Proper Testing & better use of Lux layers

[avik-pal]

Supercedes #56 and #54

[ChrisRackauckas]

Can some of the adjoints here be dropped now that NonlinearSolve.jl directly supports the adjoints?

(@frankschae )

[avik-pal]

The only difference from the upstreamed version is jacobian free backprop which is a trivial check, and the other one is it works with ps as NamedTuple using fmap. Would it be fine to add a dep on Functors?

(Initially, it made sense to have this here so that I could iterate faster, but rn there is no other blocker for upstreaming)

[ChrisRackauckas]

That could be done. We've been discussing canonicalization here: SciML/DifferentialEquations.jl#881

[codecov]

Codecov Report

Merging #58 (7718e1e) into main (58a444f) will increase coverage by 5.12%.
The diff coverage is 91.63%.

@@            Coverage Diff             @@
##             main      #58      +/-   ##
==========================================
+ Coverage   85.52%   90.65%   +5.12%     
==========================================
  Files          14       12       -2     
  Lines         456      503      +47     
==========================================
+ Hits          390      456      +66     
+ Misses         66       47      -19     

Impacted Files	Coverage Δ
src/operator.jl	80.00% <0.00%> (-20.00%)	⬇️
src/solvers/termination.jl	64.93% <70.00%> (+6.03%)	⬆️
src/adjoint.jl	79.31% <76.47%> (-1.78%)	⬇️
src/layers/jacobian_stabilization.jl	90.90% <90.90%> (+90.90%)	⬆️
src/layers/core.jl	94.28% <93.33%> (+37.14%)	⬆️
src/layers/mdeq.jl	97.84% <96.72%> (-2.16%)	⬇️
src/layers/deq.jl	100.00% <100.00%> (+4.00%)	⬆️
src/solve.jl	100.00% <100.00%> (+8.00%)	⬆️
src/solvers/discrete/broyden.jl	100.00% <100.00%> (+5.00%)	⬆️
src/solvers/discrete/limited_memory_broyden.jl	100.00% <100.00%> (ø)
... and 7 more

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[ChrisRackauckas]

CRC I would think?

[ChrisRackauckas]

Yeah I think it's fine to merge and try upstreaming some of those pieces in pieces.