Inconsistent Gradients between ForwardDiff and Zygote, MTK and Manual

[AlCap23]

Describe the bug 🐞

All gradients computed with ForwardDiff are zero when the underlying ODE is built with ModelingToolkit.
When a manual reimplementation is used, ForwardDiff returns gradients but differs from Zygote in a magnitude I would not expect. Zygote is consistent between the two cases.

Expected behavior

Consistency between MTK & Manual model using ForwardDiff. Possibly also between ForwardDiff and Zygote, but this could be within numerical accuracy.

Minimal Reproducible Example 👇

Without MRE, we would only be able to help you to a limited extent, and attention to the issue would be limited. to know more about MRE refer to wikipedia and stackoverflow.

# MWE 
using ModelingToolkit 
using OrdinaryDiffEq 
using SciMLStructures
using SciMLSensitivity
using StableRNGs
using ForwardDiff 
using Zygote 

import ModelingToolkit: t_nounits as t, D_nounits as D

## Data Generator 
@mtkmodel Lotka begin
    @variables begin
        x(t) = 1.0, [description = "Prey"]
        y(t) = 1.0, [description = "Predator"]
    end
    @parameters begin
        α = 1.5
        β = 1.0
        γ = 3.0
        δ = 1.0
    end
    @equations begin
        D(x) ~ (α - β * y) * x
        D(y) ~ (δ * x - γ) * y
    end
end


@mtkcompile lotka = Lotka()
problem = ODEProblem(lotka, [], (0.0, 10.0))
solution = solve(problem, Tsit5(), saveat=0.1)
rng = StableRNG(42)
data = (;
    t=solution.t,
    # [[y, x], :]
    measurements=Array(solution)
)
data.measurements .+= 0.05 * randn(rng, size(data.measurements))
#Debug
#f = plot(solution)
#scatter!(data.t, data.measurements[1, :])
#scatter!(data.t, data.measurements[2, :])
#display(f)

p0, repack, _ = SciMLStructures.canonicalize(SciMLStructures.Tunable(), problem.p)

objective = let  repack = repack, problem = problem 
    (p, data) -> begin
        pnew = repack(p)
        sol = solve(problem, Tsit5(), p = pnew, saveat = data.t)
        sum(abs2, sol .- data.measurements) / size(data.t, 1) 
    end 
end

# Check ≈ 0.005116418356342287 
objective(p0, data)

# Zero 
ForwardDiff.gradient(Base.Fix2(objective, data), p0)
# Not zero ([-0.21477822120509513, -0.00824858062706199, -0.13853338985792157, -0.03034598558542432],) 
Zygote.gradient(Base.Fix2(objective, data), p0)


# Non MTK version
function lv(u, p, t)
    [
        p[1] * u[1] - p[2] * u[1] * u[2],
        p[3] * u[1]*u[2] - p[4] * u[2]
    ]
end 

problem = ODEProblem(lv, [1., 1.], (0., 10.), [1.5, 1.0, 1.0, 3.0])

solution = solve(problem, Tsit5(), saveat=0.1)
# Debug
#f = plot(solution)
#scatter!(data.t, data.measurements[2, :])
#scatter!(data.t, data.measurements[1, :])
#display(f)

# For consistency
p0, repack, _ = SciMLStructures.canonicalize(SciMLStructures.Tunable(), problem.p)

objective = let  repack = repack, problem = problem
    (p, data) -> begin
        # Leaving this out changes nothing
        pnew = repack(p)
        # Switching to other Integrators, e.g. Vern7, does not improve
        sol = solve(problem, Tsit5(), p = pnew, saveat = data.t)
        sum(abs2, Array(sol) .- data.measurements[[2,1], :]) / size(data.t, 1) 
    end 
end


# Check ≈ 0.005116418356341537
objective(p0, data)

# Not Zero [ -0.3699284859839662, -0.011463508689489492, -0.2398301997277094, -0.0477244292296241] 
g1 = ForwardDiff.gradient(Base.Fix2(objective, data), p0)
# Not zero ([-0.21477822120392856, -0.008248580627071152, -0.13853338985723695, -0.03034598558526613],)
g2 = Zygote.gradient(Base.Fix2(objective, data), p0)
# Diff [ -0.15515026478003766, -0.00321492806241834, -0.10129680987047243, -0.017378443644357974]
g1 .- g2[1]

# MWE 2 
using ModelingToolkit 
using OrdinaryDiffEq 
using SciMLStructures
using SciMLSensitivity
using StableRNGs
using ForwardDiff 
using Zygote 
import ModelingToolkit: t_nounits as t, D_nounits as D

## Data Generator 
@mtkmodel Linear begin
    @variables begin
        x(t) = 1.0, [description = "Prey"]
    end
    @parameters begin
        α = 1.5
    end
    @equations begin
        D(x) ~ -α * x
    end
end


@mtkcompile linear = Linear()
problem = ODEProblem(linear, [], (0.0, 1.0))
solution = solve(problem, Tsit5(), saveat=0.1)
rng = StableRNG(42)
data = (;
    t=solution.t,
    # [[y, x], :]
    measurements=Array(solution)
)
data.measurements .+= 0.05 * randn(rng, size(data.measurements))
#Debug
#f = plot(solution)
#scatter!(data.t, data.measurements[1, :])
#scatter!(data.t, data.measurements[2, :])
#display(f)

p0, repack, _ = SciMLStructures.canonicalize(SciMLStructures.Tunable(), problem.p)

objective = let  repack = repack, problem = problem 
    (p, data) -> begin
        pnew = repack(p)
        sol = solve(problem, Tsit5(), p = pnew, saveat = data.t)
        sum(abs2, sol .- data.measurements) / size(data.t, 1) 
    end 
end

# Check 0.0031677344878386607 
objective(p0, data)

# Zero 
ForwardDiff.gradient(Base.Fix2(objective, data), p0)
# Not zero ([0.005474983463028785],) 
Zygote.gradient(Base.Fix2(objective, data), p0)


# Non MTK version
function linsys(u, p, t)
    [
        -p[1] * u[1] 
    ]
end 

problem = ODEProblem(linsys, [1.,], (0., 1.), [1.5])

solution = solve(problem, Tsit5(), saveat=0.1)
# Debug
#f = plot(solution)
#scatter!(data.t, data.measurements[2, :])
#scatter!(data.t, data.measurements[1, :])
#display(f)

# For consistency
p0, repack, _ = SciMLStructures.canonicalize(SciMLStructures.Tunable(), problem.p)

objective = let  repack = repack, problem = problem
    (p, data) -> begin
        pnew = repack(p)
        # Switching to other Integrators, e.g. Vern7, does not improve
        sol = solve(problem, Tsit5(), p = pnew, saveat = data.t)
        sum(abs2, Array(sol) .- data.measurements) / size(data.t, 1) 
    end 
end


# Check ≈ 0.0031677344878386637
objective(p0, data)

# Not Zero [0.005474949348877295] 
g1 = ForwardDiff.gradient(Base.Fix2(objective, data), p0)
# Not zero ([0.0054749834630287535],)
g2 = Zygote.gradient(Base.Fix2(objective, data), p0)
# Diff [-3.4114151458569664e-8]
g1 .- g2[1]

Error & Stacktrace ⚠️

Not applicable.

Environment (please complete the following information):

• Output of using Pkg; Pkg.status()

Status `~/MWEGradients/Project.toml`
⌅ [f6369f11] ForwardDiff v0.10.38
  [961ee093] ModelingToolkit v10.2.0
  [1dea7af3] OrdinaryDiffEq v6.98.0
  [1ed8b502] SciMLSensitivity v7.84.0
  [53ae85a6] SciMLStructures v1.7.0
  [860ef19b] StableRNGs v1.0.3
  [e88e6eb3] Zygote v0.7.9
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated`

• Output of using Pkg; Pkg.status(; mode = PKGMODE_MANIFEST)

Status `~/MWEGradients/Manifest.toml`
  [47edcb42] ADTypes v1.14.0
  [621f4979] AbstractFFTs v1.5.0
  [1520ce14] AbstractTrees v0.4.5
  [7d9f7c33] Accessors v0.1.42
  [79e6a3ab] Adapt v4.3.0
  [66dad0bd] AliasTables v1.1.3
  [ec485272] ArnoldiMethod v0.4.0
  [4fba245c] ArrayInterface v7.19.0
  [4c555306] ArrayLayouts v1.11.1
  [a9b6321e] Atomix v1.1.1
⌅ [e2ed5e7c] Bijections v0.1.10
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  [163ba53b] DiffResults v1.1.0
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⌅ [a0c0ee7d] DifferentiationInterface v0.6.54
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  [ffbed154] DocStringExtensions v0.9.5
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⌅ [f6369f11] ForwardDiff v0.10.38
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  [77dc65aa] FunctionWrappersWrappers v0.1.3
  [d9f16b24] Functors v0.5.2
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  [61eb1bfa] GPUCompiler v1.5.2
  [c145ed77] GenericSchur v0.5.5
  [c27321d9] Glob v1.3.1
  [86223c79] Graphs v1.13.0
  [076d061b] HashArrayMappedTries v0.2.0
  [34004b35] HypergeometricFunctions v0.3.28
  [7869d1d1] IRTools v0.4.14
  [615f187c] IfElse v0.1.1
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  [18e54dd8] IntegerMathUtils v0.1.2
  [8197267c] IntervalSets v0.7.11
  [3587e190] InverseFunctions v0.1.17
  [92d709cd] IrrationalConstants v0.2.4
  [82899510] IteratorInterfaceExtensions v1.0.0
  [692b3bcd] JLLWrappers v1.7.0
  [ae98c720] Jieko v0.2.1
  [98e50ef6] JuliaFormatter v2.1.2
⌅ [70703baa] JuliaSyntax v0.4.10
  [ccbc3e58] JumpProcesses v9.16.0
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  [1dea7af3] OrdinaryDiffEq v6.98.0
  [89bda076] OrdinaryDiffEqAdamsBashforthMoulton v1.2.0
  [6ad6398a] OrdinaryDiffEqBDF v1.6.0
  [bbf590c4] OrdinaryDiffEqCore v1.26.1
  [50262376] OrdinaryDiffEqDefault v1.4.0
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  [9286f039] OrdinaryDiffEqExplicitRK v1.1.0
  [e0540318] OrdinaryDiffEqExponentialRK v1.4.0
  [becaefa8] OrdinaryDiffEqExtrapolation v1.5.0
  [5960d6e9] OrdinaryDiffEqFIRK v1.12.0
  [101fe9f7] OrdinaryDiffEqFeagin v1.1.0
  [d3585ca7] OrdinaryDiffEqFunctionMap v1.1.1
  [d28bc4f8] OrdinaryDiffEqHighOrderRK v1.1.0
  [9f002381] OrdinaryDiffEqIMEXMultistep v1.3.0
  [521117fe] OrdinaryDiffEqLinear v1.3.0
  [1344f307] OrdinaryDiffEqLowOrderRK v1.2.0
  [b0944070] OrdinaryDiffEqLowStorageRK v1.3.0
  [127b3ac7] OrdinaryDiffEqNonlinearSolve v1.10.0
  [c9986a66] OrdinaryDiffEqNordsieck v1.1.0
  [5dd0a6cf] OrdinaryDiffEqPDIRK v1.3.1
  [5b33eab2] OrdinaryDiffEqPRK v1.1.0
  [04162be5] OrdinaryDiffEqQPRK v1.1.0
  [af6ede74] OrdinaryDiffEqRKN v1.1.0
  [43230ef6] OrdinaryDiffEqRosenbrock v1.11.0
  [2d112036] OrdinaryDiffEqSDIRK v1.3.0
  [669c94d9] OrdinaryDiffEqSSPRK v1.3.0
  [e3e12d00] OrdinaryDiffEqStabilizedIRK v1.3.0
  [358294b1] OrdinaryDiffEqStabilizedRK v1.1.0
  [fa646aed] OrdinaryDiffEqSymplecticRK v1.3.0
  [b1df2697] OrdinaryDiffEqTsit5 v1.1.0
  [79d7bb75] OrdinaryDiffEqVerner v1.2.0
  [90014a1f] PDMats v0.11.35
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⌅ [aea7be01] PrecompileTools v1.2.1
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  [ae029012] Requires v1.3.1
  [ae5879a3] ResettableStacks v1.1.1
  [37e2e3b7] ReverseDiff v1.16.1
  [79098fc4] Rmath v0.8.0
  [7e49a35a] RuntimeGeneratedFunctions v0.5.15
  [9dfe8606] SCCNonlinearSolve v1.2.0
  [94e857df] SIMDTypes v0.1.0
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  [c0aeaf25] SciMLOperators v1.3.1
  [431bcebd] SciMLPublic v1.0.0
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  [53ae85a6] SciMLStructures v1.7.0
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  [b77e0a4c] InteractiveUtils
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  [56ddb016] Logging
  [d6f4376e] Markdown
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  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
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  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.52.0+1
  [3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`

• Output of versioninfo()

Julia Version 1.10.9
Commit 5595d20a287 (2025-03-10 12:51 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: macOS (arm64-apple-darwin24.0.0)
  CPU: 11 × Apple M3 Pro
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, apple-m1)
Threads: 1 default, 0 interactive, 1 GC (on 5 virtual cores)
Environment:
  JULIA_EDITOR = nvim

**Additional context✱

Edit Added information for manual model with and without SciMLStructures.
Edit Added simpler MWE

[ChrisRackauckas]

@AayushSabharwal issues with the promotions?

[AlCap23]

I've added another simpler MWE. When using Zygote, the gradients match, which could indicate numerical inaccuracy.

[AlCap23]

Incrementally downgrading MTK to 9.80 did not help (didn't account for patch releases, though ).
Same for SciMLSensitivity to 7.79 .

[AlCap23]

ReverseDiff also returns zeros.

[ChrisRackauckas]

Yes this is the test failure on master. It's not good 😅 this needs some priority.