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

I wrote a blog post about a lecture introducing stability in ODE models and their numerical solution. The lecture transcript and code for figures are included.

https://nadiah.org/2025/12/04/mxb261

#mathematicalEcology #ODEs #stability #mathematics #lectureNotes #populationDynamics

#Books shipping to new homes this morning.

#BookSelling #bookstodon #novel #CharlieChan #magazine #esquire #travel #places #france #dordogne #pindar #odes #poetry #AncientGreece #ancient #classical

Odes and garden wildlife ...

https://1001species.substack.com/p/odes-and-garden-wildlife

#dragonflies #wildlife #wildlifegardening #odes #rain #nature #biodiversity #montreal #canada

'Identifiability and Asymptotics in Learning Homogeneous Linear ODE Systems from Discrete Observations', by Yuanyuan Wang, Wei Huang, Mingming Gong, Xi Geng, Tongliang Liu, Kun Zhang, Dacheng Tao.

http://jmlr.org/papers/v25/22-1159.html

#estimation #estimator #odes

'Stable Implementation of Probabilistic ODE Solvers', by Nicholas Krämer, Philipp Hennig.

http://jmlr.org/papers/v25/20-1423.html

#numerical #solvers #odes

`The fast multipole method (FMM), introduced by Rokhlin Jr. and Greengard has been said to be one of the top ten #algorithms of the 20th century. The FMM algorithm reduces the complexity of matrix-vector multiplication involving a certain type of dense #matrix which can arise out of many #physical #systems.`

https://en.wikipedia.org/wiki/Fast_multipole_method

#physics #simulation #molecularDynamics #MD #dynamics #differentialEquations #ODEs #PDEs #ODE #PDE #Coulomb #Coulombic #CoulombForce #CoulombInteraction

Faster Training of Neural ODEs Using Gauß–Legendre Quadrature

Alexander Luke Ian Norcliffe, Marc Peter Deisenroth

Action editor: Kevin Swersky.

https://openreview.net/forum?id=f0FSDAy1bU

#odes #models #quadrature

in many ways, #mathematical #physics is based on the #smoothing properties of the #integral

#integrableSystems #mathematicalPhysics #complexAnalysis #spectralTheory #harmonicAnalysis #functionalAnalysis #mathematicalAnalysis #differentialEquations #ODEs #PDEs #SDEs #DEs #equations #BrownianMotion #LangevinDynamics #dynamics #Langevin #StochasticDifferentialEquations #StochasticProcesses #WienerProcess #OrnsteinUhlenbeck #HarmonicOscillator #WaveEquation #Newton #Newtonian #Maxwell #Einstein