Support Enzyme and Mooncake in AD backends

[ErikQQY]

Part of #288

[ErikQQY]

Seems there are still some thing to do:

julia> @benchmark sol = solve(prob, MIRK4(; jac_alg = jac_alg_forwarddiff), dt = 0.05)
BenchmarkTools.Trial: 7002 samples with 1 evaluation per sample.
 Range (min … max):  378.708 μs … 444.849 ms  ┊ GC (min … max):  0.00% … 97.30%
 Time  (median):     433.500 μs               ┊ GC (median):     0.00%
 Time  (mean ± σ):   767.981 μs ±   5.955 ms  ┊ GC (mean ± σ):  21.83% ± 10.28%

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  379 μs        Histogram: log(frequency) by time       7.72 ms <

 Memory estimate: 1.46 MiB, allocs estimate: 27500.

julia> @benchmark sol = solve(prob, MIRK4(; jac_alg = jac_alg_enzyme), dt = 0.05)
BenchmarkTools.Trial: 3252 samples with 1 evaluation per sample.
 Range (min … max):  1.227 ms … 32.662 ms  ┊ GC (min … max):  0.00% … 95.57%
 Time  (median):     1.344 ms              ┊ GC (median):     0.00%
 Time  (mean ± σ):   1.536 ms ±  1.057 ms  ┊ GC (mean ± σ):  11.47% ± 14.84%

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  1.23 ms      Histogram: log(frequency) by time      5.3 ms <

 Memory estimate: 2.43 MiB, allocs estimate: 27517.

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Benchmark Results

	master	c7c3c64...	master / c7c3c64...
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK2()	7.79 ± 0.19 ms	7.69 ± 0.16 ms	1.01
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK3()	2.87 ± 0.15 ms	2.85 ± 0.13 ms	1.01
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK4()	1.01 ± 0.082 ms	0.994 ± 0.06 ms	1.01
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK5()	2.61 ± 0.44 ms	2.57 ± 0.25 ms	1.02
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK6()	1.19 ± 0.14 ms	1.17 ± 0.1 ms	1.02
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = false)	1.85 ± 0.66 ms	1.83 ± 0.69 ms	1.01
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = true)	3.14 ± 1 ms	3.09 ± 1 ms	1.02
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = false)	0.0689 ± 0.021 s	0.0698 ± 0.021 s	0.987
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = true)	0.083 ± 0.021 s	0.0825 ± 0.024 s	1.01
Simple Pendulum/IIP/Shooting(Tsit5())	0.252 ± 0.078 ms	0.247 ± 0.078 ms	1.02
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK2()	8.83 ± 0.43 ms	9.02 ± 0.48 ms	0.979
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK3()	3.3 ± 0.15 ms	3.33 ± 0.22 ms	0.991
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK4()	1.17 ± 0.089 ms	1.16 ± 0.088 ms	1.01
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK5()	2.94 ± 0.31 ms	2.97 ± 0.31 ms	0.99
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK6()	1.39 ± 0.12 ms	1.39 ± 0.11 ms	1
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = false)	3.58 ± 3 ms	3.6 ± 3 ms	0.995
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = true)	6.03 ± 5 ms	6.04 ± 5.2 ms	0.999
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = false)	0.124 ± 0.022 s	0.137 ± 0.026 s	0.906
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = true)	0.156 ± 0.028 s	0.156 ± 0.026 s	1
Simple Pendulum/OOP/Shooting(Tsit5())	0.687 ± 0.052 ms	0.692 ± 0.052 ms	0.993
time_to_load	4.96 ± 0.073 s	5.09 ± 0.056 s	0.975

Benchmark Plots

A plot of the benchmark results have been uploaded as an artifact to the workflow run for this PR.
Go to "Actions"->"Benchmark a pull request"->[the most recent run]->"Artifacts" (at the bottom).