Automatic Differentiation to Simultaneously Identify Nonlinear Dynamics and Extract Noise Probability Distributions from Data

 Sparse identification of nonlinear dynamics (SINDy) algorithm is a method that allows the identification of parsimonious system models. However, the SINDy algorithm is sensitive to large noise. To overcome this issue, we develop a variant of the SINDy algorithm that integrates automatic differentiation and recent time-stepping constrained motivated by Rudy et al. for simultaneously (i) denoising the data, (ii) learning and parametrizing the noise probability distribution, and (iii) identifying the underlying parsimonious dynamicalsystem responsible for generating the time-series data. This new noise signal separation SINDy (modified-SINDy) algorithm dramatically improves noise robustness of original SINDy algorithm and makes the identification of the noise distribution possible. We show several examples here to illustrate the effectiveness of this new member of the SINDy family. The details of the approach are in our peer reviewed paper.

Examples:

Lorenz Equations

Lorenz equation is a set of ordinary differential equations famous for having chaotic behavior. We will use the Lorenz equations to demonstrate the noise robustness of modified-SINDy. Moreover, it is also used to test the performance of modified-SINDy under different data usage.

Van der Pol Oscillator

The Van der Pol oscillator is used to demonstrate that modified-SINDy can tackle different types of noise distributions. Moreover, we show that modified-SINDy can identify the non-zero mean noise added to the signal.

Duffing and Cubic Oscillator

Duffing and Cubic oscillators are used to test the effectiveness of modified-SINDy under different noise levels.

Lotka-Volterra Equations

The Lotka-Volterra equations are usually used to describe the dynamics of the predator-prey system. We will show that modified-SINDy can identify the models of Lotka-Volterra equations given noisy measurement data.

Dependencies:

• Numpy, SciPy, Matplotlib, fitter, and TensorFlow 2.0 packages for Python are needed to run the examples.
• The noisy data needed to compare the modified-SINDy and Weak SINDy can be dowloaded here.