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

Grid-Free Approach to Partial Differential Equations on Volumetric Domains [pdf]

http://rohansawhney.io/RohanSawhneyPhDThesis.pdf

#HackerNews #GridFree #PDEs #VolumetricDomains #MathematicalModeling #ComputationalPhysics

@bradheintz I feel like I had a similar attitude towards #probability in my undergrad. It wasn’t until later on in my PhD and even only as a researcher research that I came to appreciate the beauty. Especially through the connection to measure theory and through that #analysis. It is a very interesting field full of deep results and applications crossing to several fields like random #geometry and stochastic #PDEs. Many results in #numbertheory have even been proven using probabilistic methods, I.e. showing that properties hold for a set of numbers that has a positive measure in some sense.

FortranX: Harnessing Code Generation, Portability, and Heterogeneity in Fortran

#OpenCL #HIP #CUDA #OpenMP #Fortran #CodeGeneration #PDEs

https://hgpu.org/?p=29585

A Distributed-memory Tridiagonal Solver Based on a Specialised Data Structure Optimised for CPU and GPU Architectures

#CUDA #OpenMP #Physics #ComputationalPhysics #PDEs

https://hgpu.org/?p=29552

'Boundary constrained Gaussian processes for robust physics-informed machine learning of linear partial differential equations', by David Dalton, Alan Lazarus, Hao Gao, Dirk Husmeier.

http://jmlr.org/papers/v25/23-1508.html

#boundary #pdes #gaussian

LINEAR TRANSPORT EQUATION
The linear transport equation (LTE) models the variation of the concentration of a substance flowing at constant speed and direction. It's one of the simplest partial differential equations (PDEs) and one of the few that admits an analytic solution.

Given \(\mathbf{c}\in\mathbb{R}^n\) and \(g:\mathbb{R}^n\to\mathbb{R}\), the following Cauchy problem models a substance flowing at constant speed in the direction \(\mathbf{c}\).
\[\begin{cases}
u_t+\mathbf{c}\cdot\nabla u=0,\ \mathbf{x}\in\mathbb{R}^n,\ t\in\mathbb{R}\\
u(\mathbf{x},0)=g(\mathbf{x}),\ \mathbf{x}\in\mathbb{R}^n
\end{cases}\]
If \(g\) is continuously differentiable, then \(\exists u:\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}\) solution of the Cauchy problem, and it is given by
\[u(\mathbf{x},t)=g(\mathbf{x}-\mathbf{c}t)\]

#LinearTransportEquation #LinearTransport #Cauchy #CauchyProblem #PDE #PDEs #CauchyModel #Math #Maths #Mathematics #Linear #LinearPDE #TransportEquation #DifferentialEquations

'Multilevel CNNs for Parametric PDEs', by Cosmas Heiß, Ingo Gühring, Martin Eigel.

http://jmlr.org/papers/v24/23-0421.html

#pdes #solvers #deep

'Neural Q-learning for solving PDEs', by Samuel N. Cohen, Deqing Jiang, Justin Sirignano.

http://jmlr.org/papers/v24/22-1075.html

#pdes #pde #nonlinear

`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

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

Deep Operator Learning Lessens the Curse of Dimensionality for PDEs

https://openreview.net/forum?id=zmBFzuT2DN

#pdes #pde #discretization

Beautiful PDE visualization tool!

Here's a reaction-diffusion system with a pic of Turing himself as the initial condition.

https://visualpde.com/sim/?preset=Alan

#Mathematics #DifferentialEquations #PDEs #ODEs

Learning to correct spectral methods for simulating turbulent flows

Gideon Dresdner, Dmitrii Kochkov, Peter Christian Norgaard et al.

Action editor: Ivan Oseledets.

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

#turbulent #spectral #pdes

'Neural Operator: Learning Maps Between Function Spaces With Applications to PDEs', by Nikola Kovachki et al.

http://jmlr.org/papers/v24/21-1524.html

#discretization #operators #pdes

'Learning Partial Differential Equations in Reproducing Kernel Hilbert Spaces', by George Stepaniants.

http://jmlr.org/papers/v24/21-1363.html

#regression #pdes #kernel

Are you keen to read about results in #MathematicalPhysics and Analysis of #PDEs? Take a look at the paper of our previous workshop participant! 🧐

#FunctionalAnalysis #SpectralTheory
@univienna

https://arxiv.org/pdf/2303.04527.pdf

#SoftMatter have just published the results of a project that Renato Assante, Davide Marenduzzo, Alexander Morozov, and I recently worked on together! What did we do and what’s new? Briefly…

#Microswimmer suspensions behave in a similar way to fluids containing kinesin and microtubules. Both systems can be described by the same system of three coupled nonlinear #PDEs.

A #LinearStabilityAnalysis of these equations suggests that variations in concentration across the system don’t significantly affect emergent #phaseBehaviour. How then can we explain #experiments that show visible inhomogeneities in #microtubule–#kinesin mixtures, for instance?

With increasing activity, we move away from the quiescent regime, past the onset of #SpontaneousFlow, and deeper into the active phase, where #nonlinearities become more important. What role do concentration inhomogeneities play here?

We investigated these questions, taking advantage of the #openSource #Dedalus #spectral framework to simulate the full nonlinear time evolution. This led us to predict a #novel regime of #spontaneous #microphaseSeparation into active (nematically ordered) and passive domains.

Active flow arrests macrophase separation in this regime, counteracting domain coarsening due to thermodynamic coupling between active matter concentration and #nematic order. As a result, domains reach a characteristic size that decreases with increasing activity.

This regime is one part of the #PhaseDiagram we mapped out. Along with our other findings, you can read all about it here!

low #ReynoldsNumber #turbulence #ActiveTurbulence #CahnHilliard #ActiveMatter #NavierStokes #BerisEdwards #CondensedMatter #PhaseTransitions #TheoreticalPhysics #BioPhysics #StatisticalPhysics #FluidDynamics #ComputationalPhysics #Simulation #FieldTheory #paperthread #NewPaper #science #research #ActiveGel #activeNematic #analytic #cytoskeleton #hydrodynamics #MPI #theory

How do Maxwell's equations predict that the speed of light is constant?🤔 (photo credit: Fermat's Library)

Maxwell's equations, a set of coupled partial differential equations (PDEs), describe the behaviour of electric and magnetic fields and predict that the speed of light is constant in all reference frames, which is a fundamental principle of the theory of relativity. #Electromagnetism #Relativity #SpeedOfLight #LightSpeed #Light #ElectricField #MagneticField #PDEs #Maxwell #CoupledPDEs

#SoftMatter have just published the results of a project that Renato Assante, Davide Marenduzzo, Alexander Morozov, and I recently worked on together! What did we do and what’s new? Briefly…

The #hydrodynamic behaviour of inhomogeneous #activeNematic gels (such as extensile bundles of #cytoskeletal filaments or suspensions of low #ReynoldsNumber swimmers) can be described by the time evolution of three coupled #PDEs.

Standard #ActiveGel #theory concludes, from a #LinearStabilityAnalysis of these equations, that fluctuations in concentration don’t significantly affect emergent #phaseBehaviour. However, this leaves #experimental #observations of visible inhomogeneities in #microtubule–#kinesin mixtures unexplained. As we move away from the passive (quiescent) regime, past the onset of #SpontaneousFlow, and deeper into the active phase, #nonlinearities become more important. What role do concentration inhomogeneities play here?

Alongside #analytic techniques, we used an in-house #MPI-parallel code developed within the #Dedalus #spectral framework to investigate. We predict a #novel regime of #spontaneous #microphaseSeparation into active (nematically ordered) and passive domains. In this regime, active flow arrests macrophase separation, which is itself driven by the thermodynamic coupling between active matter concentration and #nematic order. As a result, domains do not #coarsen past a typical size, which decreases with increasing activity. This regime is one part of the #PhaseDiagram we mapped out.

Along with our other findings, you can read all about it here!

#CahnHilliard #ActiveMatter #NavierStokes #BerisEdwards #CondensedMatter #PhaseTransitions #TheoreticalPhysics #BioPhysics #StatisticalPhysics #FluidDynamics #ComputationalPhysics #Simulation #FieldTheory #paperthread #NewPaper #science #research