.

The first is the diffusion bridge mixture transport of “Non-Denoising Forward-Time Diffusions” (Peluchetti, 2023). Instead of pushing data to noise and reversing the path, it starts from a forward SDE

$$X_t = \gamma_t X_0 + \beta_t Z,\qquad Z\sim\mathcal N(0,I),$$7

and conditions it on terminal endpoints through diffusion bridges. Mixtures of such bridges are then matched by another forward diffusion whose drift is

$$X_t = \gamma_t X_0 + \beta_t Z,\qquad Z\sim\mathcal N(0,I),$$8

with $$X_t = \gamma_t X_0 + \beta_t Z,\qquad Z\sim\mathcal N(0,I),$$9 given by a bridge-mixture conditional expectation involving $$q(x_t\mid x_0)=\mathcal N(\gamma_t x_0,\beta_t^2 I).$$0. If $$q(x_t\mid x_0)=\mathcal N(\gamma_t x_0,\beta_t^2 I).$$1, the terminal law is exactly the data distribution. The paper further proposes conditional-expectation training,

$$q(x_t\mid x_0)=\mathcal N(\gamma_t x_0,\beta_t^2 I).$$2

with loss

$$q(x_t\mid x_0)=\mathcal N(\gamma_t x_0,\beta_t^2 I).$$3

Sampling is then entirely forward in time: draw $$q(x_t\mid x_0)=\mathcal N(\gamma_t x_0,\beta_t^2 I).$$4 and integrate the learned SDE to $$q(x_t\mid x_0)=\mathcal N(\gamma_t x_0,\beta_t^2 I).$$5. The construction is exact in the ideal drift limit, but the paper also notes that large-scale empirical benchmarking against mature reverse-time diffusion models remains open (Peluchetti, 2023).

The second is “Forward-only Diffusion Probabilistic Models” (Luo et al., 22 May 2025), which replaces reverse denoising by a state-dependent mean-reverting SDE,

$$q(x_t\mid x_0)=\mathcal N(\gamma_t x_0,\beta_t^2 I).$$6

With $$q(x_t\mid x_0)=\mathcal N(\gamma_t x_0,\beta_t^2 I).$$7, the residual follows

$$q(x_t\mid x_0)=\mathcal N(\gamma_t x_0,\beta_t^2 I).$$8

so both the deterministic contraction and the stochastic term vanish at the clean target $$q(x_t\mid x_0)=\mathcal N(\gamma_t x_0,\beta_t^2 I).$$9. The process has closed form

$\gamma_t\to 0$0

and the residual flow is log-normal: $\gamma_t\to 0$1 Training uses stochastic flow matching,

$\gamma_t\to 0$2

and inference uses only forward transitions, either by Euler–Maruyama or by few-step Markov and non-Markov closed-form jumps. Empirically, FoD is strongest on image-conditioned restoration: on Rain100H, FoD reports PSNR $\gamma_t\to 0$3 and FID $\gamma_t\to 0$4 versus IR-SDE’s PSNR $\gamma_t\to 0$5 and FID $\gamma_t\to 0$6; on CelebA-HQ inpainting, FoD reports LPIPS $\gamma_t\to 0$7 and FID $\gamma_t\to 0$8 versus IR-SDE’s LPIPS $\gamma_t\to 0$9 and FID $\beta_t\to 1$0. On unconditional CIFAR-10, however, FoD-SDE reports FID $\beta_t\to 1$1 and FoD-ODE FID $\beta_t\to 1$2, trailing DDPM at $\beta_t\to 1$3 and Score SDE at $\beta_t\to 1$4 (Luo et al., 22 May 2025).

These two constructions establish that strict forward-only diffusion is possible, but they do so by abandoning the standard “noise then reverse” logic rather than by refining it.

3. Forward-centric but reverse-dependent model families

Most of the literature associated with the phrase does not eliminate reverse-time generation. Instead, it redesigns the forward process, the latent geometry, or the conditioning structure while preserving reverse denoising as the actual sampler.

Model	Forward-side novelty	Why it is not forward-only
Neural Flow Diffusion Models	Learnable forward marginals $\beta_t\to 1$5 and corresponding forward ODE/SDE (Bartosh et al., 2024)	Generation still starts from $\beta_t\to 1$6 and integrates a learned reverse SDE/ODE
A Flexible Diffusion Model	Learnable forward SDE via $\beta_t\to 1$7, $\beta_t\to 1$8, and $\beta_t\to 1$9, with Gaussian stationary law (Du et al., 2022)	Reverse-time score-based SDE and Euler–Maruyama sampling remain unchanged in principle
Frequency Domain Diffusion Model	RG/optimal-transport interpretation; Fourier-space UV-to-IR corruption with scale-dependent $X_0$0 (Sheshmani et al., 2024)	Practical image generation still uses DDPM/DDIM-style reverse denoising over many steps
Accelerated Image-Aware Generative Diffusion	Pixel-wise data-dependent schedule $X_0$1 and one network run predicting all reverse noises (Asthana et al., 2024)	Samples are still produced by reverse DDPM-style recursion
Groupwise Diffusion Model	Diagonal schedule $X_0$2 that diffuses one group per interval (Lee et al., 2023)	Generation is a reverse ODE solved iteratively, one group at a time
Dynamical Diffusion	Temporally aware forward latents $X_0$3 depend on $X_0$4 through $X_0$5 (Guo et al., 2 Mar 2025)	Reverse DDIM/DDPM-like denoising remains essential
GBM-based financial diffusion	GBM-inspired forward process in price space, reducing to VE diffusion in log-price space (Kim et al., 25 Jul 2025)	Reverse-time score SDE is trained “identical to the VE diffusion model”
Truncated Diffusion Probabilistic Models	Stop noising at $X_0$6 and learn an implicit prior over the hidden noisy distribution (Zheng et al., 2022)	Only the early reverse segment is replaced; a shortened reverse chain is still used

The common pattern is structural enrichment of the forward law. In NFDM, forward marginals are made learnable so that the reverse model sees easier trajectories, but unconditional generation still requires reverse-time simulation from $X_0$7 to $X_0$8 (Bartosh et al., 2024). In the flexible SDE framework, forward diffusion becomes anisotropic and partially Hamiltonian through $X_0$9 and $\rho(x,t)$0, yet the reverse generator remains the standard score-corrected reverse SDE (Du et al., 2022). In FDDM, the forward process becomes a frequency-selective transport from ultraviolet to infrared modes, motivated by RG and optimal transport, but the implemented sampler is still iterative reverse denoising in Fourier space (Sheshmani et al., 2024).

A related efficiency-oriented cluster modifies the forward schedule to reduce reverse cost without removing it. The image-aware accelerated model uses a pixel-wise schedule with exponential SNR decay and predicts the whole reverse-time noise trajectory in one network call, but still iterates a DDPM posterior update (Asthana et al., 2024). TDPM replaces the standard Gaussian start of the reverse chain by a learned hidden noisy distribution at truncation time $\rho(x,t)$1, which can reduce reverse denoising to $\rho(x,t)$2, $\rho(x,t)$3, $\rho(x,t)$4, or even $\rho(x,t)$5 steps, but the model remains a hybrid of implicit prior sampling and reverse diffusion (Zheng et al., 2022).

Finally, several models encode additional structure into the forward path. GDM makes the noising order itself sequential over groups or frequency bands, placing standard diffusion, autoregressive models, and cascaded diffusion inside one diagonal-schedule family, but sampling still solves a reverse ODE (Lee et al., 2023). DyDiff pushes temporal dependence into the forward latent recursion and correlated noise, yet the reverse sampler remains DDIM/DDPM-like (Guo et al., 2 Mar 2025). The GBM financial model uses multiplicative price-space corruption and log-space VE diffusion to respect heteroskedasticity, but it too relies on reverse-time score matching and reverse SDE sampling (Kim et al., 25 Jul 2025).

4. Forward-free, feed-forward, and forward-model-supervised adjacent notions

A separate but closely related development argues that the hand-designed forward process is not merely unnecessary but conceptually misaligned with inference. “Forward-Free Diffusion LLMs” eliminates the prescribed corruption path altogether and reformulates diffusion language modeling as recursive distribution refinement over model-generated drafts (Sun et al., 6 Jun 2026). The generative chain is

$\rho(x,t)$6

but the intermediate states are not sampled from a handcrafted $\rho(x,t)$7. Instead, FReDA learns refinement kernels

$\rho(x,t)$8

that map current draft distributions toward the target. Training uses model-generated drafts as implicit intermediate states, and inference proceeds by self-refinement or by best-of-$\rho(x,t)$9 refinement with a learned scorer. This is not forward-only diffusion; it is explicitly forward-free. Empirically, FReDA-4B reports absolute gains of up to $$\partial_t \rho = \nabla\cdot(x\rho) + \Delta \rho,$$0 over larger diffusion baselines on reasoning and coding tasks and an average $$\partial_t \rho = \nabla\cdot(x\rho) + \Delta \rho,$$1-$$\partial_t \rho = \nabla\cdot(x\rho) + \Delta \rho,$$2 speedup at matched quality (Sun et al., 6 Jun 2026).

Another adjacent use of “forward” refers not to sampling time but to the observation operator in inverse problems. “Diffusion with Forward Models” trains a conditional diffusion generator for latent signals $$\partial_t \rho = \nabla\cdot(x\rho) + \Delta \rho,$$3 that are never observed directly, using only observations $$\partial_t \rho = \nabla\cdot(x\rho) + \Delta \rho,$$4 and a known differentiable forward model (Tewari et al., 2023). Diffusion is run in observation space, while the denoiser predicts a latent signal estimate

$$\partial_t \rho = \nabla\cdot(x\rho) + \Delta \rho,$$5

which is re-rendered through the forward operator to obtain the denoised observation estimate. This is forward-model-supervised rather than forward-time generative diffusion: the supervision comes exclusively through the known measurement process, but inference still uses an iterative reverse observation-space chain (Tewari et al., 2023).

A further source of terminological confusion is “feed-forward” generation. In GANFusion, text-to-3D inference is feed-forward only in contrast to per-prompt optimization methods such as DreamFusion-style SDS; the core text-to-triplane generator still uses iterative DDIM denoising in GAN space, followed by a feed-forward renderer and upsampler (Attaiki et al., 2024). Thus “feed-forward,” “forward-model-supervised,” and “forward-free” designate adjacent but distinct departures from standard diffusion and should not be conflated with strict forward-only sampling.

5. Theoretical limitations, misconceptions, and scope of exactness

A recurring misconception is that any enriched or learnable forward process constitutes forward-only generation. The analytic literature argues the opposite for standard diffusion. In the PDE treatment, the reverse drift depends explicitly on $$\partial_t \rho = \nabla\cdot(x\rho) + \Delta \rho,$$6 through the score $$\partial_t \rho = \nabla\cdot(x\rho) + \Delta \rho,$$7, and the exact reverse flow obeys a support theorem,

$$\partial_t \rho = \nabla\cdot(x\rho) + \Delta \rho,$$8

For empirical measures, the paper derives explicit conditional expectations over training samples and shows that exact reverse dynamics converge back to the original samples; in its formulation, solving the reverse problem exactly is an overfitting regime rather than a source of novel support (Cao et al., 28 Jan 2025).

This result is often read as a general objection to diffusion-based generation, but its scope is narrower. It applies to exact reverse-time reconstruction of the standard forward diffusion. The tutorial treatment makes clear why this reverse dependence appears: the forward process

$$\partial_t \rho = \nabla\cdot(x\rho) + \Delta \rho,$$9

progressively removes information about $$d{X}_s = {X}_s\,d s + 2\,\nabla \ln \rho\left ({X}_s, T_*-s\right)\,d s + \sqrt{2}\,d B_s.$$0, so a generator cannot be obtained by simply “running the forward diffusion from noise” (Le, 2024). The impossibility is therefore tied to the information-destroying character of that forward process.

Forward-only constructions bypass this obstacle by changing the mechanism, not by discovering a hidden direct use of the standard forward semigroup. DBMT never defines generation as time reversal; it constructs a forward stochastic transport whose terminal law is exact by bridge-mixture design (Peluchetti, 2023). FoD similarly replaces noising-plus-reversal with target-attracted stochastic interpolation (Luo et al., 22 May 2025). These models do not refute the reverse-time analysis; they operate outside its assumptions.

Another misconception is that eliminating reverse-time simulation automatically improves unconditional sample quality. The available evidence does not support such a general claim. FoD is strongest on image-conditioned restoration and weaker on unconditional CIFAR-10 than leading reverse-time diffusion and rectified-flow baselines (Luo et al., 22 May 2025). DBMT offers a mathematically exact forward-time construction, but its empirical evaluation is preliminary (Peluchetti, 2023). Theoretical exactness and practical competitiveness remain separate questions.

6. Empirical landscape and research directions

The empirical record suggests that forward-only or forward-centric methods are most compelling when the source distribution already carries task structure. FoD starts from degraded images for restoration and reports consistent gains over IR-SDE across deraining, dehazing, low-light enhancement, and inpainting, indicating that target-attracted forward transport is particularly natural when the initial state is informative (Luo et al., 22 May 2025). Forward-model-supervised diffusion likewise succeeds in inverse graphics, motion prediction, and GAN inversion because the known forward operator constrains the latent space even without direct supervision on the target variable (Tewari et al., 2023). FReDA benefits from a similar alignment principle in language: drafts used in training are the same kinds of drafts seen at inference, rather than artificial corruption states (Sun et al., 6 Jun 2026).

By contrast, much of the performance gain in reverse-dependent forward-centric models comes from efficiency engineering rather than from fully new generative mechanics. FDDM reports $$d{X}_s = {X}_s\,d s + 2\,\nabla \ln \rho\left ({X}_s, T_*-s\right)\,d s + \sqrt{2}\,d B_s.$$1 to $$d{X}_s = {X}_s\,d s + 2\,\nabla \ln \rho\left ({X}_s, T_*-s\right)\,d s + \sqrt{2}\,d B_s.$$2 speedup on CelebA-64 through DCT patching and grouped frequency channels, but keeps $$d{X}_s = {X}_s\,d s + 2\,\nabla \ln \rho\left ({X}_s, T_*-s\right)\,d s + \sqrt{2}\,d B_s.$$3 training and inference steps and reports worse FID than DDPM (Sheshmani et al., 2024). The image-aware accelerated model reduces conventional step counts from around $$d{X}_s = {X}_s\,d s + 2\,\nabla \ln \rho\left ({X}_s, T_*-s\right)\,d s + \sqrt{2}\,d B_s.$$4 to $$d{X}_s = {X}_s\,d s + 2\,\nabla \ln \rho\left ({X}_s, T_*-s\right)\,d s + \sqrt{2}\,d B_s.$$5 on CIFAR10 and $$d{X}_s = {X}_s\,d s + 2\,\nabla \ln \rho\left ({X}_s, T_*-s\right)\,d s + \sqrt{2}\,d B_s.$$6 on CelebA, predicts all reverse-step noises in one network execution, and reports runtimes of $$d{X}_s = {X}_s\,d s + 2\,\nabla \ln \rho\left ({X}_s, T_*-s\right)\,d s + \sqrt{2}\,d B_s.$$7 s on CIFAR10 and $$d{X}_s = {X}_s\,d s + 2\,\nabla \ln \rho\left ({X}_s, T_*-s\right)\,d s + \sqrt{2}\,d B_s.$$8 s on CelebA, but still uses reverse DDPM-style updates (Asthana et al., 2024). TDPM reports dramatic sampling-time reductions—from $$d{X}_s = {X}_s\,d s + 2\,\nabla \ln \rho\left ({X}_s, T_*-s\right)\,d s + \sqrt{2}\,d B_s.$$9 s/image for DDPM on $\nabla \log \rho(x,t)$0 data to $\nabla \log \rho(x,t)$1 s/image at $\nabla \log \rho(x,t)$2 and $\nabla \log \rho(x,t)$3 s/image at $\nabla \log \rho(x,t)$4—by replacing early reverse steps with sampling from a learned hidden noisy prior, but this remains a hybrid between diffusion and adversarial auto-encoding (Zheng et al., 2022).

Structured-forward models also show that the forward process can carry semantics beyond noise magnitude. GDM’s groupwise schedules produce interpretable latent groups and frequency hierarchies, with CIFAR-10 FID-10K varying substantially by grouping order, such as $\nabla \log \rho(x,t)$5 for Bottom$\nabla \log \rho(x,t)$6Top versus $\nabla \log \rho(x,t)$7 for CDM at $\nabla \log \rho(x,t)$8 and 128 Euler steps (Lee et al., 2023). DyDiff reports improvements in spatiotemporal forecasting, video prediction, and time-series forecasting, including over $\nabla \log \rho(x,t)$9 CRPS reduction on Turbulence, by making each diffusion step depend on temporally previous latents (Guo et al., 2 Mar 2025). These results show that forward-process design is a meaningful inductive-bias lever even when generation remains reverse-time.

The present literature therefore supports a three-part taxonomy. First are strict forward-time generators, exemplified by bridge-mixture transport and FoD, where sampling itself is forward (Peluchetti, 2023, Luo et al., 22 May 2025). Second are forward-centric reverse-dependent models, where the forward law is learned, structured, truncated, or physically grounded, but reverse denoising remains the sampler (Bartosh et al., 2024, Sheshmani et al., 2024, Asthana et al., 2024, Lee et al., 2023, Guo et al., 2 Mar 2025, Kim et al., 25 Jul 2025, Zheng et al., 2022, Du et al., 2022). Third are adjacent reformulations that remove or repurpose the forward process entirely, such as forward-free recursive refinement in language, forward-model supervision in inverse problems, and feed-forward diffusion pipelines defined relative to optimization rather than to sampling direction (Sun et al., 6 Jun 2026, Tewari et al., 2023, Attaiki et al., 2024).

Used strictly, the term “forward-only diffusion generative model” should therefore be reserved for models in which the generative path itself is a forward transport and reverse-time denoising is absent. Used loosely, it often denotes a broader family of attempts to redesign the forward process, to make intermediate states more structured, or to weaken the dependence on explicit reversal. The distinction is not terminological only; it marks a real difference in mechanism, theory, and empirical behavior.