Lorenz Attractor Forecasting · Lorenzo Marinelli

Live attractor

The canvas below simulates the Lorenz system directly in the browser. Change ρ to see how the shape of the trajectory changes.

Why it matters: in chaotic systems, tiny prediction errors quickly compound. This is why a model can look good at one-step prediction but diverge during recursive rollouts.

Speed 5× Trail 1300 ρ parameter 28

System

The classical Lorenz system is defined by three coupled differential equations:

dx/dt = σ(y - x)
dy/dt = x(ρ - z) - y
dz/dt = xy - βz

σ = 10
ρ = variable
β = 8/3

Project notes

Goal: predict future states from previous trajectory windows.
Tried: coordinate prediction, derivative prediction, and residual-style updates.
Problem: rolling predictions can drift even when short-horizon metrics look acceptable.
Takeaway: stability matters as much as pointwise accuracy for recursive forecasting.

What I learned

The Lorenz system is a useful stress test because the ground truth is deterministic but very sensitive to initial conditions. This makes it a good playground for understanding why sequence models may perform well locally but fail globally.

In my experiments, predicting coordinates and derivatives independently surprisingly worked better than forcing derivative calculations too early in training. Residual-style updates were interesting, but recursive rollouts still exposed instability.

An early recurrent-data experiment

This was one of my first experiments with recurrent data during my first year of university. It was not a large or polished research project, but it was the first time I really saw how sequence models behave once predictions are fed back into themselves.

I am still fond of it for that reason. The Lorenz system made the lesson concrete: even a small model can look convincing over short horizons, while the recursive rollout exposes stability, drift, and error accumulation in a way that is hard to ignore.