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Automatic differentiation through inverse of coordinate transforms
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using ForwardDiff: Dual, Tag, jacobian, value, tagtype, partials, gradient | |
using LinearAlgebra | |
using NLsolve | |
using Random | |
using Test | |
Random.seed!(0) | |
const jac = jacobian | |
valueise(x) = x | |
valueise(x::T) where {T<:Dual} = valueise(value(x)) | |
function changevalue!(x::AbstractVector, y, s) | |
for i in eachindex(x) | |
x[i] == s[i] && (x[i] = y[i]) | |
end | |
return x | |
end | |
function changevalue!(x::AbstractVector{<:Dual}, y, s) | |
for i in eachindex(x) | |
x[i] = changevalue!(x[i], y[i], s[i]) | |
end | |
return x | |
end | |
function changevalue!(x::Dual{<:Tag}, y, s) where {T<:AbstractFloat, D} | |
return Dual{tagtype(x)}(changevalue!(x.value, y, s), partials(x)) | |
end | |
function changevalue!(x::Dual{<:Tag{<:Function,T}, T, D}, y::T, s::T | |
) where {T<:AbstractFloat, D} | |
return x.value == s ? Dual{tagtype(x)}(y, partials(x)) : x | |
end | |
changevalue(x, y) = changevalue!(deepcopy(x), y, valueise.(x)) | |
""" | |
Given coordinate transform f(y) -> x, find y for given x. | |
""" | |
function inversion(f::F, x::AbstractVector{T}, initial_y=ones(size(x)) | |
) where {F, T} | |
f!(m, y) = (m .= f(y) .- valueise.(x); return nothing) | |
y = NLsolve.nlsolve(f!, initial_y, autodiff=:forward, ftol=1e-12).zero | |
# What is the proper way? This only works for a single AD pass. | |
return changevalue(x, y) | |
end | |
"""Cylindrical to Cartesian""" | |
function f(rθ) | |
r, θ = rθ | |
return [r * cos(θ), r * sin(θ)] | |
end | |
"""Cartesian to Cylindrical""" | |
function f⁻¹(xy) | |
x, y = xy | |
return [sqrt(x^2 + y^2), atan(y, x)] | |
end | |
# I want to work in curvlinear coordinates R | |
# For the real problem it's only tractable to write down `f⁻¹` | |
# so I use an inversion function to go from `R` to `X` to get useful | |
# things out of `f⁻¹` in lieu of having an `f` | |
@testset "ForwardDiff" begin | |
X = 5 * (rand(2) .- 0.5) | |
R = f⁻¹(X) | |
@assert f(R) ≈ X # otherwise it'll never work | |
@assert f⁻¹(X) ≈ R # otherwise it'll never work | |
@assert R[1] > 0 | |
@assert -π <= R[2] <= π | |
@test inversion(f⁻¹, R) ≈ X | |
@test inv(jac(f, R)) ≈ jac(f⁻¹, X) # yep | |
@test inv(jac(f, R)) ≈ jac(r->f⁻¹(inversion(f⁻¹, r)), R) | |
# now apply AD on top of the jacobian calculation | |
# create arbitrary-ish function | |
foo(R) = det(jac(f, R)) # get a single number so can do gradient | |
bar(R) = det(inv(jac(r->f⁻¹(inversion(f⁻¹, r)), R))) | |
expected = gradient(foo, R) | |
@test expected ≈ [1, 0] # easy answer | |
# gradient(bar, R) actually calculates d bar / dX so need dX/dR' d bar/dX | |
@test jac(r->f⁻¹(inversion(f⁻¹, r)), R) * gradient(bar, R) ≈ expected # nearly | |
end |
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