Time-derivative of ParameterizedInterpolation

**Describe the bug 🐞**

I have a model where I use the `ParameterizedInterpolation` block as input (with the intention to optimize the parameters later to fit experimental data). In my model, the interpolation appears under a time-derivative, which leads to the error described below.

**Expected behavior**

`ParameterizedInterpolation` can be differentiated in respect to time. Especially, if the input is just `t`.

**Minimal Reproducible Example 👇**

Slightly modified example from the doc. The change is `f` to `D(f)` in `ddx * 10 ~ k * x + d * dx + D(f)`. The original example works fine. This MWE gives the same stack trace as my original model.


```julia
using ModelingToolkit
using ModelingToolkit: t_nounits as t, D_nounits as D
using ModelingToolkitStandardLibrary.Blocks
using DataInterpolations
using OrdinaryDiffEq
using DataFrames
using Plots

function MassSpringDamper(; name)
  @named input = RealInput()
  vars = @variables f(t) x(t) = 0 dx(t) [guess = 0] ddx(t)
  pars = @parameters m = 10 k = 1000 d = 1

  eqs = [f ~ input.u
    ddx * 10 ~ k * x + d * dx + D(f)
    D(x) ~ dx
    D(dx) ~ ddx]

  ODESystem(eqs, t, vars, pars; name, systems=[input])
end

function MassSpringDamperSystem(data, time; name)
  @named src = ParametrizedInterpolation(LinearInterpolation, data, time)
  @named clk = ContinuousClock()
  @named model = MassSpringDamper()

  eqs = [connect(model.input, src.output)
    connect(clk.output, src.input)]

  ODESystem(eqs, t; name, systems=[src, clk, model])
end

function generate_data()
  dt = 4e-4
  time = 0:dt:0.1
  data = sin.(2 * pi * time * 100)

  return DataFrame(; time, data)
end

df = generate_data() # example data

@named system = MassSpringDamperSystem(df.data, df.time)
sys = structural_simplify(system)
prob = ODEProblem(sys, [], (0, df.time[end]))
sol = solve(prob)

```

**Error & Stacktrace ⚠️**

```julia
ERROR: MethodError: Cannot `convert` an object of type
  MTKParameters{Array{Float64,1},Array{Num,1},Tuple,Tuple,Tuple{Array{UnionAll,1}},Tuple} to an object of type
  MTKParameters{Array{Float64,1},Array{Float64,1},Tuple,Tuple,Tuple{Array{UnionAll,1}},Tuple}
The function `convert` exists, but no method is defined for this combination of argument types.

Closest candidates are:
  (::Type{MTKParameters{T, I, D, C, N, H}} where {T, I, D, C, N, H})(::Any, ::Any, ::Any, ::Any, ::Any, ::Any)
   @ ModelingToolkit ~/.julia/packages/ModelingToolkit/Z9mEq/src/systems/parameter_buffer.jl:7
  convert(::Type{T}, ::T) where T
   @ Base Base.jl:126

Stacktrace:
  [1] setproperty!(x::OrdinaryDiffEqCore.ODEIntegrator{…}, f::Symbol, v::MTKParameters{…})
    @ Base ./Base.jl:52
  [2] _initialize_dae!(integrator::OrdinaryDiffEqCore.ODEIntegrator{…}, prob::ODEProblem{…}, alg::SciMLBase.OverrideInit{…}, isinplace::Val{…})
    @ OrdinaryDiffEqCore ~/.julia/packages/OrdinaryDiffEqCore/AYNr9/src/initialize_dae.jl:162
  [3] _initialize_dae!
    @ ~/.julia/packages/OrdinaryDiffEqCore/AYNr9/src/initialize_dae.jl:50 [inlined]
  [4] initialize_dae! (repeats 2 times)
    @ ~/.julia/packages/OrdinaryDiffEqCore/AYNr9/src/initialize_dae.jl:40 [inlined]
  [5] __init(prob::ODEProblem{…}, alg::CompositeAlgorithm{…}, timeseries_init::Tuple{}, ts_init::Tuple{}, ks_init::Tuple{}, recompile::Type{…}; saveat::Tup
le{}, tstops::Tuple{}, d_discontinuities::Tuple{}, save_idxs::Nothing, save_everystep::Bool, save_on::Bool, save_start::Bool, save_end::Nothing, callback::
Nothing, dense::Bool, calck::Bool, dt::Float64, dtmin::Float64, dtmax::Float64, force_dtmin::Bool, adaptive::Bool, gamma::Rational{…}, abstol::Nothing, rel
tol::Nothing, qmin::Rational{…}, qmax::Int64, qsteady_min::Int64, qsteady_max::Int64, beta1::Nothing, beta2::Nothing, qoldinit::Rational{…}, controller::No
thing, fullnormalize::Bool, failfactor::Int64, maxiters::Int64, internalnorm::typeof(DiffEqBase.ODE_DEFAULT_NORM), internalopnorm::typeof(LinearAlgebra.opn
orm), isoutofdomain::typeof(DiffEqBase.ODE_DEFAULT_ISOUTOFDOMAIN), unstable_check::typeof(DiffEqBase.ODE_DEFAULT_UNSTABLE_CHECK), verbose::Bool, timeseries
_errors::Bool, dense_errors::Bool, advance_to_tstop::Bool, stop_at_next_tstop::Bool, initialize_save::Bool, progress::Bool, progress_steps::Int64, progress
_name::String, progress_message::typeof(DiffEqBase.ODE_DEFAULT_PROG_MESSAGE), progress_id::Symbol, userdata::Nothing, allow_extrapolation::Bool, initialize
_integrator::Bool, alias::ODEAliasSpecifier, initializealg::OrdinaryDiffEqCore.DefaultInit, kwargs::@Kwargs{…})
    @ OrdinaryDiffEqCore ~/.julia/packages/OrdinaryDiffEqCore/AYNr9/src/solve.jl:565
  [6] __init (repeats 5 times)
    @ ~/.julia/packages/OrdinaryDiffEqCore/AYNr9/src/solve.jl:11 [inlined]
  [7] #__solve#62
    @ ~/.julia/packages/OrdinaryDiffEqCore/AYNr9/src/solve.jl:6 [inlined]
  [8] __solve
    @ ~/.julia/packages/OrdinaryDiffEqCore/AYNr9/src/solve.jl:1 [inlined]
  [9] #__solve#3
    @ ~/.julia/packages/OrdinaryDiffEqDefault/7CaO0/src/default_alg.jl:48 [inlined]
 [10] __solve
    @ ~/.julia/packages/OrdinaryDiffEqDefault/7CaO0/src/default_alg.jl:47 [inlined]
 [11] #__solve#64
    @ ~/.julia/packages/DiffEqBase/5hvMq/src/solve.jl:1538 [inlined]
 [12] __solve
    @ ~/.julia/packages/DiffEqBase/5hvMq/src/solve.jl:1529 [inlined]
 [13] #solve_call#36
    @ ~/.julia/packages/DiffEqBase/5hvMq/src/solve.jl:667 [inlined]
 [14] solve_call(::ODEProblem{Vector{…}, Tuple{…}, true, MTKParameters{…}, ODEFunction{…}, @Kwargs{}, SciMLBase.StandardODEProblem})
    @ DiffEqBase ~/.julia/packages/DiffEqBase/5hvMq/src/solve.jl:624
 [15] solve_up(::ODEProblem{…}, ::Nothing, ::Vector{…}, ::MTKParameters{…}; kwargs::@Kwargs{})
    @ DiffEqBase ~/.julia/packages/DiffEqBase/5hvMq/src/solve.jl:1183
 [16] solve_up
    @ ~/.julia/packages/DiffEqBase/5hvMq/src/solve.jl:1177 [inlined]
 [17] solve(::ODEProblem{…}; sensealg::Nothing, u0::Nothing, p::Nothing, wrap::Val{…}, kwargs::@Kwargs{})
    @ DiffEqBase ~/.julia/packages/DiffEqBase/5hvMq/src/solve.jl:1089
 [18] solve(::ODEProblem{Vector{…}, Tuple{…}, true, MTKParameters{…}, ODEFunction{…}, @Kwargs{}, SciMLBase.StandardODEProblem})
    @ DiffEqBase ~/.julia/packages/DiffEqBase/5hvMq/src/solve.jl:1079
 [19] top-level scope
    @ REPL[107]:1
Some type information was truncated. Use `show(err)` to see complete types.
```

**Environment (please complete the following information):**

  - Output of `using Pkg; Pkg.status()`

```julia
Status `~/Scratch/mwe/Project.toml`
  [a93c6f00] DataFrames v1.7.0
  [82cc6244] DataInterpolations v8.0.1
⌅ [961ee093] ModelingToolkit v9.80.5
⌃ [16a59e39] ModelingToolkitStandardLibrary v2.21.1
  [1dea7af3] OrdinaryDiffEq v6.98.0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`

```

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

```julia
Status `~/Scratch/mwe/Manifest.toml`
  [47edcb42] ADTypes v1.14.0
  [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
⌅ [e2ed5e7c] Bijections v0.1.10
  [62783981] BitTwiddlingConvenienceFunctions v0.1.6
  [8e7c35d0] BlockArrays v1.6.3
  [70df07ce] BracketingNonlinearSolve v1.3.0
  [2a0fbf3d] CPUSummary v0.2.6
  [d360d2e6] ChainRulesCore v1.25.1
  [fb6a15b2] CloseOpenIntervals v0.1.13
  [861a8166] Combinatorics v1.0.3
  [a80b9123] CommonMark v0.9.1
  [38540f10] CommonSolve v0.2.4
  [bbf7d656] CommonSubexpressions v0.3.1
  [f70d9fcc] CommonWorldInvalidations v1.0.0
  [34da2185] Compat v4.16.0
  [b152e2b5] CompositeTypes v0.1.4
  [a33af91c] CompositionsBase v0.1.2
  [2569d6c7] ConcreteStructs v0.2.3
  [187b0558] ConstructionBase v1.6.0
  [adafc99b] CpuId v0.3.1
  [a8cc5b0e] Crayons v4.1.1
  [9a962f9c] DataAPI v1.16.0
  [a93c6f00] DataFrames v1.7.0
  [82cc6244] DataInterpolations v8.0.1
  [864edb3b] DataStructures v0.18.22
  [e2d170a0] DataValueInterfaces v1.0.0
  [2b5f629d] DiffEqBase v6.176.0
  [459566f4] DiffEqCallbacks v4.6.0
  [77a26b50] DiffEqNoiseProcess v5.24.1
  [163ba53b] DiffResults v1.1.0
  [b552c78f] DiffRules v1.15.1
⌅ [a0c0ee7d] DifferentiationInterface v0.6.54
  [8d63f2c5] DispatchDoctor v0.4.19
  [31c24e10] Distributions v0.25.120
  [ffbed154] DocStringExtensions v0.9.5
  [5b8099bc] DomainSets v0.7.15
  [7c1d4256] DynamicPolynomials v0.6.2
  [06fc5a27] DynamicQuantities v1.8.0
  [4e289a0a] EnumX v1.0.5
  [f151be2c] EnzymeCore v0.8.11
  [d4d017d3] ExponentialUtilities v1.27.0
  [e2ba6199] ExprTools v0.1.10
  [55351af7] ExproniconLite v0.10.14
  [7034ab61] FastBroadcast v0.3.5
  [9aa1b823] FastClosures v0.3.2
  [442a2c76] FastGaussQuadrature v1.0.2
  [a4df4552] FastPower v1.1.2
  [1a297f60] FillArrays v1.13.0
  [64ca27bc] FindFirstFunctions v1.4.1
  [6a86dc24] FiniteDiff v2.27.0
  [1fa38f19] Format v1.3.7
⌅ [f6369f11] ForwardDiff v0.10.38
  [069b7b12] FunctionWrappers v1.1.3
  [77dc65aa] FunctionWrappersWrappers v0.1.3
  [d9f16b24] Functors v0.5.2
  [46192b85] GPUArraysCore v0.2.0
  [c145ed77] GenericSchur v0.5.5
  [c27321d9] Glob v1.3.1
  [86223c79] Graphs v1.13.0
  [34004b35] HypergeometricFunctions v0.3.28
  [615f187c] IfElse v0.1.1
  [d25df0c9] Inflate v0.1.5
  [842dd82b] InlineStrings v1.4.3
  [18e54dd8] IntegerMathUtils v0.1.2
  [8197267c] IntervalSets v0.7.11
  [3587e190] InverseFunctions v0.1.17
  [41ab1584] InvertedIndices v1.3.1
  [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
  [ba0b0d4f] Krylov v0.10.1
  [b964fa9f] LaTeXStrings v1.4.0
  [23fbe1c1] Latexify v0.16.8
  [10f19ff3] LayoutPointers v0.1.17
  [5078a376] LazyArrays v2.6.1
  [87fe0de2] LineSearch v0.1.4
  [d3d80556] LineSearches v7.4.0
  [7ed4a6bd] LinearSolve v3.17.0
  [2ab3a3ac] LogExpFunctions v0.3.29
  [d8e11817] MLStyle v0.4.17
  [1914dd2f] MacroTools v0.5.16
  [d125e4d3] ManualMemory v0.1.8
  [bb5d69b7] MaybeInplace v0.1.4
  [e1d29d7a] Missings v1.2.0
⌅ [961ee093] ModelingToolkit v9.80.5
⌃ [16a59e39] ModelingToolkitStandardLibrary v2.21.1
  [2e0e35c7] Moshi v0.3.5
  [46d2c3a1] MuladdMacro v0.2.4
  [102ac46a] MultivariatePolynomials v0.5.9
  [d8a4904e] MutableArithmetics v1.6.4
  [d41bc354] NLSolversBase v7.10.0
  [77ba4419] NaNMath v1.1.3
  [8913a72c] NonlinearSolve v4.9.0
  [be0214bd] NonlinearSolveBase v1.12.0
  [5959db7a] NonlinearSolveFirstOrder v1.5.0
  [9a2c21bd] NonlinearSolveQuasiNewton v1.6.0
  [26075421] NonlinearSolveSpectralMethods v1.2.0
  [6fe1bfb0] OffsetArrays v1.17.0
  [429524aa] Optim v1.13.2
  [bac558e1] OrderedCollections v1.8.1
  [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
  [4302a76b] OrdinaryDiffEqDifferentiation v1.10.0
  [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
  [d96e819e] Parameters v0.12.3
  [e409e4f3] PoissonRandom v0.4.4
  [f517fe37] Polyester v0.7.18
  [1d0040c9] PolyesterWeave v0.2.2
  [2dfb63ee] PooledArrays v1.4.3
  [85a6dd25] PositiveFactorizations v0.2.4
  [d236fae5] PreallocationTools v0.4.27
⌅ [aea7be01] PrecompileTools v1.2.1
  [21216c6a] Preferences v1.4.3
  [08abe8d2] PrettyTables v2.4.0
  [27ebfcd6] Primes v0.5.7
  [43287f4e] PtrArrays v1.3.0
  [1fd47b50] QuadGK v2.11.2
  [74087812] Random123 v1.7.1
  [e6cf234a] RandomNumbers v1.6.0
  [3cdcf5f2] RecipesBase v1.3.4
  [731186ca] RecursiveArrayTools v3.33.0
  [189a3867] Reexport v1.2.2
  [ae029012] Requires v1.3.1
  [ae5879a3] ResettableStacks v1.1.1
  [79098fc4] Rmath v0.8.0
  [7e49a35a] RuntimeGeneratedFunctions v0.5.15
  [9dfe8606] SCCNonlinearSolve v1.2.0
  [94e857df] SIMDTypes v0.1.0
  [0bca4576] SciMLBase v2.101.0
  [19f34311] SciMLJacobianOperators v0.1.6
  [c0aeaf25] SciMLOperators v1.3.1
  [53ae85a6] SciMLStructures v1.7.0
  [91c51154] SentinelArrays v1.4.8
  [efcf1570] Setfield v1.1.2
  [727e6d20] SimpleNonlinearSolve v2.5.0
  [699a6c99] SimpleTraits v0.9.4
  [ce78b400] SimpleUnPack v1.1.0
  [a2af1166] SortingAlgorithms v1.2.1
  [0a514795] SparseMatrixColorings v0.4.20
  [276daf66] SpecialFunctions v2.5.1
  [aedffcd0] Static v1.2.0
  [0d7ed370] StaticArrayInterface v1.8.0
  [90137ffa] StaticArrays v1.9.13
  [1e83bf80] StaticArraysCore v1.4.3
  [10745b16] Statistics v1.11.1
  [82ae8749] StatsAPI v1.7.1
  [2913bbd2] StatsBase v0.34.5
  [4c63d2b9] StatsFuns v1.5.0
  [7792a7ef] StrideArraysCore v0.5.7
  [892a3eda] StringManipulation v0.4.1
  [2efcf032] SymbolicIndexingInterface v0.3.40
  [19f23fe9] SymbolicLimits v0.2.2
  [d1185830] SymbolicUtils v3.29.0
  [0c5d862f] Symbolics v6.40.0
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.12.1
  [ed4db957] TaskLocalValues v0.1.2
  [8ea1fca8] TermInterface v2.0.0
  [1c621080] TestItems v1.0.0
  [8290d209] ThreadingUtilities v0.5.5
  [a759f4b9] TimerOutputs v0.5.29
  [410a4b4d] Tricks v0.1.10
  [781d530d] TruncatedStacktraces v1.4.0
  [5c2747f8] URIs v1.5.2
  [3a884ed6] UnPack v1.0.2
  [1986cc42] Unitful v1.23.1
  [a7c27f48] Unityper v0.1.6
  [897b6980] WeakValueDicts v0.1.0
  [1d5cc7b8] IntelOpenMP_jll v2025.0.4+0
  [856f044c] MKL_jll v2025.0.1+1
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [f50d1b31] Rmath_jll v0.5.1+0
  [1317d2d5] oneTBB_jll v2022.0.0+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.11.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.11.0
  [de0858da] Printf v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [1a1011a3] SharedArrays v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.11.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test v1.11.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.6.0+0
  [e37daf67] LibGit2_jll v1.7.2+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.6+0
  [14a3606d] MozillaCACerts_jll v2023.12.12
  [4536629a] OpenBLAS_jll v0.3.27+1
  [05823500] OpenLibm_jll v0.8.5+0
  [bea87d4a] SuiteSparse_jll v7.7.0+0
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.59.0+0
  [3f19e933] p7zip_jll v17.4.0+2

```

  - Output of `versioninfo()`

```julia
Julia Version 1.11.5
Commit 760b2e5b739 (2025-04-14 06:53 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 32 × AMD Ryzen 9 7950X3D 16-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, znver4)
Threads: 32 default, 0 interactive, 16 GC (on 32 virtual cores)
```

**Additional context**

For the `Interpolation` block I get a similar stack trace if it appears under a derivative.


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Time-derivative of ParameterizedInterpolation #397

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Time-derivative of ParameterizedInterpolation #397

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