This repository contains the code used in my thesis Discovering Equations in a Data-Driven Era: An Analysis of the SINDy Algorithm, From Theory to Advanced Applications for the BSc in Mathematical and Computing Sciences for Artificial Intelligence (BAI) at Bocconi University, Milan.

The thesis is available only upon request.

This thesis analyzes the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm, a powerful tool for discovering parsimonious dynamical systems from high-dimensional data. SINDy operates by finding sparse coefficient vectors that best fit the measured data as a linear combination of predetermined functions. The work first describes the foundations of the algorithm from a theoretical perspective. Key results are discussed, such as sampling requirements for data, noise handling for numerical differentiation, and strategies for constructing an effective library of candidate functions. Additionally, a comparison between different optimization techniques is provided, including sequential thresholding and sparse relaxed regularized regression. To demonstrate SINDy's versatility and reliability across domains, the thesis replicates the original results on the Lorenz system and explores new applications to population dynamics with the Lotka-Volterra model and to the dynamics of neurons with continuous Hopfield networks. Overall, this work illustrates how SINDy has the potential to revolutionize scientific discovery by accurately uncovering complex models directly from data.

• noise.ipynb contains code for the Data section of the thesis.
• library.ipynb contains code for the Library section of the thesis.
• optimization.ipynb contains code for the Optimization section of the thesis.
• sindy.ipynb contains the implementation of the SINDy algorithm from scratch. It replicates the identification of Lorenz from the paper
• lorenz.ipynb contains the application of SINDy on Lorenz using PySINDy. It also replicates other results from the paper.
• lotka-volterra.ipynb contains the application of SINDy on the Lotka-Volterra system.
• hopfield.ipynb contains the application of SINDy on the continuous Hopfield neural network model.