Initialization Failure when Providing Certain State Guesses (Cyclic Guesses Error)

[Nick-Hemenway]

I’ve got a simple RC circuit with two series resistors:

using ModelingToolkit, DifferentialEquations
using ModelingToolkitStandardLibrary.Electrical
using ModelingToolkitStandardLibrary.Blocks

@mtkmodel Circuit begin
    @components begin
        # Battery Model
        v_source = Constant(k=1)
        voltage = Voltage()
        resistor1 = Resistor(R=1)
        resistor2 = Resistor(R=1)
        capacitor = Capacitor(C=1)
        ground = Ground()
    end
    @equations begin
        # Input to voltage source
        connect(v_source.output, voltage.V)
        # Circuit connections
        connect(voltage.p, resistor1.p)
        connect(resistor1.n, resistor2.p)
        connect(resistor2.n, capacitor.p)
        connect(capacitor.n, ground.g, voltage.n)
    end
end

@mtkcompile sys = Circuit()

The system only has a single state since there's only a single energy storage element (the capacitor), and thus specifying an initial current or capacitor voltage should uniquely define the starting point of the system. If I create an ODEProblem by specifying the initial capacitor current, it works fine:

prob = ODEProblem(sys, [sys.capacitor.i=>0], (0,5))

But if I try to initialize the system by specifying the capacitor voltage instead,

prob = ODEProblem(sys, [sys.capacitor.v=>0], (0,5))

I get an ERROR; Cyclic guesses detected in the system... with the following full traceback:

┌ Warning: Did not converge after `maxiters = 100` substitutions. Either there is a cycle in the rules or `maxiters` needs to be higher.
└ @ Symbolics C:\Users\nhemenway\.julia\packages\Symbolics\xD5Pj\src\variable.jl:646
ERROR: Cyclic guesses detected in the system. Symbolic values were found for the following variables/parameters in the map: 
resistor1₊v(t)  => -capacitor₊i(t)
In order to resolve this, please provide additional numeric guesses so that the chain can be resolved to assign numeric values to each variable.

Stacktrace:
  [1] varmap_to_vars(varmap::Dict{…}, vars::Vector{…}; tofloat::Bool, use_union::Bool, container_type::Type, buffer_eltype::Type, toterm::Function, check::Bool, allow_symbolic::Bool, is_initializeprob::Bool, substitution_limit::Int64)
    @ ModelingToolkit C:\Users\nhemenway\.julia\packages\ModelingToolkit\KTR1R\src\systems\problem_utils.jl:408
  [2] process_SciMLProblem(constructor::Type, sys::System, op::Dict{…}; build_initializeprob::Bool, implicit_dae::Bool, t::Nothing, 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\nhemenway\.julia\packages\ModelingToolkit\KTR1R\src\systems\problem_utils.jl:1443
  [3] (NonlinearProblem{…})(sys::System, op::Dict{…}; expression::Type, check_length::Bool, check_compatibility::Bool, kwargs::@Kwargs{…})
    @ ModelingToolkit C:\Users\nhemenway\.julia\packages\ModelingToolkit\KTR1R\src\problems\nonlinearproblem.jl:66
  [4] NonlinearProblem
    @ C:\Users\nhemenway\.julia\packages\ModelingToolkit\KTR1R\src\problems\nonlinearproblem.jl:57 [inlined]
  [5] NonlinearProblem
    @ .\none:-1 [inlined]
  [6] (SCCNonlinearProblem{…} where iip)(sys::System, op::Dict{…}; eval_expression::Bool, eval_module::Module, cse::Bool, u0_constructor::Function, kwargs::@Kwargs{…})
    @ ModelingToolkit C:\Users\nhemenway\.julia\packages\ModelingToolkit\KTR1R\src\problems\sccnonlinearproblem.jl:110
  [7] InitializationProblem{…}(sys::System, t::Int64, 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\nhemenway\.julia\packages\ModelingToolkit\KTR1R\src\problems\initializationproblem.jl:138
  [8] InitializationProblem
    @ C:\Users\nhemenway\.julia\packages\ModelingToolkit\KTR1R\src\problems\initializationproblem.jl:20 [inlined]
  [9] #_#1180
    @ .\none:-1 [inlined]
 [10] maybe_build_initialization_problem(sys::System, iip::Bool, op::Dict{…}, t::Int64, 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\nhemenway\.julia\packages\ModelingToolkit\KTR1R\src\systems\problem_utils.jl:1135
 [11] process_SciMLProblem(constructor::Type, sys::System, op::Vector{…}; build_initializeprob::Bool, implicit_dae::Bool, t::Int64, 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\nhemenway\.julia\packages\ModelingToolkit\KTR1R\src\systems\problem_utils.jl:1409
 [12] (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\nhemenway\.julia\packages\ModelingToolkit\KTR1R\src\problems\odeproblem.jl:81
 [13] ODEProblem
    @ C:\Users\nhemenway\.julia\packages\ModelingToolkit\KTR1R\src\problems\odeproblem.jl:73 [inlined]
 [14] ODEProblem
    @ .\none:-1 [inlined]
 [15] ODEProblem(sys::System, op::Vector{Pair{Num, Int64}}, tspan::Tuple{Int64, Int64})
    @ ModelingToolkit .\none:-1
 [16] top-level scope
    @ c:\Users\nhemenway\OneDrive - Brunswick Corporation\Documents\01_Projects\XXXXX\TRs\TR2024-0506-005_XXXXXXX\julia\example.jl:27
Some type information was truncated. Use `show(err)` to see complete types.

If I provide an initial guess for capacitor current while specifying the initial condition for capacitor voltage, it works:

prob = ODEProblem(sys, [sys.capacitor.v=>0], (0,5),  guesses=[sys.capacitor.i=>0]) # Works

If I provide a guess for the current through resistor2 (which is the same current as the capacitor current), it works:

prob = ODEProblem(sys, [sys.capacitor.v=>0], (0,5),  guesses=[sys.resistor2.i=>0]) # Works

But, if I provide a guess for the current in resistor1, I get the same Cyclic guesses error as above.

prob = ODEProblem(sys, [sys.capacitor.v=>0], (0,5),  guesses=[sys.resistor2.i=>0]) # Does not work

Why is it that if I specify initial capacitor current, initialization works fine without providing guesses for any other variables, but if I specify the initial capacitor voltage, I have to specify guesses for additional variables for initialization to work? Further, why is it that only some variables work for initial guesses while others fail, even though the guesses all provide the same information (the current in the system). This seems like it must be a bug in the initialization code.

• Output of using Pkg; Pkg.status()

  [0c46a032] DifferentialEquations v7.17.0
⌃ [961ee093] ModelingToolkit v10.29.0
  [16a59e39] ModelingToolkitStandardLibrary v2.25.0
⌃ [91a5bcdd] Plots v1.41.1
⌅ [0c5d862f] Symbolics v6.57.0
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)

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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 1.12.2
Commit ca9b6662be (2025-11-20 16:25 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Windows (x86_64-w64-mingw32)
  CPU: 32 × 13th Gen Intel(R) Core(TM) i9-13950HX
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, alderlake)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 32 virtual cores)

[AayushSabharwal]

It's not so much a bug as a limitation of the current initialization code. Note how the variable it really needs a guess for is resistor1.v which it attempted to calculate as -capacitor.i. There's multiple different choices for how to calculate the guesses. For example, it could have tried to obtain it via resistor1.i * resistor1.R. The current system is basically a heuristic for which of the equivalent choices to take when trying to find guesses, so not all methods of specifying the initial guess actually end up working. For a simpler example, consider a system with three equations:

D(x) ~ ...
y ~ x
z ~ x

Suppose the initialization needs a guess for x, but one isn't explicitly provided. MTK works by building a dictionary of whatever it knows, and recursively substituting to find the required values. Here, it will build a dictionary of guesses with x as the key. Whether y or z is the value associated with that key is an arbitrary choice. Suppose x => y is the arbitrary choice taken. Specifying a guess for y will work. Specifying a guess for z will not.

The real way to fix this would be to use a graph algorithm instead of a dictionary.

[Nick-Hemenway]

Thank you for the response. As a user of the code, how would one come to the conclusion that resistor.v is the variable that it needs a guess for? Or more generally which variables require guesses? This is particular confusing to me given that no additional guesses are required when specifying an initial value of capacitor current (e.g. prob = ODEProblem(sys, [sys.capacitor.i=>0], (0,5)) works fine). I thought running the below code would yield some insight, but I don't see a consistent pattern between the unknowns of the system below and the behavior documented in the comments above.

julia> unknowns(sys)

2-element Vector{SymbolicUtils.BasicSymbolic{Real}}:
 capacitor₊i(t)
 capacitor₊v(t)

The system above is incredibly simple, but I'm running into issues. How would one go about systematically debugging which variables require guesses for more complex systems with far more variables? Or knowing from the forefront which variables require guesses such that you don't have to debug. I don't think this is documented clearly in the initialization documentation. It seems like this should at minimum be documented more clearly, but long term, I think it would be incredibly useful to not have the initialization be based purely on heuristics.

[AayushSabharwal]

It is admittedly a bit difficult. You need as many initial conditions as differential variables in your system. We're getting better at not requiring guesses in some situations. In your error message from building ODEProblem, it says

ERROR: Cyclic guesses detected in the system. Symbolic values were found for the following variables/parameters in the map: 
resistor1₊v(t)  => -capacitor₊i(t)
In order to resolve this, please provide additional numeric guesses so that the chain can be resolved to assign numeric values to each variable.

The pair in the middle

resistor1.v => -capacitor.i

Means that it needs a guess for resistor1.v, and tried to use observed equations/etc to figure it out but could only arrive at -capacitor.i. So providing a guess for either will satisfy it.

This is particular confusing to me given that no additional guesses are required when specifying an initial value of capacitor current (e.g. prob = ODEProblem(sys, [sys.capacitor.i=>0], (0,5)) works fine)

Yes. Since capacitor.i is the differential variable in the system, giving it an initial value directly works the best.

How would one go about systematically debugging which variables require guesses for more complex systems with far more variables?

I think the way to do so with public API at the moment is to create the ODEProblem and use the error messages. This is admittedly not great, and there are issues both for better tooling (#3418) and domain-specific initialization capabilities (#3988).

I think it would be incredibly useful to not have the initialization be based purely on heuristics.

There's some subtlety here. Initialization itself is not based on heuristics. In 90% of cases (absence of parameter initialization) the equations present in the initialization system (the constraints of initialization) are

• algebraic equations of your ODE
• observed equations of your ODE
• initial conditions obtained by combining defaults(sys) and the values passed to ODEProblem
• any additional initialization equations provided to the system.

The unknowns it's solving for are [unknowns(sys); observables(sys)]. What makes this process opaque is that this system is independent of the ODE, and ends up with a different set of unknowns. Moreover, a major part of mtkcompile itself is an NP-complete algorithm. Thus, the current algorithm is a greedy technique and depends heavily on the order and formulation of the equations of the system. This is why MTK does not guarantee the number, order or identity of unknowns/equations in a simplified system. That said, it's not completely unpredictable. We have preprocessing such that for a given system, you should consistently get the same results in the absence of changes to the core algorithm, regardless of the order of equations in your system.