Passthrough AD in ode with complex numbers

[AmitRotem]

see issue #858

[AmitRotem]

Seems that real(eltype(u0)) used for the complex case fails on other types, like unitful quantity.
Specialized promote_u0 for u0::AbstractArray{Complex}

[AmitRotem]

This should solve the complex ad test.
Not sure what is the problem with the Sundails.jl test.
Nor the format test.

[ChrisRackauckas]

The Sundials thing is unrelated. The formatting failure is because it's not following the standard formatting rules https://docs.sciml.ai/SciMLStyle/stable/, but I'll just run the formatter after merging. In the future, remember to do:

using JuliaFormatter, SomePackage
format(joinpath(dirname(pathof(SomePackage)), ".."))

[AmitRotem]

Found a small issue with complex numbers. The Dual of time in the ode function has NaN values. Meaning, in the ode function (u,p,t) -> ... t is Dual with derivative NaN
Maybe a problem with the promotion of tspan ? affecting the initial dt value ?

Example;

using LinearAlgebra, DifferentialEquations
import ForwardDiff as FD

H0 = randn(2,2)
u0 = [1.0, 0im]
Ht(u,p,t) = (H0*u)*cos(p*t)
prob0 = ODEProblem(Ht, u0, (0.0,1.0))

function loss(p)
    prob = remake(prob0; p)
    sol = solve(prob)
    lo = abs2(tr(first(sol.u)'last(sol.u)))
    lo
end

loss(rand())
FD.derivative(loss, rand()) # NaN's

This does not show up in other cases, such as;

• if the initial difference is small; e.g., Ht(u,p,t) = (H0*u)*sin(p*t)
• other random initial state; u0 = randn(ComplexF64,2)
• real initial state u0 = [1.0, 0.0]
• use some finite dt in ODEProblem(...; dt=1)

Also nan safe mode for ForwardDiff solve this.

[ChrisRackauckas]

Open a new issue for this.