Small errors become different futures.

This is the same reference used in Technical mode. Expand it whenever you need the exact numbers, assumptions or further explanation.

The next input changes the question

Let state s contain position x in metres and velocity v in metres per second. A frozen predictor takes a state and an applied force u and returns a predicted next state. Frozen means evaluation does not change its fitted parameters.

In one-step evaluation, each prediction starts from a true simulator state. In a free-running rollout, only the initial state is true; each later input is the preceding prediction. The same known action sequence is supplied to both. There is no action selection in this experiment.

The symbols with a tilde or hat are predictions. A rollout asks whether the model can keep predicting after its own errors change its inputs. A low one-step mean error does not answer that question.

A worked counterexample

Suppose the true position stays at zero and a flawed model always adds 0.1 m to its input position. Fresh-state predictions are always 0.1 m, so each one-step error is 0.1 m. Free-running positions are 0.1, 0.2, 0.3, …, 1.0 m after ten calls. This simple mechanism produces ten times the final error without changing the model.

Other models can damp, reverse or amplify an error. Errors may temporarily cancel, so the error curve need not rise at every interval. A measured average is not a per-state worst-case bound and cannot simply be multiplied by the horizon to obtain a guarantee.

Keep collection, fitting and diagnosis separate

The lab reuses the 0.1-second velocity-first discrete cart plant and the preceding dynamics model: 96 transitions from eight training episodes, a 38-parameter network, a training-fitted affine baseline and persistence. Neural updates use only measured training targets. The affine model fits those same training measurements. Persistence predicts no change.

Choose broad or gentle training actions and exact or perturbed training targets. Changing collection resets neural weights. The target-noise option does not add observations during imagination and does not perturb the reference trajectory. Changing drag changes both the teaching plant and its newly collected training data; changed-dynamics transfer is a later experiment.

The four evaluation starts are the existing public development starts. The first twelve oscillating-force transitions repeat earlier development examples; longer horizons extend that public practice. The alternative applies +2 N for the first twenty intervals and then coasts with zero force. Short horizons use the same prefix, so do not interpret them as a different random experiment.

Read the curves and the missing results

Compare absolute position error with absolute position error, and velocity with velocity. They have different units. The summary reports the mean across valid intervals and the error at the requested final interval. Incomplete runs identify their denominator; a short successful prefix must not silently replace a failed long rollout.

The plotting scales are fixed: 0–4 m for position error and 0–3 m/s for velocity error. Clipped values are counted and remain in the numerical trace. A proposed state beyond |x| = 20 m or |v| = 10 m/s, or a nonfinite value, stops free-running prediction before accepting it. These are computation limits, not physical safety limits. A stopped rollout has no final-horizon score.

Each fresh-state or free-running prediction is one model call. Rejected proposals count as calls too. Reference simulation, baseline fitting, neural fitting and rendering are different work; the count is not a timing or energy measurement.

Choose evidence for the intended use

Compare models under the same start, force sequence and horizon. A simple affine predictor can beat a poorly fitted neural model. Keep a failed long rollout as evidence about a useful operating horizon. If a controller will observe again after each action, it may tolerate errors differently from a system that imagines for many steps without correction.

This lab establishes neither a useful controller nor an independent evaluation result. The next lesson uses these predictors to select actions. Captures and exports are temporary development records, separate from the lesson check, later project assessment and human implementation review.