Extensibility of the language

DifferentialEquations

A package for solving differential equations, similar to odesolve in Matlab.

Example:

using DifferentialEquations, Plots

function lotka_volterra(du,u,p,t)
  x, y = u
  α, β, δ, γ = p
  du[1] = dx = α*x - β*x*y
  du[2] = dy = -δ*y + γ*x*y
end

u0 = [1.0,1.0]
tspan = (0.0,10.0)
p = [1.5,1.0,3.0,1.0]
prob = ODEProblem(lotka_volterra,u0,tspan,p)

sol = solve(prob)
plot(sol)

Measurements

A package defining "numbers with precision" and complete algebra on these numbers:

using Measurements

a = 4.5 ± 0.1
b = 3.8 ± 0.4

2a + b
sin(a)/cos(a) - tan(a)

It also defines recipes for Plots.jl how to plot such numbers.

Starting ODE from an interval

u0 = [1.0±0.1,1.0±0.01]

prob = ODEProblem(lotka_volterra,u0,tspan,p)
sol = solve(prob)
plot(sol,denseplot=false)

• all algebraic operations are defined,

• passes all grid refinement techniques

• plot uses the correct plotting for intervals

Who wrote the code connecting Measurements and DifferentialEquations?

Would this work with scipy.integrate.solve_ivp?

Integration with other toolkits

Flux: toolkit for modelling Neural Networks. Neural network is a function.

• integration with Measurements,

• integration with ODE (think of NN as part of the ODE)

Turing: Probabilistic modelling toolkit

• integration with Flux (NN)

• integration with ODE

• using arbitrary bijective transformations, Bijectors.jl

Back to the optimization example

optimize(z -> P(z...), z₀, Newton(); autodiff = AutoForwardDiff())

How does Optim get the gradient of P?

Hint: ForwardDiff defines a new number type, just like Measurements.

Julia from user's point of view

1. compilation of everything to as specialized as possible

   • ✅ very fast code

   • ❌ slow interaction (caching..., much improved since 1.9)

   • ❌ generating libraries is harder

     • think of fsum,

     • everything is ".h" (Eigen library)

     • less concern with standalone executable/library compilation, Application binary interface (ABI), experimental --trim since 1.12

   • ❌ debugging is different to matlab/python

2. extensibility, Multiple dispatch = multi-functions

   • ✅ allows great extensibility and code composition

   • not (yet) mainstream thinking

   • Julia is not Object-oriented

   • Julia is (not pure) functional language

   • less concern with public/private separation (public keyword since 1.11)