Learn continuous dynamics.

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

A model of change

A state records what we need to predict the next moment. Here it contains position x in metres and velocity v in metres per second. An action is an applied force u in newtons. A transition records all three inputs and the next state after 0.1 seconds. Predicting a transition is supervised learning: the next state supplies the target.

Position alone is insufficient. Two carts can pass the same mark while moving in opposite directions. Giving the model velocity distinguishes those situations. We supply exact current states in this lesson; estimating them from noisy sensors is a different problem.

The teaching environment

The fictional cart has mass 1 kg. Its discrete rule is:

Here a is acceleration in metres per second squared. The linear drag coefficient is 0.2 kg/s. The optional quadratic coefficient q is 0.6 kg/m, or zero in the linear setting. The prime means “next value,” not a derivative. Velocity updates before position. This is a defined discrete simulator, not exact integration of continuous mechanics.

For x=0.3, v=-0.5, u=1.2 and quadratic drag, acceleration is 1.2+0.1+0.15=1.45. The next velocity is -0.355 m/s and next position is 0.2645 m. The cart is still moving left even though its acceleration points right. The predictor must learn that relationship from collected transitions; it is not handed these coefficients.

Collect episodes, not shuffled answers

An episode is a short trajectory starting from a specified state. Eight training episodes supply 96 transitions. Four other starts supply 48 development transitions. Every episode has twelve steps. We assign episodes to splits before generating any rows, because neighboring rows share states and are correlated.

Broad collection applies forces up to ±2 N. Gentle collection applies up to ±0.25 N. Both development settings retain the same broader actions. Compare coverage by changing the collection setting and collecting again, rather than silently adding development rows to training.

The optional deterministic measurement perturbation changes only training next-state labels, by at most 0.01 m and 0.01 m/s. Current-state inputs and underlying motion remain exact. This is an isolated target-noise experiment, not a complete sensor-noise simulation. Reusing these public development episodes to choose settings makes them development evidence, not an independent final test.

Three predictors, the same evidence

Persistence predicts that position and velocity stay unchanged. It has no fitting cost. At a short horizon it can be competitive, so beat it before celebrating a small error.

The linear baseline fits two affine rules for state changes from the same training rows. Its four inputs are scaled position, velocity, force and a constant. A small ridge term of 10⁻⁶ on every coefficient stabilizes the solve. Linear drag produces a linear discrete transition, so this baseline can be particularly strong. No rule says a neural model must win.

The neural predictor uses three inputs, six tanh hidden units and two linear outputs: (3+1) × 6+(6+1) × 2=38 parameters. It predicts changes rather than whole next states. Each position change is divided by 0.2 m and each velocity change by 0.2 m/s when constructing training targets. Inputs use fixed scales of 2 m, 2 m/s and 2 N; these are declared unit scales, not statistics fitted on development data.

Read the comparison correctly

The table reports mean absolute position error in metres and velocity error in metres per second, separately. Both training and development diagnostics compare with exact simulator outcomes; the normalized training objective compares with the possibly perturbed training labels. Those are different measurements and need not move together.

The plot uses fresh true inputs for each row. Connecting points across episode boundaries is only a display convention. These are one-step predictions, not a rollout in which predicted states feed the next prediction. Fixed velocity axes clip values outside ±2 m/s and disclose how many; the inspector and export retain exact numbers.

A small position error does not imply accurate velocity, long-horizon prediction or successful control. Before using this model to choose actions, test free-running rollouts and compare controllers under matched scenarios and explicit work budgets. Those are subsequent Course 12 tasks.

Make a decision from evidence

Record which data you collected, how the three models compare, which errors remain and whether neural fitting earned its cost. Try linear drag and gentle collection as separate interventions. Keep an unsuccessful run: it may explain why coverage or a simpler baseline matters.

Captures and exported checkpoints are temporary practice records. They are neither authenticated training histories nor independently reviewed implementations. The lesson check, later project assessment and human review are distinct forms of evidence.