Diffusion-Based Dynamics Models

Updated 11 November 2025

Diffusion-based dynamics models are generative frameworks that encode temporal and physical evolution using stochastic noising and denoising processes.
They employ forward–reverse chains with specialized noise scheduling and cross-attention mechanisms to maintain temporal coherence and realistic state forecasting.
Empirical evidence shows these models yield improved CRPS and perceptual metrics across domains such as physics, video forecasting, and time series analysis.

A diffusion-based dynamics model refers broadly to any class of generative or predictive models in which temporal or physical dynamics are encoded via a (stochastic) diffusion process, potentially accompanied by learned or hand-crafted noise schedules and denoising mappings. Most contemporary advances focus on leveraging forward–reverse chains (either in discrete or continuous time) to encode system evolution, forecast future states, or generate realistic dynamic trajectories across spatiotemporal, physical, and control domains. This article systematically presents the theoretical constructs, principal architectures, domain-specific adaptations, and empirical impacts of diffusion-based dynamics models as found in the current literature.

1. Mathematical Foundations of Diffusion-Based Dynamics

Diffusion-based dynamics models, in their most general form, instantiate a stochastic process defined by a forward "noising" operator and a reverse-time "denoising" process. Let $x_0$ denote the clean data—this could be a spatiotemporal block (e.g. a movie or physical field snapshot), a trajectory, or a molecular conformation. The forward process incrementally corrupts $x_0$ via (e.g.) Gaussian noise, generating a latent $x_t$ after $t$ steps:

$x_t = \sqrt{\bar\alpha_t} x_0 + \sqrt{1-\bar\alpha_t} \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, I)$

where the schedule $\{\bar\alpha_t\}$ tunes the noising intensity. The reverse process then seeks to recover $x_0$ (or at least its tractable statistics) from $x_t$ —traditionally by learning a family of neural denoisers $f_\theta$ that minimize a denoising score-matching objective.

Recent advances have introduced domain-conditioned variants: conditioning on historical data for sequence forecasting, constraining reverse dynamics to physically-admissible manifolds in control, or learning the transition densities directly for molecular dynamics and population genetics.

2. Temporal and Sequential Architectures

Classical diffusion models treat each generated sample as independent. However, temporally-structured settings—forecasting, video, scientific dynamics—demand explicit modeling of temporal dependencies.

The Dynamical Diffusion (DyDiff) framework (Guo et al., 2 Mar 2025) introduces a generalized forward diffusion with explicit temporal recursion:

$x_t^s = \sqrt{\bar\gamma_t} (\sqrt{\bar\alpha_t} x_0^s + \sqrt{1-\bar\alpha_t} \epsilon_t^s ) + \sqrt{1-\bar\gamma_t} x_t^{s-1}$

for $s$ indexing prediction steps. The resultant forward kernel is a temporally-mixed Gaussian, inducing nontrivial correlations in the $s$ -dimension—i.e., future predictions (frames or steps) at each diffusion stage inherit structure from their predecessors.

The reverse denoising process is architected as a cross-attention-equipped 3D U-Net, where intermediate outputs are conditioned on both the noisy prediction block and the known input history via cross-attention. The loss retains the standard score-matching form but with noise correlations reflecting the induced temporal dependencies.

This temporal mixing mechanism is empirically essential: ablation experiments demonstrate that removal of multi-step latent correlations induces a collapse of temporal coherence, and that performance is robust to parameterization of the mixing schedule $\bar\gamma_t$ as long as it's strictly between the extremes of complete independence and full history discarding.

3. Conditional Kernels and Theoretical Guarantees

Diffusion-based dynamics models define a joint Markov process over $x_t^{1:S}$ :

$q(x_t^s | x_0^{-P:s}) = \mathcal N( \sqrt{\bar\alpha_t} \mathrm{Dynamics}(x_0^{-P:s}; \bar\gamma_t), (1-\bar\alpha_t)I )$

with covariance matrices $J_t$ encoding inter-step noise correlations via powers of $\sqrt{\bar\gamma_t}$ . Sampling and training then reduce to tractable operations, allowing the variational objective to be reduced to a form compatible with standard MSE-based score matching.

If desired, deterministic sampling can be used (DDIM-style), or full probabilistic sampling as in the original DDPM.

4. Implementation and Algorithmic Structure

Most modern diffusion-based dynamics models employ variants of the U-Net or Transformer backbone due to their strong affinity for high-dimensional spatial-temporal data. In DyDiff, the denoiser is a latent-space 3D-UNet following the architectural guidelines of Stable Video Diffusion, with group normalization, SiLU activations, and self/cross-attention over the spatio-temporal block.

A high-level training pseudocode is as follows:

for iteration in range(num_steps):
    x0_hist_future = sample_from_dataset()
    t = random_uniform(1, T)
    eps = normal_random_sample(shape=[S])
    x_dyn = Dynamics(x0_hist_future, gamma_bar_t)
    eps_dyn = Dynamics(eps, gamma_bar_t)
    x_t = sqrt(alpha_bar_t) * x_dyn + sqrt(1 - alpha_bar_t) * eps_dyn
    loss = ||eps_dyn - epsilon_theta(x_t, x0_hist, t)||^2
    update_theta(loss)

Inference proceeds by initializing $x_T^{1:S} \sim \mathcal N(0, I)$ , and applying the learned denoiser $\epsilon_\theta$ and the inversion of the dynamics-mixing transformation step-wise from $t = T$ to $1$.

5. Empirical Performance across Domains

Table: DyDiff Benchmarks and Metrics

Domain	Metric	Baseline	DyDiff	Relative Change
Turbulence (Physics)	CRPS (↓)	0.0313	0.0275	−12%
	CSI (↑)	0.896	0.900	+0.4%
SEVIR (Weather)	CRPS (↓)	8.67, 15.41	7.62, 13.56	–12%, –12%
	CSI (↑)	0.285	0.319	+12%
BAIR (Video)	FVD (↓)	48.5	45.0	–7%
RoboNet (Video)	FVD (↓)	77.0	67.7	–12%
Solar (Time Series)	CRPS (↓)	0.372	0.316	–15%
Traffic (Time Series)	CRPS (↓)	0.042	0.040	–5%

On physics and video forecasting, as well as multivariate time series, DyDiff shows systematic improvements in CRPS and perceptual error metrics, and achieves higher temporal coherence in generated sequences.

Analysis indicates that DyDiff's denoised latents converge more rapidly to the ground truth, and that correlated latent noise is crucial for effective propagation of temporal structure; models with independent noise at each prediction step degrade below the standard baseline.

6. Comparative Models and Broader Context

Several architectural philosophies have emerged in the literature to encode dynamics within the diffusion framework.

Conditional Diffusion: Models such as DYffusion (Cachay et al., 2023) couple the forward and reverse processes to explicit, task-aligned stochastic interpolators and deterministic forecasters, eschewing purely Gaussian noise and leveraging physical dynamics priors. This strategy further accelerates inference and reduces sampling error accumulation.
Physics-in-the-Loop Regularization: Recent models (e.g., Pi-fusion (Qiu et al., 6 Jun 2024)) integrate PDE residuals directly into loss objectives and sampling steps, enforcing physics-informed constraints in both the learned score function and the inference process.
Model-based Control: Formulations such as dynamics-aware diffusion for planning (Gadginmath et al., 31 Mar 2025) and D-MPC (Zhou et al., 7 Oct 2024) embed system dynamics via projections or conditional bridges within the reverse process, producing strictly admissible trajectories even in the presence of unknown dynamics.

These strategies represent a trend toward domain-aware, temporally-aware, and physically-consistent diffusion modeling.

7. Limitations and Open Challenges

The main challenges for diffusion-based dynamics models include:

Sample Complexity: Accurate modeling of long-range dependencies necessitates larger temporal context windows, which escalates computational and memory demands.
Noise Scheduling Robustness: While DyDiff is robust to a range of mixing parameters, performance deteriorates when either the temporal mixing is disabled or history is over-weighted ( $\bar\gamma_t \to 1$ ), indicating sensitivity to extremes of the schedule.
Scalability to Nonlinear or Long-Range Physics: Explicit temporal mixing is straightforward for regular, linear domains, but incorporation of complex physical laws or long-range couplings (e.g., in molecular or population dynamics) may require further architectural or theoretical generalizations.
High-Dimensional and Pixel-Based Observations: Current best practices (as in D-MPC and DyDiff) focus on low-dimensional state or latent representations. Extension to pixel-level (image or video) spaces with explicit physical constraints is an active area.
Inference Cost: Diffusion sampling remains orders of magnitude slower than closed-form or single-step predictors. Techniques such as model distillation, DDIM, or DPM-Solver can accelerate inference but with trade-offs in quality.

References

Dynamical Diffusion: Learning Temporal Dynamics with Diffusion Models (Guo et al., 2 Mar 2025)
DYffusion: A Dynamics-informed Diffusion Model for Spatiotemporal Forecasting (Cachay et al., 2023)
Pi-fusion: Physics-informed diffusion model for learning fluid dynamics (Qiu et al., 6 Jun 2024)
Diffusion Model Predictive Control (Zhou et al., 7 Oct 2024)
Dynamics-aware Diffusion Models for Planning and Control (Gadginmath et al., 31 Mar 2025)

In conclusion, diffusion-based dynamics models generalize the denoising framework to structured temporal or physical systems by injecting explicit temporal and domain-specific dependencies into both the noising and denoising processes. Carefully designed architectures and noise schedules enable accurate and temporally coherent modeling across scientific, video, and time series domains, but challenges remain in scaling, interpretability, and real-time deployment.