SINDy vs. ISINDy on the Lorenz System

Equation discovery with SINDy

SINDy (Sparse Identification of Non-linear Dynamics) discovers governing equations from data. Given state measurements $\mathbf{X}$, a library of candidate functions $\Theta(\mathbf{X})$ and a coefficient matrix $\Xi$, it looks for the sparsest $\Xi$ such that

$\dot{\mathbf{X}} = \Theta(\mathbf{X})\,\Xi$

minimising $\lVert \dot{\mathbf{X}} - \Theta(\mathbf{X})\Xi \rVert_2^2 + \lambda \lVert \Xi \rVert_1$ with the STLSQ (Sequential Thresholded Least Squares) algorithm. The test case is the Lorenz system.

The problem: noise

Standard SINDy needs the derivative $\dot{\mathbf{X}}$, and numerical differentiation amplifies high-frequency noise, so on noisy data it often fails to find the true dynamics.

The fix: Integral SINDy (ISINDy)

ISINDy integrates the equations in time instead:

$\mathbf{X}(t) - \mathbf{X}(t_0) = \Theta_{\text{int}}(\mathbf{X})\,\Xi$

Integration acts as a low-pass temporal filter and removes the need to differentiate noisy data, which makes the regression far more robust.

Part of my research internship with Prof. Saurav Sengupta, IIT (ISM) Dhanbad.