Uncovering differential equations from data with hidden variables

This repository has all the code implementation of the paper: Uncovering differential equations from data with hidden variables

Abstract

SINDy is a method for learning system of differential equations from data by solving a sparse linear regression optimization problem. In this article, we propose an extension of the SINDy method that learns systems of differential equations in cases where some of the variables are not observed. Our extension is based on regressing a higher order time derivative of a target variable onto a dictionary of functions that includes lower order time derivatives of the target variable. We evaluate our method by measuring the prediction accuracy of the learned dynamical systems on synthetic data and on a real data-set of temperature time series provided by the Réseau de Transport d'Électricité (RTE). Our method provides high quality short-term forecasts and it is orders of magnitude faster than competing methods for learning differential equations with latent variables.

Examples

The experiments performed in the paper can be found in the notebook examples/paper_experiments.ipynb. There you will find:

1. Experiments with all variables observed.
2. Experiments with only one variable observed:
   1. Oscilator variable x
   2. Rössler variable y
   3. Rössler variable x
   4. Lorenz variable x
3. Example of RTE experimental setting.

We also provide in examples/spatial_experiments some insights on prelimiar experiments using this methodology to solve PDE where some variable is not observed. In the example, a unidimentional wave equation with varying (and unknown) tension is used to generate data.

Setup for developers

Create virtual enviroment

python3.8 -m venv venv

Activate virtual enviroment

. .venv/bin/activate

Install libraries

pip install -r requirements.txt 

Jupyter notebooks

In order to be able to run jupyter notebooks:

pip install ipykernel
python -m ipykernel install --user --name venv --display-name "venv"

Source: https://ipython.readthedocs.io/en/stable/install/kernel_install.html#kernels-for-different-environments