PYTHON · NUMPY

Make it linear

Turn a nonlinear system into a linear one — not by approximating, but by changing coordinates. Meet the Koopman operator: lift the state through the right observables and the dynamics become exactly linear.

01Challenge

A linear model is a gift: it predicts forever and the entire toolbox of linear control applies. Most systems are nonlinear, so we usually give that up. Koopman's idea is that we often do not have to. A nonlinear system, viewed through the right OBSERVABLES of its state, moves linearly — the nonlinearity was in looking at only x, not, say, x². The Bench takes a nonlinear system (Brunton's slow-manifold system, whose trajectories curve onto a parabola) and lifts the state with one extra coordinate, x₁². In that lifted space the dynamics are EXACTLY linear, so ferromotion fits the Koopman operator by least squares — to machine precision. Watch: the green Koopman prediction (a linear rollout in the lifted space) sits exactly on the true nonlinear trajectory, while the best possible linear model in the raw coordinates (red) peels away immediately. One extra observable turned nonlinear into linear.

02Model

The Koopman operator is a change of perspective. Instead of tracking the state x and its nonlinear flow, track OBSERVABLES — functions of the state ψ(x) — and ask how they evolve. Koopman's theorem says this evolution is LINEAR: ψ(x_{k+1}) = A ψ(x_k), for an operator A that acts on functions. The catch is that A is infinite-dimensional in general. The practical win is that for many systems a small, well-chosen dictionary of observables closes into a finite invariant subspace where A is an ordinary matrix. Brunton's example ẋ₁ = μx₁, ẋ₂ = λ(x₂ − x₁²) is the clean case: the observables [x₁, x₂, x₁²] evolve exactly linearly, because the derivative of x₁² is 2x₁ẋ₁ = 2μx₁², which stays in the span. Extended Dynamic Mode Decomposition (EDMD) fits A from data by least squares — here it recovers the true operator to machine zero. Once you have A, the whole linear world opens: long-horizon prediction, and linear-quadratic control of a nonlinear system by controlling its lifted coordinates. This is the data-driven complement to the structured models: where SINDy discovers a sparse nonlinear equation, Koopman finds a coordinate change that makes the equation linear.

03Guided practice

Step A (see the nonlinearity): the true trajectory (gold) curves down onto the dashed slow manifold x₂ = b·x₁² — no straight-line linear model can capture that in the raw coordinates. Step B (the naive attempt): the red curve is the best possible linear model fit to the raw state (x₁, x₂). It cannot bend onto the manifold, so it diverges from the truth. Step C (lift and linearize): the green curve is the Koopman prediction — a purely LINEAR rollout, but in the lifted coordinates [x₁, x₂, x₁²]. It sits exactly on the gold truth; the operator error is machine zero. Step D (extend the horizon): drag the horizon out. The naive linear error keeps growing; the Koopman error stays at zero. The nonlinearity was never in the system — only in the coordinates you looked through.

04Feedback

PASS WHEN PASS when the Koopman rollout matches the true nonlinear trajectory (near-zero error) while the raw-coordinate linear model diverges — a nonlinear system made linear by lifting.

If your run fails, check:

Extend the horizon and watch: the green Koopman curve stays on the gold truth; the red naive-linear curve peels away.
The operator error is essentially machine zero — EDMD recovered the exact linear operator in the lifted space.

05Retrieve & space

What does the Koopman operator act on, and why is its evolution linear even for a nonlinear system?
Why does adding the single observable x₁² make Brunton's system exactly linear — what makes that dictionary 'closed'?
How do SINDy and Koopman differ in how they tame a nonlinear system?

06Mastery & project

You turned a nonlinear system into an exactly linear one by lifting it into the right observables, and fit the Koopman operator from data to machine precision. With a linear model of a nonlinear system, the entire linear prediction-and-control toolbox becomes available.

Koopman and SINDy are the data-driven, interpretable end of the spectrum. The final module puts every model you have built — structured, learned, or discovered — to work inside a controller you train by differentiating through it.

GO FURTHER · THE LIVE PLAYGROUND

ferromotion-control: Koopman / EDMD, in pure Rust ↗ — the extended dynamic mode decomposition you just ran, recovering the exact operator to machine zero.

← Discover the equation Interpretable by construction →

PYTHON · NUMPY

Make it linear

Live ferromotion-wasm: EDMD fitting the Koopman operator for a nonlinear system. Compare the lifted-linear prediction to the best raw-coordinate linear model.

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