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High performance differential equation solvers for ordinary differential equations, including neural ordinary differential equations (neural ODEs) and scientific machine learning (SciML)

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OrdinaryDiffEq.jl is a component package in the DifferentialEquations ecosystem. It holds the ordinary differential equation solvers and utilities. While completely independent and usable on its own, users interested in using this functionality should check out DifferentialEquations.jl.


Assuming that you already have Julia correctly installed, it suffices to import OrdinaryDiffEq.jl in the standard way:

import Pkg; Pkg.add("OrdinaryDiffEq")


OrdinaryDiffEq.jl is part of the SciML common interface, but can be used independently of DifferentialEquations.jl. The only requirement is that the user passes an OrdinaryDiffEq.jl algorithm to

. For example, we can solve the ODE tutorial from the docs using the
using OrdinaryDiffEq
f(u,p,t) = 1.01*u
tspan = (0.0,1.0)
prob = ODEProblem(f,u0,tspan)
sol = solve(prob,Tsit5(),reltol=1e-8,abstol=1e-8)
using Plots
plot(sol,linewidth=5,title="Solution to the linear ODE with a thick line",
     xaxis="Time (t)",yaxis="u(t) (in μm)",label="My Thick Line!") # legend=false
plot!(sol.t, t->0.5*exp(1.01t),lw=3,ls=:dash,label="True Solution!")

That example uses the out-of-place syntax

, while the inplace syntax (more efficient for systems of equations) is shown in the Lorenz example:
using OrdinaryDiffEq
function lorenz(du,u,p,t)
 du[1] = 10.0(u[2]-u[1])
 du[2] = u[1]*(28.0-u[3]) - u[2]
 du[3] = u[1]*u[2] - (8/3)*u[3]
u0 = [1.0;0.0;0.0]
tspan = (0.0,100.0)
prob = ODEProblem(lorenz,u0,tspan)
sol = solve(prob,Tsit5())
using Plots; plot(sol,vars=(1,2,3))

Very fast static array versions can be specifically compiled to the size of your model. For example:

using OrdinaryDiffEq, StaticArrays
function lorenz(u,p,t)
 SA[10.0(u[2]-u[1]),u[1]*(28.0-u[3]) - u[2],u[1]*u[2] - (8/3)*u[3]]
u0 = SA[1.0;0.0;0.0]
tspan = (0.0,100.0)
prob = ODEProblem(lorenz,u0,tspan)
sol = solve(prob,Tsit5())

For "refined ODEs", like dynamical equations and

s, refer to the DiffEqDocs. For example, in DiffEqTutorials.jl we show how to solve equations of motion using symplectic methods:
function HH_acceleration(dv,v,u,p,t)
    x,y  = u
    dx,dy = dv
    dv[1] = -x - 2x*y
    dv[2] = y^2 - y -x^2
initial_positions = [0.0,0.1]
initial_velocities = [0.5,0.0]
prob = SecondOrderODEProblem(HH_acceleration,initial_velocities,initial_positions,tspan)
sol2 = solve(prob, KahanLi8(), dt=1/10);

Other refined forms are IMEX and semi-linear ODEs (for exponential integrators).

Available Solvers

For the list of available solvers, please refer to the DifferentialEquations.jl ODE Solvers, Dynamical ODE Solvers, and the Split ODE Solvers pages.

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