[WIP] feat: add hooks for OverrideInit

[AayushSabharwal]

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the
  contributor guidelines, in particular the SciML Style Guide and
  COLPRAC.
• Any new documentation only uses public API

Additional context

Add any other context about the problem here.

[AayushSabharwal]

Suggested change

	prob, alg, args, kwargs, initializealg, ReturnCode.Success)
	prob, alg, args, kwargs, initializealg, ReturnCode.Default)

Probably should be this

[AayushSabharwal]

Should this be inside the @static_timeit?

[avik-pal]

yes

[AayushSabharwal]

Again with the @static_timeit

[AayushSabharwal]

@static_timeit

[avik-pal]

Can you briefly mention the motivation behind this? I am not too familiar with the background (other PRs for DAEs and stuff)

[AayushSabharwal]

Sure. MTK currently generates initialization problems for ODEs. SciML/ModelingToolkit.jl#3253 aims to expand this to all such problems. The basic use case is that users can specify initial conditions for variables that MTK eliminates from the system, and the initialization process solves for the states of the system. As a contrived example, consider:

y ~ x^3
x^2 + y^2 ~ 2

MTK will reduce this to a system with a single unknown x and a single equation (the latter). It will dynamically calculate y inside the f provided to NonlinearFunction. Currently, the user has to specify an initial condition for x. With OverrideInit, the user can specify an initial condition for y (say 3) and MTK will generate the following InitializationProblem (NonlinearProblem):

y ~ 3
y ~ x^3

and solve for x. As an added check, it ensures overspecified initial conditions are consistent. e.g. in the example above, if the initial condition [x => 3, y => 3] is specified, MTK generates a NonlinearLeastSquaresProblem:

x ~ 3
y ~ 3
y ~ x^3

This fails the nonlinear solve with ReturnCode.InitialFailure telling the user their initial conditions are inconsistent/unsatisfiable.

MTK can also potentially solve for parameters, such as if x + y ~ p^2 is an equation, the user can specify x and y and MTK will generate a NonlinearProblem to solve for p.

The core infrastructure is in SciMLBase, but all solver libraries need to hook into it. In most cases, this is a trivial change: https://github.com/SciML/StochasticDelayDiffEq.jl/pull/84/files. NonlinearSolve has a billion entry points, multiple different "integrators" (caches) and a whole host of other intricacies which make this PR more complicated.