Zygote gradient does not agree with finite difference

[dkweiss31]

I am simply re-running the code discussed in #938 and implemented as a test here https://github.com/SciML/SciMLSensitivity.jl/blob/d1ded754fe3b0d505d91c8ab5ea728a90be56df1/test/local_sensitivity/complex_adjoints.jl

using SciMLSensitivity, OrdinaryDiffEq, Zygote, LinearAlgebra, FiniteDiff, Test
A  = [1.0*im  2.0; 3.0 4.0]
u0 = [1.0 0.0*im; 0.0 1.0]
tspan = (0.0, 1.0)

function f(u,p,t)
    (A*u)*(p[1]*t + p[2]*t^2 + p[3]*t^3 + p[4]*t^4)
end

p = [1.5 + im, 1.0, 3.0, 1.0]
prob = ODEProblem{false}(f,u0,tspan,p)

utarget = [0.0*im 1.0; 1.0 0.0]

function loss_adjoint(p)
    ufinal = last(solve(prob, Tsit5(), p=p, abstol=1e-12, reltol=1e-12))
    loss = 1 - abs(tr(ufinal*utarget')/2)^2
    return loss
end

grad1 = Zygote.gradient(loss_adjoint,Complex{Float64}[1.5, 1.0, 3.0, 1.0])[1]
grad2 = FiniteDiff.finite_difference_gradient(loss_adjoint,Complex{Float64}[1.5, 1.0, 3.0, 1.0])
@test grad1 ≈ grad2

This appears not to be a complex variables issue as it fails even if all complex numbers are replaced with floats, e.g.

A  = [1.0  2.0; 3.0 4.0]
u0 = [1.0 0.0; 0.0 1.0]

p = [1.5, 1.0, 3.0, 1.0]

utarget = [0.0 1.0; 1.0 0.0]

grad1 = Zygote.gradient(loss_adjoint,[1.5, 1.0, 3.0, 1.0])[1]
grad2 = FiniteDiff.finite_difference_gradient(loss_adjoint,[1.5, 1.0, 3.0, 1.0])

In the former case I obtain

Test Failed at /Users/danielweiss/VScode_projects/JuliaQOC/JuliaQOC/src/main_2.jl:68
  Expression: grad1 ≈ grad2
   Evaluated: ComplexF64[0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im] ≈ ComplexF64[-9.601871252960366e8 + 3.389795607486498e7im, -6.401247503556682e8 + 2.2598637655570008e7im, -4.800935625331817e8 + 1.694897852876889e7im, -3.8407484806169194e8 + 1.3559182705553723e7im]
ERROR: There was an error during testing

In the latter case I obtain

Test Failed at /Users/danielweiss/VScode_projects/JuliaQOC/JuliaQOC/src/main_2.jl:68
  Expression: grad1 ≈ grad2
   Evaluated: [0.0, 0.0, 0.0, 0.0] ≈ [-3.1309304303724656e9, -2.0872869526760757e9, -1.565465215635736e9, -1.2523721720643353e9]
ERROR: There was an error during testing

I apologize in advance if I am doing something foolish, I am a relatively new Julia user.

Environment

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  [0ac62f75] libass_jll v0.15.1+0
  [2db6ffa8] libevdev_jll v1.11.0+0
  [f638f0a6] libfdk_aac_jll v2.0.2+0
  [36db933b] libinput_jll v1.18.0+0
  [b53b4c65] libpng_jll v1.6.40+0
  [f27f6e37] libvorbis_jll v1.3.7+1
  [009596ad] mtdev_jll v1.1.6+0
  [1270edf5] x264_jll v2021.5.5+0
  [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+1
  [0dad84c5] ArgTools
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8bb1440f] DelimitedFiles
  [8ba89e20] Distributed
  [f43a241f] Downloads
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions
  [44cfe95a] Pkg
  [de0858da] Printf
  [9abbd945] Profile
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML
  [a4e569a6] Tar
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll
  [deac9b47] LibCURL_jll
  [29816b5a] LibSSH2_jll
  [c8ffd9c3] MbedTLS_jll
  [14a3606d] MozillaCACerts_jll
  [4536629a] OpenBLAS_jll
  [05823500] OpenLibm_jll
  [efcefdf7] PCRE2_jll
  [bea87d4a] SuiteSparse_jll
  [83775a58] Zlib_jll
  [8e850b90] libblastrampoline_jll
  [8e850ede] nghttp2_jll
  [3f19e933] p7zip_jll

• Output of versioninfo()

Julia Version 1.7.3
Commit 742b9abb4d (2022-05-06 12:58 UTC)
Platform Info:
  OS: macOS (x86_64-apple-darwin21.4.0)
  CPU: Apple M1 Pro
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-12.0.1 (ORCJIT, westmere)
Environment:
  JULIA_EDITOR = code
  JULIA_NUM_THREADS = 

[ChrisRackauckas]

You left out the pretty fundamental detail that it prints out a warning that the backwards pass is unstable for the chosen solver. Indeed, you should not hardcode a solver and ignore the warnings about when it fails! If you heed the warning it works perfectly fine:

using SciMLSensitivity, OrdinaryDiffEq, Zygote, LinearAlgebra, FiniteDiff, Test
A  = [1.0*im  2.0; 3.0 4.0]
u0 = [1.0 0.0*im; 0.0 1.0]
tspan = (0.0, 1.0)

function f(u,p,t)
    (A*u)*(p[1]*t + p[2]*t^2 + p[3]*t^3 + p[4]*t^4)
end

p = [1.5 + im, 1.0, 3.0, 1.0]
prob = ODEProblem{false}(f,u0,tspan,p)

utarget = [0.0*im 1.0; 1.0 0.0]

function loss_adjoint(p)
    ufinal = last(solve(prob, VCABM(), p=p, abstol=1e-12, reltol=1e-12))
    loss = 1 - abs(tr(ufinal*utarget')/2)^2
    return loss
end

grad1 = Zygote.gradient(loss_adjoint,Complex{Float64}[1.5, 1.0, 3.0, 1.0])[1]
grad2 = FiniteDiff.finite_difference_gradient(loss_adjoint,Complex{Float64}[1.5, 1.0, 3.0, 1.0])
@test grad1 ≈ grad2 rtol=1e-5

The warning:

┌ Warning: Interrupted. Larger maxiters is needed. If you are using an integrator for non-stiff ODEs or an automatic switching algorithm (the default), you may want to consider using a method for stiff equations. See the solver pages for more details (e.g. https://docs.sciml.ai/DiffEqDocs/stable/solvers/ode_solve/#Stiff-Problems).
└ @ SciMLBase C:\Users\accou\.julia\packages\SciMLBase\Dwomw\src\integrator_interface.jl:600

Note that if the adjoint pass is interrupted you need to check the return code.