Lorenz System

anypinn create my-project --template lorenz

Chaotic 3-field ODE. Recovers \(\sigma\), \(\rho\), and \(\beta\) simultaneously from trajectory observations.

Background

The Lorenz system, derived by Edward Lorenz (1963) as a simplified model of atmospheric convection, is one of the earliest and most studied examples of deterministic chaos. For the classical parameter values (\(\sigma = 10\), \(\rho = 28\), \(\beta = 8/3\)) the system exhibits extreme sensitivity to initial conditions (the "butterfly effect"). Trajectories settle onto a fractal strange attractor that never repeats. Recovering all three parameters simultaneously from a noisy, chaotic trajectory tests the robustness of the inverse solver; Huber loss is used to mitigate the effect of outlier residuals.

Governing Equations

\[ \begin{cases} \dot{x} = \sigma\,(y - x) \\[6pt] \dot{y} = x\,(\rho - z) - y \\[6pt] \dot{z} = x\,y - \beta\,z \end{cases} \]

where:

• \(x(t)\): proportional to convective circulation intensity
• \(y(t)\): proportional to temperature difference between ascending and descending currents
• \(z(t)\): proportional to deviation from linear vertical temperature profile
• \(\sigma\): Prandtl number (to recover)
• \(\rho\): normalized Rayleigh number (to recover)
• \(\beta\): geometric factor (to recover)

Default Configuration

The generated template uses the following values.

Parameters to recover:

Symbol	Code constant	True value
\(\sigma\)	TRUE_SIGMA	\(10.0\)
\(\rho\)	TRUE_RHO	\(28.0\)
\(\beta\)	TRUE_BETA	\(8/3 \approx 2.667\)

Initial conditions: \(x(0) = -8, \quad y(0) = 7, \quad z(0) = 27\)

Domain: \(t \in [0, 3]\)

Scaling: state variables are divided by \(S = 20\) in the training ODE.

Features Demonstrated

• Multi-parameter recovery (3 simultaneous parameters)
• Huber loss via PINNHyperparameters.criterion for robustness to chaotic trajectories
• 3-field system

Results