PYTHON · NUMPY

A model that bends

Give the world model a nonlinear basis so it can fit the curved parts of physics a straight line misses.

01Challenge

Fit a nonlinear (random-Fourier-feature) world model and show it imagines the pendulum's curved dynamics more faithfully than the linear one.

02Model

The linear model has a ceiling: the real world curves — a pendulum's restoring force follows the sine of its angle — and a straight line can't bend to it. Real world models fix this with depth and nonlinearity. You'll do it with a random Fourier basis: lift the input through a bank of cosines, then learn a linear read-out over that curved basis. It is still a fit-by-solving model, no fragile training loop, but now it can represent the sine the pendulum lives on. The test that matters is the rollout: over many imagined steps, the nonlinear model drifts less, because it got each step a little more right.

03Guided practice

1. The linear baseline and its rollout error. 2. A nonlinear basis. cos(x·Ω + φ) turns the input into curved features; a linear read-out over them is a nonlinear model. 3. The verdict. On the imagined rollout, the nonlinear model beats the linear one — capturing the curve is what lets it dream the real world.

04Feedback

PASS WHEN PASS when the nonlinear model's imagined-rollout error is below the linear model's.

If your run fails, check:

A cell errored — run them in order.
The nonlinear model should win on the multi-step rollout.

05Retrieve & space

Why does a linear world model hit a ceiling on a pendulum? (its dynamics curve through sin(theta)).
How do random Fourier features add nonlinearity? (a fixed cosine basis the read-out is linear over).
Why judge on the rollout, not one step? (rollout compounds small errors, so accuracy shows up over the horizon).

06Mastery & project

You can extend a world model with a nonlinear basis and verify the gain where it matters.

When a world model drifts, ask whether it can even represent the shape of the real dynamics.

← Imagine a trajectory Imagine in a latent →

PYTHON · NUMPY

A model that bends

Fit a nonlinear world model and beat the linear one on the rollout.

You read

the arrays and values already in scope

You change

the code you write in each cell

Fixed

the dataset and the checks that grade you