`structural_simplify` fails to propagate variable simplification to noise equations

**Describe the bug 🐞**

When using `structural_simplify` on an SDE system with an input variable that gets simplified out, the noise equations still retain a reference to the simplified variable, causing the solver to fail with an undefined variable error.

**Expected behavior**

The `structural_simplify` function should properly propagate the simplification to the noise equations, ensuring that any simplified variables are also removed from the noise terms.

**Minimal Reproducible Example 👇**

```julia
using ModelingToolkit, StochasticDiffEq
using ModelingToolkit: t_nounits as t, D_nounits as D

sts = @variables begin 
    x(t)=1.0
    input(t)
    [input=true]
end
ps = @parameters a=2
@brownian η

eqs = [D(x) ~ -a*x + (input + 1) * η
       input ~ 0.0]
        
sys = System(eqs, t, sts, ps; name = :name)
sys = structural_simplify(sys)

# The system equations are properly simplified
equations(sys)  # Returns: [Differential(t)(x(t)) ~ -a*x(t)]

# `input` is removed from unknowns
getfield(sys, :unknowns)  # Returns: [x(t)]

# However, it still appears in noise equations
getfield(sys, :noiseeqs)  # Returns: [1 + input(t)]

# Thus, solving fails due to undefined variable
prob = SDEProblem(sys, [], (0.0, 1.0), [])
sol = solve(prob, RKMil()) # This fails with UndefVarError: `input` not defined
```

<details>
  <summary> Error & Stacktrace ⚠️ </summary>

```julia
ERROR: UndefVarError: `input` not defined
Stacktrace:
  [1] macro expansion
    @ ~/.julia/packages/SymbolicUtils/jf8aQ/src/code.jl:430 [inlined]
  [2] macro expansion
    @ ~/.julia/packages/Symbolics/2khHV/src/build_function.jl:546 [inlined]
  [3] macro expansion
    @ ~/.julia/packages/SymbolicUtils/jf8aQ/src/code.jl:387 [inlined]
  [4] macro expansion
    @ ~/.julia/packages/RuntimeGeneratedFunctions/M9ZX8/src/RuntimeGeneratedFunctions.jl:163 [inlined]
  [5] macro expansion
    @ ./none:0 [inlined]
  [6] generated_callfunc
    @ ./none:0 [inlined]
  [7] (::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{…})(::Vector{…}, ::Vector{…}, ::Vector{…}, ::Float64)
    @ RuntimeGeneratedFunctions ~/.julia/packages/RuntimeGeneratedFunctions/M9ZX8/src/RuntimeGeneratedFunctions.jl:150
  [8] g
    @ ~/.julia/packages/ModelingToolkit/zfOUk/src/systems/diffeqs/sdesystem.jl:502 [inlined]
  [9] sde_determine_initdt(u0::Vector{…}, t::Float64, tdir::Float64, dtmax::Float64, abstol::Float64, reltol::Float64, internalnorm::typeof(DiffEqBase.ODE_DEFAULT_NORM), prob::SDEProblem{…}, order::Rational{…}, integrator::StochasticDiffEq.SDEIntegrator{…})
    @ StochasticDiffEq ~/.julia/packages/StochasticDiffEq/G9rbc/src/initdt.jl:34
 [10] auto_dt_reset!
    @ ~/.julia/packages/StochasticDiffEq/G9rbc/src/integrators/integrator_interface.jl:355 [inlined]
 [11] handle_dt!(integrator::StochasticDiffEq.SDEIntegrator{…})
    @ StochasticDiffEq ~/.julia/packages/StochasticDiffEq/G9rbc/src/solve.jl:655
 [12] __init(_prob::SDEProblem{…}, alg::RKMil{…}, timeseries_init::Vector{…}, ts_init::Vector{…}, ks_init::Type, recompile::Type{…}; saveat::Tuple{}, tstops::Tuple{}, d_discontinuities::Tuple{}, save_idxs::Nothing, save_everystep::Bool, save_noise::Bool, save_on::Bool, save_start::Bool, save_end::Nothing, callback::Nothing, dense::Bool, calck::Bool, dt::Float64, adaptive::Bool, gamma::Rational{…}, abstol::Nothing, reltol::Nothing, qmin::Rational{…}, qmax::Rational{…}, qsteady_min::Int64, qsteady_max::Int64, beta2::Nothing, beta1::Nothing, qoldinit::Rational{…}, controller::Nothing, fullnormalize::Bool, failfactor::Int64, delta::Rational{…}, maxiters::Int64, dtmax::Float64, dtmin::Float64, internalnorm::typeof(DiffEqBase.ODE_DEFAULT_NORM), isoutofdomain::typeof(DiffEqBase.ODE_DEFAULT_ISOUTOFDOMAIN), unstable_check::typeof(DiffEqBase.ODE_DEFAULT_UNSTABLE_CHECK), verbose::Bool, force_dtmin::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, initialize_integrator::Bool, seed::UInt64, alias_u0::Bool, alias_jumps::Bool, kwargs::@Kwargs{})
    @ StochasticDiffEq ~/.julia/packages/StochasticDiffEq/G9rbc/src/solve.jl:608
 [13] __init (repeats 2 times)
    @ ~/.julia/packages/StochasticDiffEq/G9rbc/src/solve.jl:18 [inlined]
 [14] __solve(prob::SDEProblem{…}, alg::RKMil{…}, timeseries::Vector{…}, ts::Vector{…}, ks::Nothing, recompile::Type{…}; kwargs::@Kwargs{…})
    @ StochasticDiffEq ~/.julia/packages/StochasticDiffEq/G9rbc/src/solve.jl:6
 [15] __solve (repeats 5 times)
    @ ~/.julia/packages/StochasticDiffEq/G9rbc/src/solve.jl:1 [inlined]
 [16] #solve_call#44
    @ ~/.julia/packages/DiffEqBase/HW4ge/src/solve.jl:632 [inlined]
 [17] solve_call
    @ ~/.julia/packages/DiffEqBase/HW4ge/src/solve.jl:589 [inlined]
 [18] solve_up(prob::SDEProblem{…}, sensealg::Nothing, u0::Vector{…}, p::MTKParameters{…}, args::RKMil{…}; kwargs::@Kwargs{})
    @ DiffEqBase ~/.julia/packages/DiffEqBase/HW4ge/src/solve.jl:1120
 [19] solve_up
    @ ~/.julia/packages/DiffEqBase/HW4ge/src/solve.jl:1099 [inlined]
 [20] #solve#51
    @ ~/.julia/packages/DiffEqBase/HW4ge/src/solve.jl:1036 [inlined]
 [21] solve(prob::SDEProblem{…}, args::RKMil{…})
    @ DiffEqBase ~/.julia/packages/DiffEqBase/HW4ge/src/solve.jl:1026
 [22] top-level scope
    @ ~/Documents/MTK_noise_issue_MWE/MWE.jl:24
Some type information was truncated. Use `show(err)` to see complete types.
```
</details>

<details>
  <summary> Environment </summary>

  - Output of `using Pkg; Pkg.status()`

```julia
julia> using Pkg; Pkg.status()
Status `~/Documents/MTK_noise_issue_MWE/Project.toml`
  [961ee093] ModelingToolkit v9.50.0
  [789caeaf] StochasticDiffEq v6.71.0
```

  - Output of `using Pkg; Pkg.status(; mode = PKGMODE_MANIFEST)`

```julia
julia> using Pkg; Pkg.status(; mode = PKGMODE_MANIFEST)
Status `~/Documents/MTK_noise_issue_MWE/Manifest.toml`
  [47edcb42] ADTypes v1.10.0
  [1520ce14] AbstractTrees v0.4.5
  [7d9f7c33] Accessors v0.1.38
  [79e6a3ab] Adapt v4.1.1
  [66dad0bd] AliasTables v1.1.3
  [ec485272] ArnoldiMethod v0.4.0
  [4fba245c] ArrayInterface v7.17.0
  [4c555306] ArrayLayouts v1.10.4
  [e2ed5e7c] Bijections v0.1.9
  [62783981] BitTwiddlingConvenienceFunctions v0.1.6
  [8e7c35d0] BlockArrays v1.1.1
  [70df07ce] BracketingNonlinearSolve v1.1.0
  [2a0fbf3d] CPUSummary v0.2.6
  [00ebfdb7] CSTParser v3.4.3
  [7057c7e9] Cassette v0.3.14
  [d360d2e6] ChainRulesCore v1.25.0
  [fb6a15b2] CloseOpenIntervals v0.1.13
  [861a8166] Combinatorics v1.0.2
  [a80b9123] CommonMark v0.8.15
  [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 v1.5.8
  [adafc99b] CpuId v0.3.1
  [a8cc5b0e] Crayons v4.1.1
  [9a962f9c] DataAPI v1.16.0
  [864edb3b] DataStructures v0.18.20
  [e2d170a0] DataValueInterfaces v1.0.0
  [2b5f629d] DiffEqBase v6.160.0
  [459566f4] DiffEqCallbacks v4.1.0
  [77a26b50] DiffEqNoiseProcess v5.23.0
  [163ba53b] DiffResults v1.1.0
  [b552c78f] DiffRules v1.15.1
  [a0c0ee7d] DifferentiationInterface v0.6.22
  [8d63f2c5] DispatchDoctor v0.4.17
  [b4f34e82] Distances v0.10.12
  [31c24e10] Distributions v0.25.113
  [ffbed154] DocStringExtensions v0.9.3
  [5b8099bc] DomainSets v0.7.14
  [7c1d4256] DynamicPolynomials v0.6.0
  [06fc5a27] DynamicQuantities v1.3.0
  [4e289a0a] EnumX v1.0.4
  [f151be2c] EnzymeCore v0.8.6
  [d4d017d3] ExponentialUtilities v1.26.1
  [e2ba6199] ExprTools v0.1.10
⌅ [6b7a57c9] Expronicon v0.8.5
  [7034ab61] FastBroadcast v0.3.5
  [9aa1b823] FastClosures v0.3.2
  [29a986be] FastLapackInterface v2.0.4
  [a4df4552] FastPower v1.1.1
  [1a297f60] FillArrays v1.13.0
  [64ca27bc] FindFirstFunctions v1.4.1
  [6a86dc24] FiniteDiff v2.26.0
  [1fa38f19] Format v1.3.7
  [f6369f11] ForwardDiff v0.10.38
  [f62d2435] FunctionProperties v0.1.2
  [069b7b12] FunctionWrappers v1.1.3
  [77dc65aa] FunctionWrappersWrappers v0.1.3
⌅ [d9f16b24] Functors v0.4.12
⌅ [46192b85] GPUArraysCore v0.1.6
  [14197337] GenericLinearAlgebra v0.3.14
  [c145ed77] GenericSchur v0.5.4
  [c27321d9] Glob v1.3.1
  [86223c79] Graphs v1.12.0
  [3e5b6fbb] HostCPUFeatures v0.1.17
  [34004b35] HypergeometricFunctions v0.3.25
  [615f187c] IfElse v0.1.1
  [d25df0c9] Inflate v0.1.5
  [18e54dd8] IntegerMathUtils v0.1.2
  [8197267c] IntervalSets v0.7.10
  [3587e190] InverseFunctions v0.1.17
  [92d709cd] IrrationalConstants v0.2.2
  [82899510] IteratorInterfaceExtensions v1.0.0
  [692b3bcd] JLLWrappers v1.6.1
  [98e50ef6] JuliaFormatter v1.0.62
  [ccbc3e58] JumpProcesses v9.14.0
  [ef3ab10e] KLU v0.6.0
  [ba0b0d4f] Krylov v0.9.8
  [b964fa9f] LaTeXStrings v1.4.0
  [23fbe1c1] Latexify v0.16.5
  [10f19ff3] LayoutPointers v0.1.17
  [5078a376] LazyArrays v2.2.2
  [2d8b4e74] LevyArea v1.0.0
  [87fe0de2] LineSearch v0.1.4
  [d3d80556] LineSearches v7.3.0
  [7ed4a6bd] LinearSolve v2.36.2
  [2ab3a3ac] LogExpFunctions v0.3.28
  [bdcacae8] LoopVectorization v0.12.171
  [d8e11817] MLStyle v0.4.17
  [1914dd2f] MacroTools v0.5.13
  [d125e4d3] ManualMemory v0.1.8
  [bb5d69b7] MaybeInplace v0.1.4
  [e1d29d7a] Missings v1.2.0
  [961ee093] ModelingToolkit v9.50.0
  [46d2c3a1] MuladdMacro v0.2.4
  [102ac46a] MultivariatePolynomials v0.5.7
  [d8a4904e] MutableArithmetics v1.5.2
  [d41bc354] NLSolversBase v7.8.3
  [2774e3e8] NLsolve v4.5.1
  [77ba4419] NaNMath v1.0.2
  [8913a72c] NonlinearSolve v4.1.0
  [be0214bd] NonlinearSolveBase v1.3.3
  [5959db7a] NonlinearSolveFirstOrder v1.0.0
  [9a2c21bd] NonlinearSolveQuasiNewton v1.0.0
  [26075421] NonlinearSolveSpectralMethods v1.0.0
  [6fe1bfb0] OffsetArrays v1.14.1
  [429524aa] Optim v1.10.0
  [bac558e1] OrderedCollections v1.6.3
  [1dea7af3] OrdinaryDiffEq v6.90.1
  [89bda076] OrdinaryDiffEqAdamsBashforthMoulton v1.1.0
  [6ad6398a] OrdinaryDiffEqBDF v1.1.2
  [bbf590c4] OrdinaryDiffEqCore v1.10.2
  [50262376] OrdinaryDiffEqDefault v1.1.0
  [4302a76b] OrdinaryDiffEqDifferentiation v1.2.0
  [9286f039] OrdinaryDiffEqExplicitRK v1.1.0
  [e0540318] OrdinaryDiffEqExponentialRK v1.1.0
  [becaefa8] OrdinaryDiffEqExtrapolation v1.2.1
  [5960d6e9] OrdinaryDiffEqFIRK v1.3.0
  [101fe9f7] OrdinaryDiffEqFeagin v1.1.0
  [d3585ca7] OrdinaryDiffEqFunctionMap v1.1.1
  [d28bc4f8] OrdinaryDiffEqHighOrderRK v1.1.0
  [9f002381] OrdinaryDiffEqIMEXMultistep v1.1.0
  [521117fe] OrdinaryDiffEqLinear v1.1.0
  [1344f307] OrdinaryDiffEqLowOrderRK v1.2.0
  [b0944070] OrdinaryDiffEqLowStorageRK v1.2.1
  [127b3ac7] OrdinaryDiffEqNonlinearSolve v1.2.4
  [c9986a66] OrdinaryDiffEqNordsieck v1.1.0
  [5dd0a6cf] OrdinaryDiffEqPDIRK v1.1.0
  [5b33eab2] OrdinaryDiffEqPRK v1.1.0
  [04162be5] OrdinaryDiffEqQPRK v1.1.0
  [af6ede74] OrdinaryDiffEqRKN v1.1.0
  [43230ef6] OrdinaryDiffEqRosenbrock v1.3.1
  [2d112036] OrdinaryDiffEqSDIRK v1.1.0
  [669c94d9] OrdinaryDiffEqSSPRK v1.2.0
  [e3e12d00] OrdinaryDiffEqStabilizedIRK v1.1.0
  [358294b1] OrdinaryDiffEqStabilizedRK v1.1.0
  [fa646aed] OrdinaryDiffEqSymplecticRK v1.1.0
  [b1df2697] OrdinaryDiffEqTsit5 v1.1.0
  [79d7bb75] OrdinaryDiffEqVerner v1.1.1
  [90014a1f] PDMats v0.11.31
  [65ce6f38] PackageExtensionCompat v1.0.2
  [d96e819e] Parameters v0.12.3
  [e409e4f3] PoissonRandom v0.4.4
  [f517fe37] Polyester v0.7.16
  [1d0040c9] PolyesterWeave v0.2.2
  [f27b6e38] Polynomials v4.0.11
  [85a6dd25] PositiveFactorizations v0.2.4
  [d236fae5] PreallocationTools v0.4.24
  [aea7be01] PrecompileTools v1.2.1
  [21216c6a] Preferences v1.4.3
  [27ebfcd6] Primes v0.5.6
  [43287f4e] PtrArrays v1.2.1
  [1fd47b50] QuadGK v2.11.1
  [74087812] Random123 v1.7.0
  [e6cf234a] RandomNumbers v1.6.0
  [3cdcf5f2] RecipesBase v1.3.4
  [731186ca] RecursiveArrayTools v3.27.3
  [f2c3362d] RecursiveFactorization v0.2.23
  [189a3867] Reexport v1.2.2
  [ae029012] Requires v1.3.0
  [ae5879a3] ResettableStacks v1.1.1
  [79098fc4] Rmath v0.8.0
  [47965b36] RootedTrees v2.23.1
  [7e49a35a] RuntimeGeneratedFunctions v0.5.13
  [94e857df] SIMDTypes v0.1.0
  [476501e8] SLEEFPirates v0.6.43
  [0bca4576] SciMLBase v2.60.0
  [19f34311] SciMLJacobianOperators v0.1.1
  [c0aeaf25] SciMLOperators v0.3.12
  [53ae85a6] SciMLStructures v1.5.0
  [efcf1570] Setfield v1.1.1
  [727e6d20] SimpleNonlinearSolve v2.0.0
  [699a6c99] SimpleTraits v0.9.4
  [ce78b400] SimpleUnPack v1.1.0
  [a2af1166] SortingAlgorithms v1.2.1
  [47a9eef4] SparseDiffTools v2.23.0
  [0a514795] SparseMatrixColorings v0.4.10
  [e56a9233] Sparspak v0.3.9
  [276daf66] SpecialFunctions v2.4.0
  [aedffcd0] Static v1.1.1
  [0d7ed370] StaticArrayInterface v1.8.0
  [90137ffa] StaticArrays v1.9.8
  [1e83bf80] StaticArraysCore v1.4.3
  [82ae8749] StatsAPI v1.7.0
  [2913bbd2] StatsBase v0.34.3
  [4c63d2b9] StatsFuns v1.3.2
  [789caeaf] StochasticDiffEq v6.71.0
  [7792a7ef] StrideArraysCore v0.5.7
  [2efcf032] SymbolicIndexingInterface v0.3.35
  [19f23fe9] SymbolicLimits v0.2.2
  [d1185830] SymbolicUtils v3.7.2
  [0c5d862f] Symbolics v6.19.0
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.12.0
  [8ea1fca8] TermInterface v2.0.0
  [1c621080] TestItems v1.0.0
  [8290d209] ThreadingUtilities v0.5.2
  [a759f4b9] TimerOutputs v0.5.25
  [0796e94c] Tokenize v0.5.29
  [d5829a12] TriangularSolve v0.2.1
  [410a4b4d] Tricks v0.1.9
  [781d530d] TruncatedStacktraces v1.4.0
  [5c2747f8] URIs v1.5.1
  [3a884ed6] UnPack v1.0.2
  [1986cc42] Unitful v1.21.0
  [a7c27f48] Unityper v0.1.6
  [3d5dd08c] VectorizationBase v0.21.71
  [19fa3120] VertexSafeGraphs v0.2.0
  [1d5cc7b8] IntelOpenMP_jll v2024.2.1+0
  [856f044c] MKL_jll v2024.2.0+0
  [efe28fd5] OpenSpecFun_jll v0.5.5+0
  [f50d1b31] Rmath_jll v0.5.1+0
  [1317d2d5] oneTBB_jll v2021.12.0+0
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.10.0
  [de0858da] Printf
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays v1.10.0
  [10745b16] Statistics v1.10.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.4.0+0
  [e37daf67] LibGit2_jll v1.6.4+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.2+1
  [14a3606d] MozillaCACerts_jll v2023.1.10
  [4536629a] OpenBLAS_jll v0.3.23+4
  [05823500] OpenLibm_jll v0.8.1+2
  [bea87d4a] SuiteSparse_jll v7.2.1+1
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.8.0+1
  [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
julia> versioninfo()
Julia Version 1.10.4
Commit 48d4fd48430 (2024-06-04 10:41 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: macOS (arm64-apple-darwin22.4.0)
  CPU: 8 × Apple M1 Pro
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, apple-m1)
Threads: 6 default, 0 interactive, 3 GC (on 6 virtual cores)
Environment:
  JULIA_EDITOR = code
```
</details>

**Additional context**

Note: The issue occurs when *multiplying* noise by terms containing the simplified variable. The following works fine:
```julia
D(x) ~ -a*x + (input + 1) + η  # Addition works
```

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`structural_simplify` fails to propagate variable simplification to noise equations #3212

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structural_simplify fails to propagate variable simplification to noise equations #3212

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`structural_simplify` fails to propagate variable simplification to noise equations #3212