Update 1D heateq example

[bgroenks96]

• Fixes a bug in the enthalpy inverse function
• Initial implementation of a benchmarking loop for comparing adjoint performance

TODO: why is checkpointing always slower and requiring more allocations than no checkpointing??

[bgroenks96]

@ChrisRackauckas and @facusapienza21 Could you guys tell me if I am doing something stupid here? For some reason, enabling checkpointing with InterpolatingAdjoint seems to always increase the runtime and allocations by roughly a factor of 2. Maybe there is some kind of issue with my benchmarking setup? Or maybe there is a type instability issue with the checkpointing implementation?

I separated the forward solve from the adjoint calculation using adjoint_sensitivities to make sure we isolate only the adjoint calculation... but this doesn't seem to change much.

Note that I reduced the size of the problem to make testing easier. It's now only ~150 MOL equations. This can be easily adjusted with the grid_edges variable.

Sorry for the messy code... still a WIP.

[facusapienza21]

@bgroenks96 What do you want to do with this PR?

[bgroenks96]

I guess just close it? It's a shame that it didn't make it into the paper, but I unfortunately don't really have the time (or motivation) to dig further into the details of benchmarking checkpointing 🤷‍♂️

[bgroenks96]

Oh wait, I forgot that it fixed a bug... maybe we should merge it actually, if possible.