Changing SciML from Zygote-first to Mooncake-first

[ChrisRackauckas]

Cover the main solvers:

• DiffEq, done!
• NonlinearSolve, Mooncake for NonlinearSolve's Adjoints. NonlinearSolve.jl#719
• LinearSolve, Chainrules rrules for Mooncake, LinearSolve integration LinearSolve.jl#801
• Integrals: https://github.com/SciML/Integrals.jl/blob/409acc3a639c84cf066d448f1491c5027c15a32c/ext/IntegralsZygoteExt.jl#L5

Get all of the packages with ChainRules to make sure what is required to move to Mooncake does:

• https://github.com/SciML/DiffEqGPU.jl/blob/a802496aff598650fb30123490b0e6bec15b3533/src/algorithms.jl#L173-L179
• https://github.com/SciML/ComponentArrays.jl/blob/e6714d8d65a580b1b2bca533e562615c05dbfe81/src/compat/chainrulescore.jl#L9
• https://github.com/SciML/ModelingToolkit.jl/blob/cc3f4bd88f84ac4f4fe48758ebb9bc017b32e6fc/src/adjoints.jl#L2
• https://github.com/SciML/DataInterpolations.jl/blob/912d889c28c856802df0db93f423685604857013/ext/DataInterpolationsChainRulesCoreExt.jl#L8
• https://github.com/SciML/SciMLBase.jl/blob/b59a86d47aac4709d4605c0de96e6c41663fbf74/ext/SciMLBaseZygoteExt.jl#L6
• https://github.com/SciML/RecursiveArrayTools.jl/blob/3710526066702c8163b60b248e4ce9192fec2fa2/ext/RecursiveArrayToolsZygoteExt.jl#L6
• https://github.com/SciML/ModelingToolkitStandardLibrary.jl/blob/0e75c63c866819aa7dc23a703d7c2768a52fd52e/src/Blocks/sources.jl#L2

Make sure Lux.jl is compatible enough.

Once all of those are hit, we update the docs to use Mooncake instead of Zygote everywhere.

[SebastianM-C]

Should Optimization also be on the list?

[ChrisRackauckas]

Well it doesn't have rules to be covered. It just needs to change its docs to tell people to use Mooncake instead of Zygote