Re-enable conservation laws for NonlinearSystems

[TorkelE]

Needs some checking, but this should re-enable the use of conservation law elimination for NonlinearSystems using the approach in SciML/ModelingToolkit.jl#3458 (comment). Basically requires:

• Re-adding the conservation law to the equations.
• Removing the conservation law from the observables list.
• Creating initialization_eqs with the conservation laws (but with species replaced with their initial value parameter.
• Adding the eliminated species back as species of the system.

EDIT, this is not actually the case, my mistake: The updated approach will not enable us to deal with singular Jacobians, which sucks. In the case of Bifurcation diagrams and homotopy continuation this is handled separately though so no problem there (but a problem for normal non-linear system solving).

[ChrisRackauckas]

The updated approach will not enable us to deal with singular Jacobians, which sucks.

But it gets rid of singular Jacobians?

[TorkelE]

No, you are right, I just didn't look through things properly yet 😅

[ChrisRackauckas]

Have you tried using the standard MTK HC interface? I would hope the Catalyst one isn't needed anymore, and it should make things like hill functions now handled through the rationalization passes.

[TorkelE]

Doesn't that still eliminate all but one of the solutions though? Last time I checked it only returned a single solution, which mostly defeated the purpose of using it (it might have been updated since, but cannot find anything about it). Long-term using the MTK interface is still the plan though.

[isaacsas]

@ChrisRackauckas can you point us to the official documentation on the MTK interface? Searching the last MTK release's docs for homotopy doesn't seem to give any results?

[isaacsas]

In either case, we can keep the Catalyst interface for now, and when the MTK interface is documented update the tutorial to show its usage too (or recommend it as the preferred approach if it encompasses all the Catalyst functionality). Adding that wouldn't be breaking, so can wait to be handled till post V15 being released.

[ChrisRackauckas]

It's now just a solver alg in NonlinearSolve.jl

https://docs.sciml.ai/NonlinearSolve/stable/api/homotopycontinuation/

We do need to figure out how to document that better. The AllRoots option controls the return @TorkelE .

[ChrisRackauckas]

Though for the depolynomialization you'll want to target it through https://github.com/SciML/ModelingToolkit.jl/blob/master/src/systems/nonlinear/homotopy_continuation.jl#L536.

I'll chalk this up to the fact that MTK kind of needs a complete doc structure overhaul. Maybe we can dedicate one of the upcoming symbolic numerics thrusday meetings to that and just get it done.

@TorkelE did you de-register the Hill functions? Since that would be required for the rationalization.

[isaacsas]

I'll chalk this up to the fact that MTK kind of needs a complete doc structure overhaul. Maybe we can dedicate one of the upcoming symbolic numerics thrusday meetings to that and just get it done.

That would be great, especially in clarifying what is public API for Catalyst (or other libraries) to use, and what is internal MTK API.

[TorkelE]

@TorkelE did you de-register the Hill functions? Since that would be required for the rationalization.

still registered, but we can probably change. Not sure what the wider implications for various cases would be though.

[TorkelE]

I'll chalk this up to the fact that MTK kind of needs a complete doc structure overhaul. Maybe we can dedicate one of the upcoming symbolic numerics thrusday meetings to that and just get it done.

If one happen, drop me a note and I can come by as well

[isaacsas]

Note that if we deregister it does mean we will no longer have the future ability to trivially detect these functions in symbolic expressions.

Perhaps we should instead expand these functions by default when converting to other system types?

[isaacsas]

(Also, we should probably move these discussions to issues.)

[isaacsas]

@TorkelE I had some merge issues on the tests when updating to master given the divergence between the PR and master now, please double check the tests are ok now.

[TorkelE]

Right, yeah, I probably should make a commit making these tests up to date (also running some of the new remake tests on NonlinearSystems as well, now when those works). When you are done with the update I will do one more update solely going through that the tests are good.

[isaacsas]

I think I'm done, but now I'm not sure if tests will fail due to them being out of sync with the code or something. I guess we shall see!

[TorkelE]

Yes! I will have a look at the tests.

[isaacsas]

@TorkelE if everything passes and the doc build looks ok I am happy to merge this (and will do so).