Default values of variables are not converted to equations when building initialization systems.

[zalanzsiboras]

The bug 🐞

When generating the initialization system of an MTK model, the default values (of variables) are not converted to equations. This results in unbalanced errors when simplifying the generated init. system with mtkcompile(), or errors or an underdetermined init. system when one generates the init. problem directly with InitializationProblem().

Expected behavior

When generating the init. system of a model an equation should be added for each variable with a default value. This would avoid the aforementioned errors and underdetermined systems.

Minimal Reproducible Example 👇

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

@mtkmodel ModelWithDefault_1 begin
    @parameters begin
        x_inf
    end
    @variables begin
        x(t) = 10.0
    end
    @equations begin
        D(x) ~ x_inf - x
    end
end

@mtkmodel ModelWithDefault_2 begin
    @parameters begin
        x_inf
        y_inf
    end
    @variables begin
        x(t) = 10.0
        y(t)
    end
    @equations begin
        D(x) ~ x_inf - x
        D(y) ~ y_inf - y
    end
end

@mtkmodel ModelWithDefault_3 begin
    @parameters begin
        x_inf
        y_inf
        z_sum
    end
    @variables begin
        x(t) = 10.0
        y(t)
        z(t)
    end
    @equations begin
        D(x) ~ x_inf - x
        D(y) ~ y_inf - y
        z ~ z_sum - (x + y)
    end
end


@mtkbuild sys1 = ModelWithDefault_1()
op1 = Dict([sys1.x_inf => 5.0])
init_system_1 = generate_initializesystem(sys1; op = op1)
equations(init_system_1)
# Empty?
iprob_1 = InitializationProblem(sys1, 0.0, op1)
# ERROR: Initialization incomplete. [...] Missing variables: Any[x(t)]

@mtkbuild sys2 = ModelWithDefault_2()
op2 = Dict([sys2.y => 11.0, sys2.x_inf => 5.0, sys2.y_inf => 6.0])
init_system_2 = generate_initializesystem(sys2; op = op2)
equations(init_system_2)
# Only has an equation for y.
iprob_2 = InitializationProblem(sys2, 0.0, op2)
# ERROR: Initialization incomplete. [...] Missing variables: Any[x(t)]

@mtkbuild sys3 = ModelWithDefault_3()
op3 = Dict([sys3.y => 11.0, sys3.x_inf => 5.0, sys3.y_inf => 6.0, sys3.z_sum => 21.0])
init_system_3 = generate_initializesystem(sys3; op = op3)
equations(init_system_3)
# Only has an equation for y and z.
mtkcompile(init_system_3)
# ERROR: ExtraVariablesSystemException: The system is unbalanced. There are 3 highest order derivative variables and 2 equations.
iprob_3 = InitializationProblem(sys3, 0.0, op3)
# This results in an underdetermined init. system!

Errors & Stacktraces ⚠️

InitializationProblem() ERROR.

ERROR: Initialization incomplete. Not all of the state variables of the
DAE system can be determined by the initialization. Missing
variables:

Any[x(t)]

Stacktrace:
 [1] InitializationProblem{…}(sys::System, t::Float64, op::Dict{…}; guesses::Vector{…}, 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 ..\.julia\packages\ModelingToolkit\hlGut\src\problems\initializationproblem.jl:87
 [2] InitializationProblem
   @..\.julia\packages\ModelingToolkit\hlGut\src\problems\initializationproblem.jl:20 [inlined]
 [3] InitializationProblem
   @ .\none:0 [inlined]
 [4] InitializationProblem(sys::System, t::Float64, op::Dict{Num, Float64})
   @ ModelingToolkit .\none:0
 [5] top-level scope
   @ ...\bug_init_problem_gen.jl:65

mtkcompile() ERROR.

ERROR: ExtraVariablesSystemException: The system is unbalanced. There are 3 highest order derivative variables and 2 equations.
More variables than equations, here are the potential extra variable(s):
 z(t)
Note that the process of determining extra variables is a best-effort heuristic. The true extra variables are dependent on the model and may not be in this list.
Stacktrace:
 [1] _mtkcompile!(state::TearingState{…}; simplify::Bool, check_consistency::Bool, fully_determined::Bool, warn_initialize_determined::Bool, dummy_derivative::Bool, inputs::Vector{…}, outputs::Vector{…}, disturbance_inputs::Vector{…}, kwargs::@Kwargs{…})
   @ ModelingToolkit ..\.julia\packages\ModelingToolkit\hlGut\src\systems\systemstructure.jl:1002
 [2] _mtkcompile!
   @ ..\.julia\packages\ModelingToolkit\hlGut\src\systems\systemstructure.jl:985 [inlined]
 [3] mtkcompile!(state::TearingState{…}; simplify::Bool, check_consistency::Bool, fully_determined::Bool, warn_initialize_determined::Bool, inputs::Vector{…}, outputs::Vector{…}, disturbance_inputs::Vector{…}, kwargs::@Kwargs{…})
   @ ModelingToolkit ..\.julia\packages\ModelingToolkit\hlGut\src\systems\systemstructure.jl:919
 [4] __mtkcompile(sys::System; simplify::Bool, inputs::Vector{…}, outputs::Vector{…}, disturbance_inputs::Vector{…}, sort_eqs::Bool, kwargs::@Kwargs{…})
   @ ModelingToolkit ..\.julia\packages\ModelingToolkit\hlGut\src\systems\systems.jl:96 
 [5] __mtkcompile
   @ ..\.julia\packages\ModelingToolkit\hlGut\src\systems\systems.jl:62 [inlined]       
 [6] mtkcompile(sys::System; additional_passes::Vector{…}, simplify::Bool, split::Bool, allow_symbolic::Bool, allow_parameter::Bool, conservative::Bool, fully_determined::Bool, inputs::Vector{…}, outputs::Vector{…}, disturbance_inputs::Vector{…}, kwargs::@Kwargs{})
   @ ModelingToolkit ..\.julia\packages\ModelingToolkit\hlGut\src\systems\systems.jl:37 
 [7] mtkcompile(sys::System)
   @ ModelingToolkit ..\.julia\packages\ModelingToolkit\hlGut\src\systems\systems.jl:30 
 [8] top-level scope
   @ ...\bug_init_problem_gen.jl:73

Environment (please complete the following information):

• Output of using Pkg; Pkg.status()

Status `...\Project.toml`
  [961ee093] ModelingToolkit v10.21.0

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

Status `...\Manifest.toml`
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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)
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, alderlake)
Threads: 1 default, 0 interactive, 1 GC
Environment:
  JULIA_EDITOR = code
  JULIA_VSCODE_REPL = 1
  JULIA_NUM_THREADS = 0

[AayushSabharwal]

I'd recommend against using InitializationProblem and generate_initializesystem directly. With the way things have grown, they rely a lot on preprocessing of their inputs to produce the correct initialization, and that preprocessing happens elsewhere (during ODEProblem construction). The easiest way I can recommend to reliably get the initialization problem/system is to build the ODEProblem and access prob.f.initialization_data.initializeprob. Note that this isn't public API, so shouldn't be relied on for production code.

[zalanzsiboras]

Thank you for the answer and the disclaimer.

For my use case I would need a solution that is production ready and generates the same (or essentially the same) initialization problem as ODEProblem construction would as I would like to solve the initialization problem as a separate step.

As a workaround I could add the default values manually:

@mtkbuild sys1 = ModelWithDefault_1()
op1 = Dict([sys1.x_inf => 5.0])

defs1 = defaults(sys1)
plus_defs1 = Dict([var => defs1[var] for var in unknowns(sys1) if var ∈ keys(defs1)])
op1_plus_defs1 = merge(op1, plus_defs1)

init_system_1 = generate_initializesystem(sys1; op = op1_plus_defs1)
equations(init_system_1)
# Now contains an equation for "x(t)".
iprob_1 = InitializationProblem(sys1, 0.0, op1_plus_defs1)
# Builds successfully.

Could this fix be implemented for InitializationProblem and generate_initializesystem for now?

Additionally, if InitializationProblem and generate_initializesystem don't give the same result as the ODEProblem construction it should be pointed out in the initialization tutorial and their docstrings.