How to tell ModelingToolkit that two variables have the same initial conditions without computing them

Suppose that we have a differential-algebraic system of equations, with two variables f(t) and g(t) such that f(0) = g(0). Suppose moreover that g(t) is algebraic and g(0) could be explicitly calculated.
Is it possible to avoid the latter calculation and let MTK figure it out by itself?

Here is a MWE of what we are trying to do, which returns an error:

using ModelingToolkit, DifferentialEquations

@variables t
D = Differential(t)

function system_f(; name)
    @variables f(t) g(t)
    eqs = [D(f) ~ g - f]
    ODESystem(eqs; name)

function system_g(; name)
    @variables g(t)
    eqs = [g ~ t]
    ODESystem(eqs; name)

@named f = system_f()
@named g = system_g()

connected_eqs = [
    f.g ~ g.g,

@named _model = ODESystem(connected_eqs, t)
@named model = compose(_model, [f, g])

model = structural_simplify(model)
prob = ODEProblem(model, [f.f => g.g], (0.0, 10.0))
sol = solve(prob)

In the above example, we have that g(t)=t, thus f(0)=g(0)=0.
When g(t) is much more complicated, is there a way to avoid having to provide g(0) to MTK explicitly?


Define a parameter and make both initial conditions equal to that parameter.

Hi Chris, thanks for your answer.

Regarding your proposal, how can we set the parameter value to be dependent on g(t_0)?

The initial value of g is automatically computed and dependent on t_0 (from the timespan passed to ODEProblem), thus we cannot assign a constant value to it.
Since we don’t need to provide an initial value of g, we also don’t want to provide an initial value to f (or to a parameter p), because it can be clearly derived from the problem.

One -quite inelegant- way I’m managing to address the problem is to solve the system 2 times, where the first time is merely used to get the initial value of g, as follows:

  • first set both initial conditions equal to an arbitrary value
  • solve the system, getting the solution pre_sol,
  • solve again the system, this time with initial condition f(0)=\text{pre_sol}.g(0)

For example, here are the lines to substitute to the last two ones in the initial MWE:

pre_prob = ODEProblem(model, [f.f=>42,g.g=>42], (0.0, 10.0)) 
pre_sol = solve(pre_prob)
prob = ODEProblem(model, [f.f=>pre_sol[g.g, 1],g.g=>pre_sol[g.g, 1]], (0.0, 10.0))
sol = solve(prob)

Of course, I suppose that there’s a more direct solution.

Open an issue.

Opened: Solving a system with initial condition of the form $f(0) = g(0)$ · Issue #1911 · SciML/ModelingToolkit.jl · GitHub

Just a small update: we added a simpler MWE in the MTK issue.
We’ve been hitting this limitation again lately, in trying to port some famous models from Vensim to a Julia library.