Differentiating through DataInterpolationsND functions throws error

[mcgarnett]

Describe the bug 🐞

It is not possible to "differentiate through" an interpolating function from DataInterpolationsND, and
mtkcompile() return a complete system. An error is returned about encountering an operator with a different independent variable than time

Encountered operator `Differential(p(t), 1)` which has different independent variable than the one used in the system `t`.

If the DataInterpolationND is replaced with a "dummy" function that does the same interpolation, the system compiles. This indicates that there is not anything wrong with the system of equations as posed, but the problem somehow lies in applying the chain rule to a DataInterpolationND function

Expected behavior

I would expect mtkcompile() to be able to compile the system, using the partial differentials of the interpolating
functions liq_PT_2_U, liq_PT_2_D, gas_PT_2_U, gas_PT_2_D.

Minimal Reproducible Example 👇

using ModelingToolkit
using ModelingToolkit: t_nounits as t, D_nounits as D
using OrdinaryDiffEq
using DataInterpolationsND

itp_dims = (LinearInterpolationDimension([0.1, 10.0]), LinearInterpolationDimension([300.0, 310.0]))

pars = @parameters begin
    V_internal = 10.0, []
    (liq_PT_2_U_itp::NDInterpolation)(..) = NDInterpolation([112.55 154.35; 111.74 153.25], itp_dims), []
    (liq_PT_2_D_itp::NDInterpolation)(..) = NDInterpolation([996.56 993.38; 1001.0 997.7], itp_dims), []
    (gas_PT_2_D_itp::NDInterpolation)(..) = NDInterpolation([939.84 971.00; 940.13 971.30], itp_dims), []
    (gas_PT_2_U_itp::NDInterpolation)(..) = NDInterpolation([0.16039 0.15522; 1.5971 1.5458], itp_dims), []
end

p_init = 0.3
T_liq_init = 300
T_gas_init = 300
m_liq_init = 1000
d_liq_init = liq_PT_2_D_itp(p_init, T_liq_init)
V_liq_init = m_liq_init / d_liq_init
V_ull_init = V_internal - V_liq_init
U_liq_init = m_liq_init * liq_PT_2_U_itp(p_init, T_liq_init)
d_gas_init = gas_PT_2_D_itp(p_init, T_gas_init)
m_gas_init = d_gas_init * V_ull_init
U_gas_init = m_gas_init * gas_PT_2_U_itp(p_init, T_gas_init)

vars = @variables begin
    V_ull(t), [description = "ullage volume"]
    V_liq(t), [description = "liquid volume"]
    p(t) = p_init, [description = "pressure"]
    m_gas(t) = m_gas_init, [description = "fluid mass in volume"]
    U_gas(t) = U_gas_init, [description = "extensive internal energy of gas"]
    T_gas(t), [description = "temperature of gas"]
    d_gas(t), [description = "density of gas"]
    m_liq(t) = m_liq_init, [description = "fluid mass in volume"]
    U_liq(t) = U_liq_init, [description = "extensive internal energy of liquid"]
    T_liq(t), [description = "temperature of liquid"]
    d_liq(t), [description = "density of liquid"]
end

eqs = [
    V_internal ~ V_ull + V_liq,
    d_gas ~ m_gas / V_ull,
    d_gas ~ gas_PT_2_D_itp(p, T_gas),
    U_gas ~ m_gas * gas_PT_2_U_itp(p, T_gas),
    D(m_gas) ~ 0.0,
    D(U_gas) ~ 1.0e3 * p * D(V_liq),
    d_liq ~ liq_PT_2_D_itp(p, T_liq),
    U_liq ~ m_liq * liq_PT_2_U_itp(p, T_liq),
    D(m_liq) ~ 10.0,
    D(U_liq) ~ 10.0 * 150.0 - 1.0e3 * p * D(V_liq),
    V_liq ~ m_liq / d_liq,
]

@named _sys = System(eqs, t, vars, pars)
# this works, resulting in system with 10 equations and 10 unknowns.
# sys = mtkcompile(_sys; fully_determined=false)

# this errors out, resulting in taking derivative wrt T_gas and p_gas instead of time:
# Encountered operator `Differential(p(t), 1)` which has different independent variable than the one used in the system `t`.

sys = mtkcompile(_sys; fully_determined=true)

Bypassing the DataInterpolationND, doing a similar type of interpolation, it can be compiled

using ModelingToolkit
using ModelingToolkit: t_nounits as t, D_nounits as D
using OrdinaryDiffEq

p_data = [0.1, 10.0]
T_data = [300.0, 310.0]
liq_U_data = [112.55 154.35; 111.74 153.25]
liq_D_data = [996.56 993.38; 1001.0 997.7]
gas_U_data = [939.84 971.00; 940.13 971.30]
gas_D_data = [0.16039 0.15522; 1.5971 1.5458]

function itp(x, y, X, Y, Q)
    ([X[2] - x, x - X[1]]' * Q * [Y[2] - y, y - Y[1]]) / ((X[2] - X[1]) * (Y[2] - Y[1]))
end

liq_PT_2_U_itp(p, T) = itp(p, T, p_data, T_data, liq_U_data)
liq_PT_2_D_itp(p, T) = itp(p, T, p_data, T_data, liq_D_data)
gas_PT_2_D_itp(p, T) = itp(p, T, p_data, T_data, gas_U_data)
gas_PT_2_U_itp(p, T) = itp(p, T, p_data, T_data, gas_D_data)


# extrap=(ExtendExtrap(), ExtendExtrap())
pars = @parameters begin
    V_internal = 10.0, []
end

p_init = 0.3
T_liq_init = 300
T_gas_init = 300
m_liq_init = 1000
d_liq_init = liq_PT_2_D_itp(p_init, T_liq_init)
V_liq_init = m_liq_init / d_liq_init
V_ull_init = V_internal - V_liq_init
U_liq_init = m_liq_init * liq_PT_2_U_itp(p_init, T_liq_init)
d_gas_init = gas_PT_2_D_itp(p_init, T_gas_init)
m_gas_init = d_gas_init * V_ull_init
U_gas_init = m_gas_init * gas_PT_2_U_itp(p_init, T_gas_init)

vars = @variables begin
    V_ull(t), [description = "ullage volume"]
    V_liq(t), [description = "liquid volume"]
    p(t) = p_init, [description = "pressure"]
    m_gas(t) = m_gas_init, [description = "fluid mass in volume"]
    U_gas(t) = U_gas_init, [description = "extensive internal energy of gas"]
    T_gas(t), [description = "temperature of gas"]
    d_gas(t), [description = "density of gas"]
    m_liq(t) = m_liq_init, [description = "fluid mass in volume"]
    U_liq(t) = U_liq_init, [description = "extensive internal energy of liquid"]
    T_liq(t), [description = "temperature of liquid"]
    d_liq(t), [description = "density of liquid"]


end

eqs = [
    V_internal ~ V_ull + V_liq,
    d_gas ~ m_gas / V_ull,
    d_gas ~ gas_PT_2_D_itp(p, T_gas),
    U_gas ~ m_gas * gas_PT_2_U_itp(p, T_gas),
    D(m_gas) ~ 0.1,
    D(U_gas) ~ 1.0e3 * p * D(V_liq),
    d_liq ~ liq_PT_2_D_itp(p, T_liq),
    U_liq ~ m_liq * liq_PT_2_U_itp(p, T_liq),
    D(m_liq) ~ 10.0,
    D(U_liq) ~ 10.0 * 150.0 - 1.0e3 * p * D(V_liq),

    V_liq ~ m_liq / d_liq,
]

@named _sys = System(eqs, t, vars, pars)
# now it works, resulting in system with 9 equations and 9 unknowns.

sys = mtkcompile(_sys; fully_determined=true)

Error & Stacktrace ⚠️

julia> 
ERROR: Encountered operator `Differential(p(t), 1)` which has different independent variable than the one used in the system `t`.

Context:
pˍt(t)*Differential(p(t), 1)(liq_PT_2_D_itp(p(t), T_liq(t))) + T_liqˍt(t)*Differential(T_liq(t), 1)(liq_PT_2_D_itp(p(t), T_liq(t)))
Stacktrace:
  [1] validate_operator(op::Differential, args::SymbolicUtils.SmallVec{…}, iv::SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{…}; context::SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{…})
    @ ModelingToolkitBase C:\Users\mark.garnett\.julia\packages\ModelingToolkitBase\Rcink\src\utils.jl:722
  [2] validate_operator
    @ C:\Users\mark.garnett\.julia\packages\ModelingToolkitBase\Rcink\src\utils.jl:721 [inlined]
  [3] collect_vars!(unknowns::OrderedCollections.OrderedSet{…}, parameters::OrderedCollections.OrderedSet{…}, expr::SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{…}, iv::SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{…}, ::Type{…}; depth::Int64)
    @ ModelingToolkitBase C:\Users\mark.garnett\.julia\packages\ModelingToolkitBase\Rcink\src\utils.jl:833
  [4] collect_vars!
    @ C:\Users\mark.garnett\.julia\packages\ModelingToolkitBase\Rcink\src\utils.jl:798 [inlined]
  [5] #collect_vars!#37
    @ C:\Users\mark.garnett\.julia\packages\ModelingToolkitBase\Rcink\src\utils.jl:892 [inlined]
  [6] collect_vars!
    @ C:\Users\mark.garnett\.julia\packages\ModelingToolkitBase\Rcink\src\utils.jl:887 [inlined]
  [7] collect_scoped_vars!(unknowns::OrderedCollections.OrderedSet{…}, parameters::OrderedCollections.OrderedSet{…}, sys::System, iv::SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{…}, ::Type{…}; depth::Int64)
    @ ModelingToolkitBase C:\Users\mark.garnett\.julia\packages\ModelingToolkitBase\Rcink\src\utils.jl:688
  [8] discover_globalscoped(sys::System)
    @ ModelingToolkitBase C:\Users\mark.garnett\.julia\packages\ModelingToolkitBase\Rcink\src\systems\abstractsystem.jl:643
  [9] complete(sys::System; split::Bool, flatten::Bool, add_initial_parameters::Bool)
    @ ModelingToolkitBase C:\Users\mark.garnett\.julia\packages\ModelingToolkitBase\Rcink\src\systems\abstractsystem.jl:680
 [10] complete
    @ C:\Users\mark.garnett\.julia\packages\ModelingToolkitBase\Rcink\src\systems\abstractsystem.jl:677 [inlined]
 [11] mtkcompile(sys::System; additional_passes::Tuple{}, inputs::Vector{…}, outputs::Vector{…}, disturbance_inputs::Vector{…}, split::Bool, kwargs::@Kwargs{…})
    @ ModelingToolkitBase C:\Users\mark.garnett\.julia\packages\ModelingToolkitBase\Rcink\src\systems\systems.jl:104
 [12] top-level scope
    @ c:\Users\mark.garnett\EngineModeling\ignore\chain_rule_bug\bug_for_github.jl:63
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> Pkg.status()

  [a93c6f00] DataFrames v1.8.2
  [4f1ef021] DataInterpolationsND v0.1.2
  [8bb1440f] DelimitedFiles v1.9.1
⌅ [0c46a032] DifferentialEquations v7.17.0
⌅ [5b8099bc] DomainSets v0.7.18
⌃ [e9467ef8] GLMakie v0.13.10
⌃ [c27321d9] Glob v1.4.0
  [a98d9a8b] Interpolations v0.16.2
⌃ [7ed4a6bd] LinearSolve v3.76.0
⌃ [961ee093] ModelingToolkit v11.26.0
  [16a59e39] ModelingToolkitStandardLibrary v2.28.0
  [15e1cf62] NPZ v0.4.3
⌃ [8913a72c] NonlinearSolve v4.19.0
⌅ [1dea7af3] OrdinaryDiffEq v6.111.0
⌅ [127b3ac7] OrdinaryDiffEqNonlinearSolve v1.28.0
  [91a5bcdd] Plots v1.41.6
  [efcf1570] Setfield v1.1.2
  [1986cc42] Unitful v1.28.0
  [de0858da] Printf v1.11.0

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

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⌃ [56b672f2] BoundaryValueDiffEqCore v2.3.1
⌃ [85d9eb09] BoundaryValueDiffEqFIRK v1.14.0
⌃ [1a22d4ce] BoundaryValueDiffEqMIRK v1.14.0
⌃ [9255f1d6] BoundaryValueDiffEqMIRKN v1.13.0
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• Output of versioninfo()

Julia Version 1.12.5
Commit 5fe89b8ddc (2026-02-09 16:05 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Windows (x86_64-w64-mingw32)
  CPU: 20 × Intel(R) Core(TM) Ultra 7 265HX
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, arrowlake-s)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 20 virtual cores)
Environment:
  JULIA_EDITOR = code
  JULIA_VSCODE_REPL = 1

Additional context

Add any other context about the problem here.

[ChrisRackauckas]

The bug is that the interpolation needs symbolic derivatives defined.

[mcgarnett]

Thank you Chris should I be doing that in my code? I thought it looked like it was being taken care of in here [https://github.com/SciML/DataInterpolationsND.jl/blob/main/ext/DataInterpolationsNDSymbolicsExt.jl]
around line 79

[ChrisRackauckas-Claude]

I dug into this. You're right that ext/DataInterpolationsNDSymbolicsExt.jl intends to define the symbolic derivatives — but those definitions have been dead code since the Symbolics v7 update, and there is a second, deeper problem that is specific to how you're using the interpolations as MTK callable parameters. Details:

1. The extension is non-functional on Symbolics v7 (fixed in #72)

The extension still uses the Symbolics v6 extension APIs:

• The Symbolics.derivative(interp::NDInterpolation, args, ::Val{I}) overloads around line 79 hook into an API that was removed in Symbolics v7 (replaced by @register_derivative/derivative_rule). Defining methods on it does nothing, so no derivative rule ever fires.
• Scalar-output terms (your case: 2-D table, scalar value) were built without shape information, which expand_derivatives rejects outright (Differentiation with array expressions is not yet supported).
• Array-output calls are even worse: Symbolics.array_term doesn't exist in any v7 release, so they throw UndefVarError.

This went unnoticed because the extension's tests are in an Extensions test group that was never registered in test/test_groups.toml, so CI never ran them. #72 ports the extension to the v7 APIs, adds the test group to CI, and adds scalar-output + chain-rule + second-derivative tests.

2. Callable parameters can't currently get symbolic derivatives at all — upstream gap

Even with #72, your MWE as written still fails, and this part is not a DataInterpolationsND bug. When you declare

(liq_PT_2_D_itp::NDInterpolation)(..) = NDInterpolation(...)

the thing that appears in your equations is a symbolic stand-in for the parameter (a symbolic function with symtype FnType{Tuple, Real, NDInterpolation}), not the NDInterpolation object itself — the object is just the parameter's default value. Symbolics' derivative-rule lookup dispatches on the operation value, so rules registered for NDInterpolation instances can never fire for the symbolic stand-in, and expand_derivatives leaves Differential(p(t))(liq_PT_2_D_itp(p(t), T_liq(t))) unexpanded — which is exactly the operator mtkcompile then rejects.

This affects 1-D DataInterpolations.jl identically (a callable parameter applied to an unknown rather than t fails at ODEProblem construction). I've filed JuliaSymbolics/Symbolics.jl#1890 proposing a dispatch hook on the FnType's callable type so interpolation packages can register rules that work for callable parameters.

Workaround that works today (with #72)

Embed the interpolation objects directly in the equations instead of declaring them as callable parameters:

liq_PT_2_D_itp = NDInterpolation([996.56 993.38; 1001.0 997.7], itp_dims)  # plain Julia object
# ... no @parameters entry for it, just use liq_PT_2_D_itp(p, T_liq) in the equations

With the #72 branch, your full MWE rewritten this way compiles (mtkcompile with fully_determined = true gives the expected 9-equation system) and solves end-to-end (verified with Rodas5P). The trade-off is that the interpolation data is baked into the compiled system rather than being a swappable parameter; if you need to swap tables you'll have to wait for the Symbolics-level fix.

🤖 Generated with Claude Code