Problem construction throws invalid unit validation error in InitializationProblem construction stage for some systems

[bspanoghe]

Describe the bug 🐞

Some systems throw an invalid unit validation error during problem construction (specifically in the initialization problem construction stage), seemingly caused by automatically generated intermediate variables. The sphere-shaped system described in the MRE throws an error, while the equivalent cuboid-shaped system described in additional context runs perfectly.

Expected behavior

The problem should be constructed without unit validation error.

Minimal Reproducible Example 👇

I encounter the problem in a large system of connected hydraulic components that can change dimensions under influence of pressure. Below is a heavily simplified version of a single, spherical component.

using ModelingToolkit, DynamicQuantities

@independent_variables t, [description = "Time", unit = u"s"];
d = Differential(t);

function Hydraulics(; name, P, D)
    @parameters begin
        ϵ_D = 1.0, [description = "Elasticity", unit = u"Pa"]
        K = 1.0, [description = "Hydraulic conductivity", unit = u"m^3 / s / Pa"]
    end

    @variables begin
        P(t) = P, [description = "Water pressure", unit = u"Pa"]
        D(t) = D, [description = "Dimension (radius)", unit = u"m"]
        V(t), [description = "Volume", unit = u"m^3"]
    end

    eqs = [
        V ~ 4/3 * pi * D^3, # (m^3) = (m)^3
        d(V) ~ -K*P, # (m^3 / s) = (m^3 / s / Pa) * Pa
        d(D) ~ 1/ϵ_D * D*d(P), # (m / s) = (1 / Pa) * (m) * (Pa / s)
    ]
    return System(eqs, t; name)
end

@mtkcompile system = Hydraulics(P = 1.0, D = 1.0);
prob = ODEProblem(system, [], (0.0, 10.0));

Error & Stacktrace ⚠️

┌ Warning:  in eq. #1right, in sum var"####subexpr#233#234" + (var"####subexpr#231#232"*Pˍt(t)) / ϵ_D, units [1.0 s⁻¹, 1.0 ] do not match.
└ @ ModelingToolkit C:\Users\bspanogh\.julia\packages\ModelingToolkit\80wTI\src\systems\unit_check.jl:181
ERROR: ModelingToolkit.ValidationError("Some equations had invalid units. See warnings for details.")

Click for stacktrace

Stacktrace:
  [1] check_units
    @ C:\Users\bspanogh\.julia\packages\ModelingToolkit\80wTI\src\systems\unit_check.jl:318 [inlined]
  [2] System(tag::UInt64, eqs::Vector{…}, noise_eqs::Nothing, jumps::Vector{…}, constraints::Vector{…}, costs::Vector{…}, consolidate::typeof(ModelingToolkit.default_consolidate), unknowns::Vector{…}, ps::Vector{…}, brownians::Vector{…}, iv::Nothing, observed::Vector{…}, parameter_dependencies::Vector{…}, var_to_name::Dict{…}, name::Symbol, description::String, defaults::Dict{…}, guesses::Dict{…}, systems::Vector{…}, initialization_eqs::Vector{…}, continuous_events::Vector{…}, discrete_events::Vector{…}, connector_type::Nothing, assertions::Dict{…}, metadata::Base.ImmutableDict{…}, gui_metadata::Nothing, is_dde::Bool, tstops::Vector{…}, inputs::OrderedCollections.OrderedSet{…}, outputs::OrderedCollections.OrderedSet{…}, tearing_state::Nothing, namespacing::Bool, complete::Bool, index_cache::Nothing, ignored_connections::Nothing, preface::Vector{…}, parent::Nothing, initializesystem::Nothing, is_initializesystem::Bool, is_discrete::Bool, isscheduled::Bool, schedule::Nothing; checks::Bool)
    @ ModelingToolkit C:\Users\bspanogh\.julia\packages\ModelingToolkit\80wTI\src\systems\system.jl:296
  [3] System
    @ C:\Users\bspanogh\.julia\packages\ModelingToolkit\80wTI\src\systems\system.jl:263 [inlined]
  [4] System(eqs::Vector{…}, iv::Nothing, dvs::Vector{…}, ps::Vector{…}, brownians::Vector{…}; constraints::Vector{…}, noise_eqs::Nothing, jumps::Vector{…}, costs::Vector{…}, consolidate::Function, observed::Vector{…}, parameter_dependencies::Vector{…}, defaults::Dict{…}, guesses::Dict{…}, systems::Vector{…}, initialization_eqs::Vector{…}, continuous_events::Vector{…}, discrete_events::Vector{…}, connector_type::Nothing, assertions::Dict{…}, metadata::Base.ImmutableDict{…}, gui_metadata::Nothing, is_dde::Nothing, tstops::Vector{…}, inputs::OrderedCollections.OrderedSet{…}, outputs::OrderedCollections.OrderedSet{…}, tearing_state::Nothing, ignored_connections::Nothing, parent::Nothing, description::String, name::Symbol, discover_from_metadata::Bool, initializesystem::Nothing, is_initializesystem::Bool, is_discrete::Bool, preface::Vector{…}, checks::Bool)
    @ ModelingToolkit C:\Users\bspanogh\.julia\packages\ModelingToolkit\80wTI\src\systems\system.jl:452
  [5] System
    @ C:\Users\bspanogh\.julia\packages\ModelingToolkit\80wTI\src\systems\system.jl:339 [inlined]
  [6] #System#720
    @ C:\Users\bspanogh\.julia\packages\ModelingToolkit\80wTI\src\systems\system.jl:468 [inlined]
  [7] ModelingToolkit.SCCNonlinearFunction{…}(sys::System, _eqs::Vector{…}, _dvs::Vector{…}, _obs::Vector{…}, cachesyms::Tuple{…}, op::Dict{…}; eval_expression::Bool, eval_module::Module, cse::Bool, kwargs::@Kwargs{…})
    @ ModelingToolkit C:\Users\bspanogh\.julia\packages\ModelingToolkit\80wTI\src\problems\sccnonlinearproblem.jl:45
  [8] (SCCNonlinearProblem{…} where iip)(sys::System, op::Dict{…}; eval_expression::Bool, eval_module::Module, cse::Bool, u0_constructor::Function,
 kwargs::@Kwargs{…})
    @ ModelingToolkit C:\Users\bspanogh\.julia\packages\ModelingToolkit\80wTI\src\problems\sccnonlinearproblem.jl:231
  [9] InitializationProblem{…}(sys::System, t::Float64, op::Dict{…}; guesses::Dict{…}, check_length::Bool, warn_initialize_determined::Bool, initialization_eqs::Vector{…}, fully_determined::Nothing, check_units::Bool, use_scc::Bool, allow_incomplete::Bool, algebraic_only::Bool, time_dependent_init::Bool, kwargs::@Kwargs{…})
    @ ModelingToolkit C:\Users\bspanogh\.julia\packages\ModelingToolkit\80wTI\src\problems\initializationproblem.jl:138
 [10] InitializationProblem
    @ C:\Users\bspanogh\.julia\packages\ModelingToolkit\80wTI\src\problems\initializationproblem.jl:20 [inlined]
 [11] #_#1109
    @ .\none:0 [inlined]
 [12] maybe_build_initialization_problem(sys::System, iip::Bool, op::Dict{…}, t::Float64, defs::Dict{…}, guesses::Dict{…}, missing_unknowns::Set{…}; implicit_dae::Bool, time_dependent_init::Bool, u0_constructor::Function, p_constructor::Function, floatT::Type, initialization_eqs::Vector{…}, use_scc::Bool, eval_expression::Bool, eval_module::Module, kwargs::@Kwargs{…})
    @ ModelingToolkit C:\Users\bspanogh\.julia\packages\ModelingToolkit\80wTI\src\systems\problem_utils.jl:1128
 [13] process_SciMLProblem(constructor::Type, sys::System, op::Vector{…}; build_initializeprob::Bool, implicit_dae::Bool, t::Float64, guesses::Dict{…}, warn_initialize_determined::Bool, initialization_eqs::Vector{…}, eval_expression::Bool, eval_module::Module, fully_determined::Nothing, check_initialization_units::Bool, u0_eltype::Nothing, tofloat::Bool, u0_constructor::typeof(identity), p_constructor::typeof(identity), check_length::Bool, symbolic_u0::Bool, warn_cyclic_dependency::Bool, circular_dependency_max_cycle_length::Int64, circular_dependency_max_cycles::Int64, substitution_limit::Int64, use_scc::Bool, time_dependent_init::Bool, algebraic_only::Bool, allow_incomplete::Bool, is_initializeprob::Bool, kwargs::@Kwargs{…})
    @ ModelingToolkit C:\Users\bspanogh\.julia\packages\ModelingToolkit\80wTI\src\systems\problem_utils.jl:1400
 [14] (ODEProblem{…})(sys::System, op::Vector{…}, tspan::Tuple{…}; callback::Nothing, check_length::Bool, eval_expression::Bool, expression::Type, eval_module::Module, check_compatibility::Bool, kwargs::@Kwargs{})
    @ ModelingToolkit C:\Users\bspanogh\.julia\packages\ModelingToolkit\80wTI\src\problems\odeproblem.jl:81
 [15] ODEProblem
    @ C:\Users\bspanogh\.julia\packages\ModelingToolkit\80wTI\src\problems\odeproblem.jl:73 [inlined]
 [16] ODEProblem
    @ .\none:0 [inlined]
 [17] ODEProblem(sys::System, op::Vector{Any}, tspan::Tuple{Float64, Float64})
    @ ModelingToolkit .\none:0
 [18] top-level scope
ivative.jl:1

Environment (please complete the following information):

• Output of using Pkg; Pkg.status()

Status `~\Project.toml`
  [06fc5a27] DynamicQuantities v1.10.0
  [961ee093] ModelingToolkit v10.26.0

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

Click for manifest

Status `~\Manifest.toml`
  [47edcb42] ADTypes v1.18.0
  [1520ce14] AbstractTrees v0.4.5
  [7d9f7c33] Accessors v0.1.42
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⌅ [861a8166] Combinatorics v1.0.2
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⌅ [70703baa] JuliaSyntax v0.4.10
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  [102ac46a] MultivariatePolynomials v0.5.13
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  [77ba4419] NaNMath v1.1.3
  [be0214bd] NonlinearSolveBase v2.0.0
  [6fe1bfb0] OffsetArrays v1.17.0
  [429524aa] Optim v1.13.2
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  [e409e4f3] PoissonRandom v0.4.7
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  [85a6dd25] PositiveFactorizations v0.2.4
  [d236fae5] PreallocationTools v0.4.34
⌅ [aea7be01] PrecompileTools v1.2.1
  [21216c6a] Preferences v1.5.0
  [27ebfcd6] Primes v0.5.7
  [43287f4e] PtrArrays v1.3.0
  [1fd47b50] QuadGK v2.11.2
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⌃ [19f23fe9] SymbolicLimits v0.2.3
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  [bea87d4a] SuiteSparse_jll v7.7.0+0
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+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 -m`

• Output of versioninfo()

Julia Version 1.11.6
Commit 9615af0f26 (2025-07-09 12:58 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Windows (x86_64-w64-mingw32)
  CPU: 12 × 12th Gen Intel(R) Core(TM) i7-1265U
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, alderlake)
Threads: 1 default, 0 interactive, 1 GC (on 12 virtual cores)
Environment:
  JULIA_EDITOR = code
  JULIA_VSCODE_REPL = 1
  JULIA_NUM_THREADS = 1

Additional context

• The system runs perfectly when defined without units.

• These are the equations generated by the initializationsystem:

julia> generate_initializesystem(system) |> equations
5-element Vector{Equation}:
 0 ~ -V(t) + 4.1887902047863905(D(t)^3)
 0 ~ K*P(t) + 12.566370614359172(D(t)^2)*Dˍt(t)
 0 ~ (Pˍt(t)*D(t)) / ϵ_D - Dˍt(t)
 D(t) ~ Initial(D(t))
 P(t) ~ Initial(P(t))

Based on the error message Warning: in eq. #1right, in sum var"####subexpr#233#234" + (var"####subexpr#231#232"*Pˍt(t)) / ϵ_D, units [1.0 s⁻¹, 1.0 ] do not match., it seems that something goes wrong in the third equation.

• The same system defined for a cuboid shape runs without error:

using ModelingToolkit, DynamicQuantities

@independent_variables t, [description = "Time", unit = u"s"];
d = Differential(t);

function Hydraulics(; name, P, D)
    @parameters begin
        ϵ_D[1:3] = [1.0, 2.0, 3.0], [description = "Elasticity", unit = u"Pa"]
        K = 1.0, [description = "Hydraulic conductivity", unit = u"m^3 / s / Pa"]
    end

    @variables begin
        P(t) = P, [description = "Water pressure", unit = u"Pa"]
        D(t)[1:3] = D, [description = "Dimensions (length, width, height)", unit = u"m"]
        V(t), [description = "Volume", unit = u"m^3"]
    end

    eqs = [
        V ~ D[1]*D[2]*D[3],
        d(V) ~ -K*P,
        d(D[1]) ~ 1/ϵ_D[1] * D[1]*d(P),
        d(D[2]) ~ 1/ϵ_D[2] * D[2]*d(P),
        d(D[3]) ~ 1/ϵ_D[3] * D[3]*d(P),
    ]
    return System(eqs, t; name)
end

@mtkcompile system = Hydraulics(P = 1.0, D = [1.0, 1.0, 1.0]);
prob = ODEProblem(system, [], (0.0, 10.0));

• For some extra context: the compartments can have any shape, so D(t) is generally a 1 to 3 dimensional vector with a function to calculate its volume based on the shape. Every dimension of the compartment can extend or contract with a different elasticity, so the equation defining change in size based on pressure is defined for every dimension separately.