FERROMOTION · RUST → WASM

Discover the equation

You have only noisy position data from a swinging pendulum and no equations. Can an algorithm recover the actual governing law from the data alone?

Recover a system's governing equation directly from data, not a black-box fit, but an actual formula. Meet SINDy and the sparsity knob that separates the real terms from the noise.

01Challenge

Every method so far assumed some structure and learned the rest. SINDy assumes almost nothing and discovers the structure itself. Give it measured states and their rates of change, hand it a library of candidate terms (1, x, y, x², xy, …, x³) and it solves for the coefficients that reproduce the dynamics. The trick is a sparsity prior: the true physics uses only a FEW terms, so it sequentially thresholds away the small coefficients and refits. The Bench runs real SINDy on noisy data from the Duffing oscillator. The coefficient grid starts dense, twelve terms, most of them spurious noise. Turn the threshold λ up and watch the grid sparsify: the junk coefficients vanish until, in the sweet spot, exactly four survive: the four terms of the true Duffing system ẋ = y, ẏ = −x − 0.3x³ − 0.1y, each coefficient within 10% of the truth, because the data carries measurement noise. Push λ too far and it kills the real terms too. You have discovered an equation.

Predict firstYou have only noisy position data from a swinging pendulum and no equations. Can an algorithm recover the actual governing law from the data alone?

02Model

SINDy, Sparse Identification of Nonlinear Dynamics, is a different flavor of physics-informed learning: no network, just linear algebra with a sparsity prior. Stack the measured derivatives ẋ into a vector, build a matrix Θ(x) whose columns are candidate functions evaluated at every sample, and solve ẋ = Θ(x)Ξ for the coefficients Ξ. Ordinary least squares gives a dense Ξ, every candidate gets a nonzero coefficient, most of them fitting noise. The insight is that real dynamical laws are sparse: nature writes short equations. So SINDy uses sequentially thresholded least squares (fit, zero every coefficient below λ, refit on the survivors, repeat) which drives Ξ to just the few terms that matter. λ is the knob between an overfit dense model and an oversparse one, and its sweet spot recovers the exact governing equation, coefficients included. Because the result is a formula over named terms, it is interpretable and extrapolates, unlike a black box, you can read what it learned and check it against physics. The catch is the library: SINDy can only discover terms you thought to include, and it needs the derivatives, which are noisy when estimated from data, which is exactly why the threshold matters.

03Guided practice

Step A (dense start): at the opening λ the grid is full, twelve active terms with small spurious coefficients fighting the noise. Read the discovered equations: cluttered with junk. Step B (threshold up): drag λ higher. Watch spurious cells wink out as their coefficients fall below the threshold. Step C (the sweet spot): around λ ≈ 0.08 exactly four terms remain and the discovered equations match the true Duffing system term for term, with coefficients that noise leaves within 10% (PASS. Step D (too far): push λ higher still and the smallest real term, −0.1y, dies first, then −0.3x³) the model is now wrong in the other direction. The right sparsity is a window, and finding it is the craft of equation discovery.

04Feedback

PASS WHEN PASS when the sparsity threshold isolates exactly the true terms and the discovered equations match the Duffing system.

If your run fails, check:

Too many active terms? Raise λ to threshold away the spurious ones.
Too few? λ is so large it is killing real terms, the small −0.1y goes first. Lower it.

05Retrieve & space

Why does ordinary least squares give a dense, overfit model, and how does sequentially thresholded least squares fix it?
What does the threshold λ trade off, and why is the right value a window rather than a single number?
What are the two things SINDy needs that limit it, and why is the derivative estimate the reason λ matters?

06Mastery & project

You discovered a governing equation from data (an actual formula, coefficients and all) by combining a candidate library with a sparsity prior. Discovery, not fitting: the output is something you can read, check, and trust.

SINDy discovers dynamics that are sparse in the coordinates you chose. When they are not, you change coordinates instead, the Koopman idea in the next lesson, lifting a nonlinear system into a space where it becomes linear.

GO FURTHER · THE LIVE PLAYGROUND

ferromotion-learn: SINDy, in pure Rust ↗ · the candidate library and sequentially-thresholded least squares you just ran, on-device.

← Learn the flow, predict the future Make it linear →

FERROMOTION · RUST → WASM

Discover the equation

Live ferromotion-wasm: SINDy discovering the Duffing equation from noisy data. Turn λ to sparsify the coefficient grid until only the true terms survive.

You read

the readout: what the Rust crate computes, recomputed as you change things

You change

the bench controls, and with them the assumption the measurement rests on

Fixed

the ferromotion crate doing the computing, and the check that decides pass