Sparse Identification of Nonlinear Dynamics

## Project description

PySINDy is a sparse regression package with several implementations for the Sparse Identification of Nonlinear Dynamical systems (SINDy) method introduced in Brunton et al. (2016a), including the unified optimization approach of Champion et al. (2019), SINDy with control from Brunton et al. (2016b), Trapping SINDy from Kaptanoglu et al. (2021), SINDy-PI from Kaheman et al. (2020), PDE-FIND from Rudy et al. (2017), and so on. A comprehensive literature review is given in de Silva et al. (2020) and Kaptanoglu, de Silva et al. (2021).

## System identification

System identification refers to the process of leveraging measurement data to infer governing equations, in the form of dynamical systems, describing the data. Once discovered, these equations can make predictions about future states, can inform control inputs, or can enable the theoretical study using analytical techniques. Dynamical systems are a flexible, well-studied class of mathematical objects for modeling systems evolving in time. SINDy is a model discovery method which uses sparse regression to infer nonlinear dynamical systems from measurement data. The resulting models are inherently interpretable and generalizable.

### How it works

Suppose, for some physical system of interest, we have measurements of state variables x(t) (a vector of length n) at different points in time. Examples of state variables include the position, velocity, or acceleration of objects; lift, drag, or angle of attack of aerodynamic objects; and concentrations of different chemical species. If we suspect that the system could be well-modeled by a dynamical system of the form

```x'(t) = f(x(t)),
```

then we can use SINDy to learn f(x) from the data (x'(t) denotes the time derivative of x(t)). Note that both f(x) and x(t) are typically vectors. The fundamental assumption SINDy employs is that each component of f(x), f_i(x) can be represented as a sparse linear combination of basis functions theta_j(x)

```f_i(x) = theta_1(x) * xi_{1,i} + theta_2(x) * xi_{2,i} + ... + theta_k * xi{k,i}
```

Concatenating all the objects into matrices (denoted with capitalized names) helps to simplify things. To this end we place all measurements of the state variables into a data matrix X (with a row per time measurement and a column per variable), the derivatives of the state variables into a matrix X', all basis functions evaluated at all points in time into a matrix Theta(X) (each basis function gets a column), and all coefficients into a third matrix Xi (one column per state variable). The approximation problem to be solved can then be compactly written as

```X' = Theta(X) * Xi.
```

Each row of this matrix equation corresponds to one coordinate function of f(x). SINDy employs sparse regression techniques to find a solution Xi with sparse column vectors. For a more in-depth look at the mathematical foundations of SINDy, please see our introduction to SINDy.

### Relation to PySINDy

The PySINDy package revolves around the SINDy class which consists of three primary components; one for each term in the above matrix approximation problem.

• differentiation_method: computes X', though if derivatives are known or measured directly, they can be used instead
• feature_library: specifies the candidate basis functions to be used to construct Theta(X)
• optimizer: implements a sparse regression method for solving for Xi

Once a SINDy object has been created it must be fit to measurement data, similar to a scikit-learn model. It can then be used to predict derivatives given new measurements, evolve novel initial conditions forward in time, and more. PySINDy has been written to be as compatible with scikit-learn objects and methods as possible.

### Example

Suppose we have measurements of the position of a particle obeying the following dynamical system at different points in time

```x' = -2x
y' = y
```

Note that this system of differential equations decouples into two differential equations whose solutions are simply x(t) = x_0 * exp(-2 * t) and y(t) = y_0 * exp(t), where x_0 = x(0) and y_0 = y(0) are the initial conditions.

Using the initial conditions x_0 = 3 and y_0 = 0.5, we construct the data matrix X.

```import numpy as np
import pysindy as ps

t = np.linspace(0, 1, 100)
x = 3 * np.exp(-2 * t)
y = 0.5 * np.exp(t)
X = np.stack((x, y), axis=-1)  # First column is x, second is y
```

To instantiate a SINDy object with the default differentiation method, feature library, and optimizer and then fit it to the data, we invoke

```model = ps.SINDy(feature_names=["x", "y"])
model.fit(X, t=t)
```

We use the feature_names argument so that the model prints out the correct labels for x and y. We can inspect the governing equations discovered by the model and check whether they seem reasonable with the print function.

```model.print()
```

which prints the following

```x' = -2.000 x
y' = 1.000 y
```

PySINDy provides numerous other features not shown here. We recommend the feature overview section of the documentation for a more exhaustive summary of additional features.

## Installation

### Installing with pip

If you are using Linux or macOS you can install PySINDy with pip:

```pip install pysindy
```

### Installing from source

First clone this repository:

```git clone https://github.com/dynamicslab/pysindy.git
```

Then, to install the package, run

```pip install .
```

If you do not have pip you can instead use

```python setup.py install
```

If you do not have root access, you should add the --user option to the above lines.

### Caveats

If you would like to use the SINDy-PI optimizer, the Trapping SINDy optimizer (TrappingSR3), or the other SR3 optimizations with inequality constraints, you will also need to install the cvxpy package, e.g. with pip install cvxpy.

To run the unit tests, example notebooks, or build a local copy of the documentation, you should install the additional dependencies in requirements-dev.txt

```pip install -r requirements-dev.txt
```

## Documentation

The documentation site for PySINDy can be found here. There are numerous examples of PySINDy in action to help you get started. Examples are also available as Jupyter notebooks. A video overview of PySINDy can be found on Youtube. We have also created a video playlist with practical PySINDy tips.

PySINDy implements a lot of advanced functionality that may be overwhelming for new users or folks who are unfamiliar with these methods. Below (see here if image does not render https://github.com/dynamicslab/pysindy/blob/master/docs/JOSS2/Fig3.png), we provide a helpful flowchart for figuring out which methods to use, given the characteristics of your dataset:

This flow chart summarizes how PySINDy users can start with a dataset and systematically choose the proper candidate library and sparse regression optimizer that are tailored for a specific scientific task. The GeneralizedLibrary class allows for tensoring, concatenating, and otherwise combining many different candidate libraries.

## Community guidelines

### Contributing examples

We love seeing examples of PySINDy being used to solve interesting problems! If you would like to contribute an example, reach out to us by creating an issue.

### Contributing code

We welcome contributions to PySINDy. To contribute a new feature please submit a pull request. To get started we recommend installing the packages in requirements-dev.txt via

```pip install -r requirements-dev.txt
```

This will allow you to run unit tests and automatically format your code. To be accepted your code should conform to PEP8 and pass all unit tests. Code can be tested by invoking

```pytest
```

We recommend using pre-commit to format your code. Once you have staged changes to commit

```git add path/to/changed/file.py
```

you can run the following to automatically reformat your staged code

```pre-commit
```

Note that you will then need to re-stage any changes pre-commit made to your code.

There are a number of SINDy variants and advanced functionality that would be great to implement in future releases:

1. Bayesian SINDy, for instance that from Hirsh, Seth M., David A. Barajas-Solano, and J. Nathan Kutz. “Sparsifying Priors for Bayesian Uncertainty Quantification in Model Discovery.” arXiv preprint arXiv:2107.02107 (2021).
2. Tensor SINDy, using the methods in Gelß, Patrick, et al. “Multidimensional approximation of nonlinear dynamical systems.” Journal of Computational and Nonlinear Dynamics 14.6 (2019).
3. Stochastic SINDy, using the methods in Brückner, David B., Pierre Ronceray, and Chase P. Broedersz. “Inferring the dynamics of underdamped stochastic systems.” Physical review letters 125.5 (2020): 058103.
4. Integration of PySINDy with a Python model-predictive control (MPC) code.
5. The PySINDy weak formulation is based on the work in Reinbold, Patrick AK, Daniel R. Gurevich, and Roman O. Grigoriev. “Using noisy or incomplete data to discover models of spatiotemporal dynamics.” Physical Review E 101.1 (2020): 010203. It might be useful to additionally implement the weak formulation from Messenger, Daniel A., and David M. Bortz. “Weak SINDy for partial differential equations.” Journal of Computational Physics (2021): 110525. The weak formulation in PySINDy is also fairly slow and computationally intensive, so finding ways to speed up the code would be great.
6. The blended conditional gradients (BCG) algorithm for solving the constrained LASSO problem, Carderera, Alejandro, et al. “CINDy: Conditional gradient-based Identification of Non-linear Dynamics–Noise-robust recovery.” arXiv preprint arXiv:2101.02630 (2021).

### Reporting issues or bugs

If you find a bug in the code or want to request a new feature, please open an issue.

### Getting help

For help using PySINDy please consult the documentation and/or our examples, or create an issue.

## Citing PySINDy

PySINDy has been published in the Journal of Open Source Software (JOSS). The paper can be found here.

If you use PySINDy in your work, please cite it using the following two references:

Brian M. de Silva, Kathleen Champion, Markus Quade, Jean-Christophe Loiseau, J. Nathan Kutz, and Steven L. Brunton., (2020). PySINDy: A Python package for the sparse identification of nonlinear dynamical systems from data. Journal of Open Source Software, 5(49), 2104, https://doi.org/10.21105/joss.02104

Alan A. Kaptanoglu, Brian M. de Silva, Urban Fasel, Kadierdan Kaheman, Jared L. Callaham, Charles B. Delahunt, Zachary G. Nicolaou, Kathleen Champion, Jean-Christophe Loiseau,J. Nathan Kutz, and Steven L. Brunton. PySINDy: A comprehensive Python packagefor robust sparse system identification. arXiv preprint arXiv:2111.08481, 2021.

Bibtex:

```@article{desilva2020,
doi = {10.21105/joss.02104},
url = {https://doi.org/10.21105/joss.02104},
year = {2020},
publisher = {The Open Journal},
volume = {5},
number = {49},
pages = {2104},
author = {Brian de Silva and Kathleen Champion and Markus Quade and Jean-Christophe Loiseau and J. Kutz and Steven Brunton},
title = {PySINDy: A Python package for the sparse identification of nonlinear dynamical systems from data},
journal = {Journal of Open Source Software}
}
```

Bibtex:

```@article{kaptanoglu2021pysindy,
title={PySINDy: A comprehensive Python package for robust sparse system identification},
author={Alan A. Kaptanoglu and Brian M. de Silva and Urban Fasel and Kadierdan Kaheman and Jared L. Callaham and Charles B. Delahunt and Zachary G. Nicolaou and Kathleen Champion and Jean-Christophe Loiseau and J. Nathan Kutz and Steven L. Brunton},
year={2021},
Journal = {arXiv preprint arXiv:2111.08481},
}
```

## References

• de Silva, Brian M., Kathleen Champion, Markus Quade, Jean-Christophe Loiseau, J. Nathan Kutz, and Steven L. Brunton. PySINDy: a Python package for the sparse identification of nonlinear dynamics from data. arXiv preprint arXiv:2004.08424 (2020) [arXiv]
• Kaptanoglu, Alan A., Brian M. de Silva, Urban Fasel, Kadierdan Kaheman, Jared L. Callaham, Charles B. Delahunt, Zachary G. Nicolaou, Kathleen Champion, Jean-Christophe Loiseau, J. Nathan Kutz, and Steven L. Brunton. PySINDy: A comprehensive Python package for robust sparse system identification. arXiv preprint arXiv:2111.08481 (2021). [arXiv]
• Brunton, Steven L., Joshua L. Proctor, and J. Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences 113.15 (2016): 3932-3937. [DOI]
• Champion, K., Zheng, P., Aravkin, A. Y., Brunton, S. L., & Kutz, J. N. (2020). A unified sparse optimization framework to learn parsimonious physics-informed models from data. IEEE Access, 8, 169259-169271. [DOI]
• Brunton, Steven L., Joshua L. Proctor, and J. Nathan Kutz. Sparse identification of nonlinear dynamics with control (SINDYc). IFAC-PapersOnLine 49.18 (2016): 710-715. [DOI]
• Kaheman, K., Kutz, J. N., & Brunton, S. L. (2020). SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics. Proceedings of the Royal Society A, 476(2242), 20200279. [DOI]
• Kaptanoglu, A. A., Callaham, J. L., Aravkin, A., Hansen, C. J., & Brunton, S. L. (2021). Promoting global stability in data-driven models of quadratic nonlinear dynamics. Physical Review Fluids, 6(9), 094401. [DOI]

## Contributors

Thanks to the members of the community who have contributed to PySINDy!

 billtubbs Bug fix #68 kopytjuk Concatenation feature for libraries #72 andgoldschmidt derivative package for numerical differentiation #85