@b3m2a1 Experimental`NumericalFunction most (all?) of the numerical solvers create such a function as a canonical representation of the objective function for the particular problem at hand. For DEs, it represents the "right-hand side" of the system converted to a first-order system. It is created by NDSolve`ProcessEquations and stored in the NDSolve`StateData object it constructs. Alexey Popkov investigated it and asked a few questions, such as the following one:

I included it because it seems to be one of the steps carried out by ProcessEquations, and it takes some time. If one is to write one's own ProcessEquations to work with NDSolve, then constructing the NumericalFunction should be one of the steps.

@xzczd I've been mucking around with modifying NDSolve`StateData internals and came across your answer here: mathematica.stackexchange.com/a/129193/6358

Unfortunately that answer doesn't seem to work for me in v12, so maybe something has changed?

(doublechecked -- works in v10.3)

@ChrisK Oh… Seems that it breaks at least from v11.3. The reason is probably NDSolve`StateData[…] becomes an atom now.

That might explain why I'm having a hard time accessing those NDSolve`StateData internals!

@xzczd So, forgive my ignorance, but does that mean there is no way to change the internals of an atom?

or can I create an expression that is atomic (e.g. if I wanted to make my own NDSolve`StateData)?

@ChrisK Solutions here might help, I'm still checking: mathematica.stackexchange.com/q/96225/1871

This answer doesn't seem to work, NDSolve`ProcessSolutions will fail then: mathematica.stackexchange.com/a/97332/1871

@xzczd Thanks for the leads! Every day is an adventure...

BTW even in v10.3, AtomQ[NDSolve`StateData] is True . ??

Perhaps it's another example for the How well this "full form" integrates into the rest of the language varies from case to case. part. (From Szabolcs's answer. )

I see. Well, that's enough Mathematica internals for me for one day! thanks again

@ChrisK Seems that modification for `` NDSolveStateData`` always causes failure in the ``NDSolveProcessSolutions`` stage from v11.3, can't figure out a workaround. Time to ask a question in the main site.

@xzczd It'd be great if you could ask it, I'm so ignorant I don't know if I could even formulate a good question!

@xzczd thanks!

@ChrisK Just test in v11.2, NDSolve`StateData[…] is not an atom there, funny.

And the LinkWrite trick doesn't even work in v9, so something may be broken in the conversion process.

@halirutan I was able to get an edu-rate license for your plugin :D very excited to mess around with this. @b3m2a1 your EasyIDE is crazy cool!! I hope I can have a semester soon where I end up just programming in all of these under research credits or something--give it my full attention! Many thanks again to all of you :D:D:D:D:D

@xzczd @ChrisK there might be some invisible bits set on them. Do you two have a quick way I can get a StateData object? I can play with it and see.

This might be part of the problem:

state // System`Private`MDataQ

True

1 // System`Private`MDataQ

False

@b3m2a1 NDSolve`ProcessEquations[{x'[t] == 0, x[0] == 0}, x, t][[1]]

invisible bits?! sounds serious :)

@CATrevillian I know. I was asked by sales if they can provide you with one :)

@CATrevillian In future it will be the same. Everyone from academia, students, teachers, etc.. get it for free anyway.

Regardless :) thank you @halirutan