#pdes — Public Fediverse posts

Today we are at the Julia4PDEs workshop, organized in collaboration with the @eScienceCenter for two days of talks about the various projects tackling partial differential equations in Julia.

#Julia #HPC #PDEs #GPU #Julia4PDEs

“Looking back, my path has not been linear, and I changed direction more than once; however, what has stayed constant is curiosity, even when the topics and the places were changing. This is also what I like most about mathematics: there is room for many different trajectories, as long as you keep following questions that genuinely interest you.” - Mikaela Iacobelli

➡️ https://hermathsstory.eu/mikaela-iacobelli

#Academia #PhD #AssociateProfessor #MathematicalPhysics #PDEs #KineticTheory #MathsJourney #WomenInSTEM #WomenInMaths #HerMathsStory

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

'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

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

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

On my way to Perth today for the Trilateral meeting on #nonlinear #PDEs …

The #WaveEquation in 2 spatial dimensions. A membrane stretched over a square frame is set to vibrating after some initial displacement. Warmer colors indicate greater displacement in the positive z-axis. #PDEs #mathtodon #math

#Poisson’s equation in 2 spatial dimensions. Heat is generated at each point within a rectangular plate at a uniform rate. The top and bottom edges are held constant at a certain temperature, while heat is lost out of the left and right edges; the boundary conditions are given in terms of both the temperature, and spatial derivatives of the temperature, along the edges of the plate. The solution gives the steady-state temperatures at each location within the plate. #PDEs #math

The #HeatEquation (left) and #LaplaceEquation (right) solved under the same boundary conditions over a square plate. Over time, the solution to the #HeatEquation approaches that of the #LaplaceEquation - in this case, giving the steady-state temperature distribution at each interior point of a square plate whose adjacent edges are held either at 100 degrees C or 0 degrees C (blue shows colder regions; red shows hotter regions; the plate is insulated so that heat does not escape).
#PDEs #math

Mastodon #introduction: TuxRiders is a journey to research experiences using free and #opensource scientific computing programs, which aimed to demonstrate their power for real-world scientific research.

In our #YouTube channel, we regularly talk about #FiniteElement, #EngineeringMath, #PDEs, #NumericalMethods, #FreeFEM, #FEniCS, #ParaView, #C++, #Python, #OpenFOAM, and #Linux.

Check out our YT channel: https://www.youtube.com/TuxRiders

For those interested, here is the poster for the #trilateral meeting on #nonlinear #PDEs in January 2023, in Perth, Western Australia

I’ll be in #Perth for the trilateral meeting on #nonlinear #PDEs in January, 2023. Anyone else going to be there?