This year, Simon Prince, Professor of Computer Science at UCL, published a series of tutorials on ordinary differential equations (ODEs) and stochastic differential equations (SDEs) in machine learning for RBC Borealis. These are intended for readers with no background in these areas and require only basic calculus.
Article 1 describes what ODEs and SDEs are and their applications in machine learning.
https://rbcborealis.com/research-blogs/odes-and-sdes-for-machine-learning
Article 2 describes ODEs, vector ODEs and PDEs and defines associated terminology. They develop several categories of ODE and discuss how their solutions are related to one another. They discuss the necessary conditions for an ODE to have a solution.
https://rbcborealis.com/research-blogs/introduction-ordinary-differential-equations
Article 3 describes methods for solving first-order ODEs in closed form. They categorise ODEs into distinct families and develop a method to solve each family.
https://rbcborealis.com/research-blogs/closed-form-solutions-for-odes
For many ODEs, there is no known closed-form solution.
Article 4 considers numerical methods, which can be used to approximate the solution of any ODE regardless of its tractability.
https://rbcborealis.com/research-blogs/numerical-methods-for-odes
This concludes their treatment of ODEs. In the coming weeks, we will focus on SDEs. They will describe stochastic processes and SDEs, and show how to solve SDEs using either direct stochastic integration or Ito's lemma. They will introduce the Fokker-Planck equation, which transforms a stochastic differential equation into the PDE governing the evolving probability density of the solution. They also consider Andersen's theorem, which allows us to reverse the direction of SDEs.
#ODEs #PDEs #SDEs #ODE #PDE #SDE #Calculus #ML #DL #VectorCalculus #LectureSeries #Tutorials
