SINDy_example

Sparse Identification of Nonlinear Dynamical Systems (SINDy) is an algorithm to discover governing dynamical equations for time series 𝐱(𝑡) .

The key idea is to construct a differential equation:

$$\frac{d{\bf x}}{d t} = \Theta({\bf x}^T) {\bf x}$$ or like this: $$\frac{d{\bf x}}{d t^2} = \Theta({\bf x}^T) {\bf x}$$ if we are looking for a second-order derivative.

where the time derivative is computed from the time series data and Θ(𝐱) is a non-linear basis.

We will follow the next steps: Steps:

Compute time derivative: Perhaps the trickiest part, especially for noisy time series although one can use total variation regularized derivatives for such cases as here

Choose basis: We will choose a non-linear basis constructed from the time series data, consisting of a polynomial basis and trigonometrical functions.

Use regression: We apply regularized linear regression Lasso/Ridge with one step or we can perform sequential thresholding least squares [1].

Once we discover the underlying dynamical equations, it will be a lot easier to forecast. Next moves:

In reality, it is difficult to know the right coordinates and basis functions to use. The interesting extension of SINDy so far has been to use latent basis / coordinates instead of physical space basis. In latent space,

$$ {\bf x} \longrightarrow {\bf z} = Encoder({\bf x})$$

A non-linear basis in 𝐳 is used to perform SINDy. The full cost function also takes into account the physical space SINDy expression. This approach is developed in articles [2] and [3]

Discovering governing equations from data by sparse identification of nonlinear dynamical systems Steven L. Brunton & others https://www.pnas.org/content/113/15/3932
Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders Joseph Bakarji, Kathleen Champion https://arxiv.org/pdf/2201.05136.pdf
Enhancing Computational Fluid Dynamics with Machine Learning Ricardo Vinuesa1, and Steven L. Brunton https://arxiv.org/pdf/2110.02085.pdf

CostiaB/SINDy_example

SINDy_example