ForwardDiff2 = ForwardDiff.jl + ChainRules.jl + Struct of arrays
User API:
D(f)(x) returns a lazy representation of the derivative.
D(f)(x) * v computes df(x)/dx ⋅ v, taking advantage of the laziness in D(f)(x).
DI(f)(x) is a convenience function to materialize the derivative, gradient or
Jacobian of f at x.
julia> using Random; Random.seed!(123);
julia> using ForwardDiff2: D, DI; using LinearAlgebra
julia> D(sin)(10) * 11 === cos(10) * 11
true
julia> x = rand(3);
julia> v = rand(3);
julia> D(prod)(x) # lazy gradient
D(prod)([0.768448, 0.940515, 0.673959])
julia> D(prod)(x) * I # materialize the gradient by multiplying by I
1×3 Adjoint{Float64,SArray{Tuple{3},Float64,1,3}} with indices SOneTo(1)×SOneTo(3):
0.633868 0.517902 0.722737
julia> D(cumsum)(x) * I # Jacobian
3×3 Adjoint{Float64,Array{Float64,2}}:
1.0 0.0 0.0
1.0 1.0 0.0
1.0 1.0 1.0
julia> D(cumsum)(x) * v # Jacobian-vector product
3-element Array{Float64,1}:
0.3954531123351086
0.7086970681426272
1.3712518846162807
julia> DI(DI(prod))(x) # Hessian
3×3 SArray{Tuple{3,3},Float64,2,9} with indices SOneTo(3)×SOneTo(3):
0.0 0.673959 0.940515
0.673959 0.0 0.768448
0.940515 0.768448 0.0
julia> DI(DI(prod))(x) * v # Hessian-vector product
3-element SArray{Tuple{3},Float64,1,3} with indices SOneTo(3):
0.8342562312269415
0.775657771761718
0.6126411738403423Note that ForwardDiff2.jl also works with ModelingToolkit.jl:
julia> using ModelingToolkit
julia> @variables x[1:3] v[1:3]
(Operation[x₁, x₂, x₃], Operation[v₁, v₂, v₃])
julia> D(sin)(x[1]) * 11
cos(x₁) * 11
julia> D(prod)(x) * I # gradient
1×3 Adjoint{Operation,StaticArrays.SArray{Tuple{3},Operation,1,3}} with indices SOneTo(1)×SOneTo(3):
conj((identity(1) * x₂ + x₁ * identity(0)) * x₃ + (x₁ * x₂) * identity(0)) … conj((identity(0) * x₂ + x₁ * identity(0)) * x₃ + (x₁ * x₂) * 1)
julia> D(cumsum)(x) * I # Jacobian
3×3 Adjoint{Operation,Array{Expression,2}}:
conj(1) conj(identity(0)) conj(identity(0))
conj(identity(0) + 1) conj(1 + identity(0)) conj(identity(0) + identity(0))
conj((identity(0) + identity(0)) + 1) conj((identity(0) + 1) + identity(0)) conj((1 + identity(0)) + identity(0))
julia> D(cumsum)(x) * [1, 2, 3] # Jacobian-vector product
3-element Array{Int64,1}:
1
3
6
julia> D(cumsum)(x) * v # Jacobian-vector product
3-element Array{Operation,1}:
v₁
v₂ + v₁
(v₃ + v₂) + v₁
julia> DI(DI(prod))(x) # Hessian
3×3 StaticArrays.SArray{Tuple{3,3},Operation,2,9} with indices SOneTo(3)×SOneTo(3):
conj(((identity(1) * x₂ + x₁ * identity(0)) * identity(identity(0)) + (x₁ * x₂) * 0) + ((identity(1) * x₂ + x₁ * identity(0)) * identity(0) + x₃ * ((identity(1) * identity(identity(0)) + x₁ * 0) + (identity(1) * identity(0) + x₂ * 0)))) … conj(((identity(0) * x₂ + x₁ * identity(0)) * identity(identity(0)) + (x₁ * x₂) * 0) + ((identity(1) * x₂ + x₁ * identity(0)) * identity(1) + x₃ * ((identity(0) * identity(identity(0)) + x₁ * 0) + (identity(1) * identity(0) + x₂ * 0))))
conj(((identity(1) * x₂ + x₁ * identity(0)) * identity(identity(0)) + (x₁ * x₂) * 0) + ((identity(identity(0)) * x₂ + x₁ * 1) * identity(0) + x₃ * ((identity(1) * identity(1) + x₁ * 0) + (identity(identity(0)) * identity(0) + x₂ * 0)))) conj(((identity(0) * x₂ + x₁ * identity(0)) * identity(identity(0)) + (x₁ * x₂) * 0) + ((identity(identity(0)) * x₂ + x₁ * 1) * identity(1) + x₃ * ((identity(0) * identity(1) + x₁ * 0) + (identity(identity(0)) * identity(0) + x₂ * 0))))
conj(((identity(1) * x₂ + x₁ * identity(0)) * identity(1) + (x₁ * x₂) * 0) + ((identity(identity(0)) * x₂ + x₁ * identity(0)) * identity(0) + x₃ * ((identity(1) * identity(identity(0)) + x₁ * 0) + (identity(identity(0)) * identity(0) + x₂ * 0)))) conj(((identity(0) * x₂ + x₁ * identity(0)) * identity(1) + (x₁ * x₂) * 0) + ((identity(identity(0)) * x₂ + x₁ * identity(0)) * identity(1) + x₃ * ((identity(0) * identity(identity(0)) + x₁ * 0) + (identity(identity(0)) * identity(0) + x₂ * 0))))
julia> DI(DI(prod))(x) * v # Hessian-vector product
3-element StaticArrays.SArray{Tuple{3},Operation,1,3} with indices SOneTo(3):
(conj(((identity(1) * x₂ + x₁ * identity(0)) * identity(identity(0)) + (x₁ * x₂) * 0) + ((identity(1) * x₂ + x₁ * identity(0)) * identity(0) + x₃ * ((identity(1) * identity(identity(0)) + x₁ * 0) + (identity(1) * identity(0) + x₂ * 0)))) * v₁ + conj(((identity(0) * x₂ + x₁ * 1) * identity(identity(0)) + (x₁ * x₂) * 0) + ((identity(1) * x₂ + x₁ * identity(0)) * identity(0) + x₃ * ((identity(0) * identity(identity(0)) + x₁ * 0) + (identity(1) * identity(1) + x₂ * 0)))) * v₂) + conj(((identity(0) * x₂ + x₁ * identity(0)) * identity(identity(0)) + (x₁ * x₂) * 0) + ((identity(1) * x₂ + x₁ * identity(0)) * identity(1) + x₃ * ((identity(0) * identity(identity(0)) + x₁ * 0) + (identity(1) * identity(0) + x₂ * 0)))) * v₃
(conj(((identity(1) * x₂ + x₁ * identity(0)) * identity(identity(0)) + (x₁ * x₂) * 0) + ((identity(identity(0)) * x₂ + x₁ * 1) * identity(0) + x₃ * ((identity(1) * identity(1) + x₁ * 0) + (identity(identity(0)) * identity(0) + x₂ * 0)))) * v₁ + conj(((identity(0) * x₂ + x₁ * 1) * identity(identity(0)) + (x₁ * x₂) * 0) + ((identity(identity(0)) * x₂ + x₁ * 1) * identity(0) + x₃ * ((identity(0) * identity(1) + x₁ * 0) + (identity(identity(0)) * identity(1) + x₂ * 0)))) * v₂) + conj(((identity(0) * x₂ + x₁ * identity(0)) * identity(identity(0)) + (x₁ * x₂) * 0) + ((identity(identity(0)) * x₂ + x₁ * 1) * identity(1) + x₃ * ((identity(0) * identity(1) + x₁ * 0) + (identity(identity(0)) * identity(0) + x₂ * 0)))) * v₃
(conj(((identity(1) * x₂ + x₁ * identity(0)) * identity(1) + (x₁ * x₂) * 0) + ((identity(identity(0)) * x₂ + x₁ * identity(0)) * identity(0) + x₃ * ((identity(1) * identity(identity(0)) + x₁ * 0) + (identity(identity(0)) * identity(0) + x₂ * 0)))) * v₁ + conj(((identity(0) * x₂ + x₁ * 1) * identity(1) + (x₁ * x₂) * 0) + ((identity(identity(0)) * x₂ + x₁ * identity(0)) * identity(0) + x₃ * ((identity(0) * identity(identity(0)) + x₁ * 0) + (identity(identity(0)) * identity(1) + x₂ * 0)))) * v₂) + conj(((identity(0) * x₂ + x₁ * identity(0)) * identity(1) + (x₁ * x₂) * 0) + ((identity(identity(0)) * x₂ + x₁ * identity(0)) * identity(1) + x₃ * ((identity(0) * identity(identity(0)) + x₁ * 0) + (identity(identity(0)) * identity(0) + x₂ * 0)))) * v₃Planned features:
- works both on GPU and CPU
- Dual cache
- user-extensible scalar and tensor derivative definitions
- in-place function
- sparsity exploitation (color vector support)
- complex differentiation (use
ChainRules.jl) - plays nicely with Zygote
The ForwardDiff2 source code follows the YASGuide.