Forward-Only Diffusion (FoD)

• Forward-Only Diffusion is a generative modeling framework that uses a direct forward stochastic process, replacing reverse denoising steps to match the target distribution exactly.
• It employs principled SDE-based and one-step formulations that enhance efficiency and flexibility in diverse applications such as image restoration and robot control.
• Empirical evaluations show that FoD achieves competitive performance with standard diffusion models while reducing sampling complexity and discretization errors.

Forward-Only Diffusion (FoD) defines a class of generative modeling frameworks that eschew the conventional forward-backward (denoise-invert) structure in favor of a direct, analytically tractable, and often accelerated forward-only stochastic process whose terminal law matches the data distribution. Recent developments in FoD have introduced both principled SDE-based approaches for data generation, as well as efficient one-step formulations for domains such as robot control, all without resorting to time reversal or explicit score estimation. These frameworks yield performance competitive with, or exceeding, traditional diffusion models, while providing reduced sampling complexity and higher flexibility in the choice of diffusion dynamics (Luo et al., 22 May 2025, Peluchetti, 2023, Shi et al., 25 Mar 2026).

1. Mathematical Foundations of Forward-Only Diffusion

FoD models generatively transport a source distribution toward a target data distribution over continuous time, typically via a state-dependent stochastic differential equation (SDE) with mean-reverting dynamics. The canonical instance is a linear, mean-reverting SDE:

$$d\mathbf{x}_t = \theta(t) (\mu - \mathbf{x}_t) dt + \sigma(t) (\mathbf{x}_t - \mu) d\mathbf{w}_t,$$

where

• $\mathbf{x}_t \in \mathbb{R}^d$ denotes the process state,
• $\mu \sim p_{\mathrm{data}}$ is the target sample from the empirical distribution,
• $\theta(t) > 0$ is the mean-reversion rate,
• $\sigma(t) \geq 0$ is the diffusion schedule,
• $\mathbf{w}_t$ is standard $d$-dimensional Brownian motion (Luo et al., 22 May 2025).

Both drift and diffusion drive $\mathbf{x}_t$ toward $\mu$; as $\mathbf{x}_t$ approaches $\mathbf{x}_t \in \mathbb{R}^d$0, stochasticity is suppressed and the process converges deterministically. Analytical solutions are available: the process admits a log-normal noise structure,

$\mathbf{x}_t \in \mathbb{R}^d$1

with convergence to $\mathbf{x}_t \in \mathbb{R}^d$2 in mean-square as $\mathbf{x}_t \in \mathbb{R}^d$3 (Luo et al., 22 May 2025).

In alternative approaches, FoD is formulated using mixtures of diffusion bridges. Given any joint coupling $\mathbf{x}_t \in \mathbb{R}^d$4 of the data distribution $\mathbf{x}_t \in \mathbb{R}^d$5 with itself, the FoD process comprises a mixture of bridge SDEs, ensuring the terminal law exactly matches $\mathbf{x}_t \in \mathbb{R}^d$6. The local drift becomes a weighted mixture over the bridge adjustments: $\mathbf{x}_t \in \mathbb{R}^d$7 with

$\mathbf{x}_t \in \mathbb{R}^d$8

where $\mathbf{x}_t \in \mathbb{R}^d$9 is a data-dependent expectation over bridge scores (Peluchetti, 2023).

2. Training Objectives and Learning Algorithms

FoD models obviate backward SDE simulation and instead fit forward transition kernels or conditional expectations via quadratic losses. In the mean-reverting SDE framework, the Stochastic Flow Matching (SFM) objective minimizes

$\mu \sim p_{\mathrm{data}}$0

where $\mu \sim p_{\mathrm{data}}$1 predicts the vector flow from $\mu \sim p_{\mathrm{data}}$2 to $\mu \sim p_{\mathrm{data}}$3 (Luo et al., 22 May 2025).

The bridge-mixing (DBMT) FoD form supports several objectives:

• Score Matching (Fisher divergence):

$\mu \sim p_{\mathrm{data}}$4

• Conditional Expectation (Direct Denoising):

$\mu \sim p_{\mathrm{data}}$5

The latter directly regresses the conditional expectation $\mu \sim p_{\mathrm{data}}$6, training the model to denoise any intermediate state to the terminal data sample (Peluchetti, 2023).

In single-step generative robot control via trajectory parameterization, as in FODMP, multi-step score-based denoising is distilled into a consistency objective for parameter regression: $\mu \sim p_{\mathrm{data}}$7 training the student network $\mu \sim p_{\mathrm{data}}$8 to produce data-consistent primitives from single noisy inputs (Shi et al., 25 Mar 2026).

3. Sampling Procedures: Markovian and Non-Markovian

FoD SDEs admit closed-form kernels for arbitrary intervals, enabling sampling at arbitrary time increments. Two variants are prevalent:

• Markov chain: Each update depends only on the immediate past state,

$\mu \sim p_{\mathrm{data}}$9

• Non-Markov chain: Each update depends explicitly on the original starting state $\theta(t) > 0$0,

$\theta(t) > 0$1

Non-Markovian sampling, by repeatedly resetting to the clean path from $\theta(t) > 0$2, suppresses compounding discretization error and often yields higher sample fidelity in few-step regimes (Luo et al., 22 May 2025).

Bridge-mixing FoD also supports direct sampling via the closed-form structure of the bridge law, facilitating efficient simulation (Peluchetti, 2023).

4. Empirical Results and Applications

Empirical evaluation demonstrates that FoD approaches attain or surpass the performance of standard (reverse-solved) diffusion models, particularly in regimes emphasizing efficiency or flexibility.

Image Restoration: On Rain100H, RESIDE, LOL, and CelebA-HQ, FoD with 10-step fast sampling (non-Markov) achieves PSNR ranging from 23.05–33.63, SSIM from 0.855–0.941, and competitive FID/LPIPS, outperforming both forward-backward SDEs and flow-matching baselines (Luo et al., 22 May 2025).

Unconditional Generation: On CIFAR-10, FoD-SDE attains FID 7.89, with the ODE variant yielding FID 5.01, outperforming other forward-only/flow-matching methods, but still behind state-of-the-art DDPM and Score SDE (Luo et al., 22 May 2025, Peluchetti, 2023).

Generative Robot Motion: FODMP distills multi-step diffusions over ProDMP trajectory parameters into a single-step mapping, resulting in inference speeds of ≈17 ms and success rates up to 86.3% on “Medium” difficulty MetaWorld/ManiSkill, outperforming both action chunking and multi-step approaches (Shi et al., 25 Mar 2026).

Application Domain	Task	FoD Variant	Main Metric(s)	Performance
Image Restoration	Rain100H/RESIDE/LOL/etc.	Mean-reverting SDE (SFM)	PSNR, SSIM, FID	SOTA or superior to IR-SDE/ReFlow/PMRF
Uncond. Generation	CIFAR-10	FoD SDE/ODE, DBMT	FID	FID 5.01–7.89 (ODE best FoD variant)
Robot Control	MetaWorld/ManiSkill	FODMP (one-step)	Success %, time (ms)	78.2% Avg., 17 ms inference

5. Theoretical Guarantees and Structural Properties

Mean-reverting FoD SDEs feature log-normal noise accumulation and exponential error contraction. Explicit limiting behavior as $\theta(t) > 0$3 recovers deterministic flow-matching/Rectified Flow ODEs:

$\theta(t) > 0$4

with explicit linear interpolation between source and target (Luo et al., 22 May 2025).

Mixture-of-bridge (DBMT) FoD possesses an inherent exactness property: the terminal law is exactly the empirical data distribution for any finite sample size (up to discretization error), circumventing the need for an approximate time-reversal SDE and simplifying matching the data law (Peluchetti, 2023). The Diffusion Mixture Representation theorem ensures that an arbitrary mixture of Itô processes yields another Itô process with the desired marginals, supporting theoretical soundness and architectural flexibility.

6. Implementation Considerations

FoD models require only standard architectures, e.g., convolutional U-Nets (without attention for efficiency) for image domains, and simple MLP-based regressors for trajectory parameterization in control. Schedules for $\theta(t) > 0$5 and $\theta(t) > 0$6 can be chosen for integrability and normalized energy; typical values employ cosine for drift and linear for diffusion variance.

Sampling is feasible with very few steps (5–20), and non-Markovian step policies further mitigate discretization artifacts. Optimizers are standard (e.g., AdamW), with training scalable to large datasets/GPU clusters. In DBMT, high-dimensional covariance structures ($\theta(t) > 0$7) necessary for image domains are managed efficiently via circulant embedding and FFT (Peluchetti, 2023). In FODMP, closed-form ODE-based decoding preserves full temporal behavior for robot primitives (Shi et al., 25 Mar 2026).

Terminal conditions and step discretization are selected such that convergence proceeds to within a prescribed tolerance (e.g., $\theta(t) > 0$8), yielding near-deterministic recovery of target data (Luo et al., 22 May 2025).

7. Extensions and Related Developments

FoD’s framework opens the way to exact, efficient, and non-denoising generative modeling for a broad class of modalities. Key avenues include:

• Extending mixture-of-bridges DBMT to arbitrary covariance structures and spatio-temporal couplings (Peluchetti, 2023).
• Integrating consistency distillation and explicit cost objectives in single-step FoD, especially in time-dependent control applications (Shi et al., 25 Mar 2026).
• Exploring new trajectory representations (e.g., B-splines, Fourier features) and adapting closed-form FoD policies for large-scale, real-world domains.
• Analytical tractability in high-dimensional domains via structured Gaussian processes and scalable sampling.

Across diverse application regimes, FoD offers a competitive and theoretically robust alternative to classical diffusion modeling, characterized by direct forward mapping, reduced complexity, and exact or near-exact convergence to target distributions.

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Key References:

• "Forward-only Diffusion Probabilistic Models" (Luo et al., 22 May 2025)
• "Non-Denoising Forward-Time Diffusions" (Peluchetti, 2023)
• "FODMP: Fast One-Step Diffusion of Movement Primitives Generation for Time-Dependent Robot Actions" (Shi et al., 25 Mar 2026)