Diffusion and Flow World Models › Diffusion and Flow World Models

Imagine a tabletop robot trying to flick a puck into one of two goals. Suppose its idealized simulator has a deterministic transition once the full physical state, realized contact impulse, surface parameters, and boundary conditions are fixed. Suppose also that an unobserved left-or-right intention selects the corresponding goal with substantial probability, while residual contact variation is small relative to the distance between the goals. The observer records only a coarse command $a_t$, not that intention, exact contact, or other hidden physical variables. Given the recorded position $s_t$ and command $a_t$, the conditional next-position distribution can then have two goal-centered modes with very little probability between them. Mean-square-error regression targets the conditional mean of these possibilities, which is the prediction at the population optimum. For separated, similarly weighted outcomes, that mean can lie in a low-density region that is a poor representative prediction. Point regression then washes away discrete future structure exactly when the distinction matters for a decision.

Diffusion and flow models offer a flexible modern way to address this problem. Instead of predicting one point, they map a random base sample into a draw from a learned conditional distribution over futures. Ancestral diffusion uses stochastic reverse updates, whereas zero-noise Denoising Diffusion Implicit Models (DDIM; Song, Meng, and Ermon, 2021) with $\eta=0$ and flow ODE integration are deterministic once their initial sample, conditioning, model, and solver settings are fixed. All three obtain diversity from different random base samples. When the learned conditional transport represents both modes, those samples can map to distinct outcomes rather than a single average; ancestral diffusion can inject additional randomness during its reverse updates.

Distributional world models predate diffusion. As discussed in Part III, Ch 3: Latent-Variable Representations, latent-variable models and mixture-density transition models can also represent multimodal futures. Some VAE-based systems use compressed stochastic latents and simple priors, which can reduce modeling or sampling cost; lossy compression may discard task-relevant detail, and total sampling cost still depends on the decoder. Autoregressive token models (see Part V, Ch 3: Autoregressive Token World Models) represent multimodality through sequential token sampling, so decoding is serial and earlier sampling errors can feed later predictions. Diffusion and flow instead update all coordinates of their chosen target through repeated denoiser or vector-field evaluations; conditioning may be injected at every evaluation. They do not require a compact latent bottleneck or token-by-token decoder, although particular variants may use either, and iterative sampling adds latency.

The chapter unfolds in four moves. First, we develop conditional denoising dynamics used by many diffusion world models: the corruption process, the training objective, and learned reverse-time sampling. Second, we distinguish diffusing over one-step states from diffusing over whole trajectories and examine rollout cost and planning by inpainting. Third, flow matching recasts generation as numerical integration of an ordinary differential equation, while rectified flow targets straighter transport paths. Fourth, we separate control mechanisms from the latency constraints that decide whether a sampler fits inside a control loop. A worked example then compares point regression, ancestral diffusion, DDIM, and flow matching on a synthetic bimodal transition.