Taxonomies of World Models › Reading the Results

The headline result is the growth of multi-step error with horizon, plotted on a logarithmic scale so that the early errors remain visible beside the much larger long-horizon errors.

The pattern confirms the taxonomy's central claim: a representation is useful only together with dynamics that fit its coordinates. All three contestants see the same episodes, action sequences, and evaluation cases. The observation-space baseline asks one linear map to express nonlinear dynamics in sixteen sensor coordinates, so its structural error compounds. The other two contestants use the pendulum's structured transition in three dynamics-relevant coordinates. Their difference is initialization: the learned model estimates those coordinates from an observation, while the oracle receives them directly.

The horizon changes the ranking. At one step, the observation-space baseline has about 0.008 radians of angle RMSE and appears better than the learned latent model at about 0.021. At sixty steps, the baseline reaches about 1.55 radians while the learned latent model remains near 0.15, a reversal of roughly a factor of ten. The oracle-feature reference is slightly lower at about 0.14. A next-step score therefore answers a different question from the one a planner cares about: whether repeated applications remain useful. The bar chart makes that reversal explicit.

To see what the aggregate curves mean at the level of individual imaginations, we overlay one predicted trajectory against ground truth for the two learned models.

Two details deserve attention beyond the obvious difference in tracking quality. First, the true trajectory is visibly irregular while both predictions are smooth: the process noise that agitates the real pendulum is absent from deterministic imaginations, exactly the mean-seeking behavior described earlier. Second, the latent model's errors are not random jitter but slow drift, the signature of a slightly wrong but structurally reasonable model. Drift of this kind is what model-based controllers can sometimes correct through feedback, whereas the observation-space model's errors are too violent and too early for any planner to exploit its imagination productively. This distinction between error modes, drift versus chaos, matters practically: slow drift can be diagnosed, corrected through feedback, and often traced to specific neglected effects, while chaotic divergence tells you only that the model class was wrong and gives no hint of how to fix it.

Last, the stochastic variant shows what the deterministic/stochastic axis contributes independently of representation quality.

Read this figure as a portrait of honest ignorance. The deterministic model draws one line and asserts it. The stochastic model draws a cloud whose width grows with horizon, reflecting accumulating uncertainty, and the true trajectories travel inside the cloud rather than along its center. Nothing about the cloud tells a planner what to do; converting sampled futures into risk-sensitive decisions is the business of Part VII. But the cloud is the raw material that risk-sensitive planning consumes, and a deterministic model cannot manufacture it after the fact. Note also what the figure does not show: any guarantee that the cloud's width is correct. With only five reference trajectories overlaid, we can check bracketing informally, but rigorous calibration assessment would demand many more trials and a proper scoring rule, which is exactly the methodology Part XI develops.

It is worth being precise about what this toy demonstrates and what it does not. The feature-space model enjoys an unfair advantage we handed it: access to the sufficient statistics. The learned encoder also receives supervised state targets, a privileged diagnostic that isolates state-estimation error from transition-model error rather than a general recipe for representation learning. Part III develops methods that try to recover useful state from observations without such labels, using reconstruction, self-supervised dynamics, rewards, or combinations of objectives. Token-based and object-centered variants make still different tradeoffs. The pendulum fixes the shape of the argument, not the engineering frontier. What transfers from toy to frontier is the logic: identify which taxonomic position a design occupies, predict its characteristic failure from that position, and check the prediction against rollout behavior rather than one-step accuracy.