Sensitivity Analysis and Bifurcation Plots

[ChrisRackauckas]

I would like to be able to perform a sensitivity analysis as well on a given ODEProblem. SciML/ODE.jl#22 would be interested and could be done using ForwardDiff. The ODEInterface sensitivity analysis should be parsed to be compatible.

[ChrisRackauckas]

This would require that "special parameters" to be specified in the ODEProblem. For example, not any old parameter could do this because, for example, numbers that are inlined into the the function would not be able to be "found". There would need to be a field params of type ODEParameters <: DEParameters and the user would have to construct a type for this, though it could be made easier with a good constructor. The analysis could then loop through the fields of the params, jiggle them around, and give a sensitivity output.

This same setup allows for bifurcation plotting as well.

[ChrisRackauckas]

Quick implementations could plug into existing tools like PyDST

[ChrisRackauckas]

#41 is related.

[ChrisRackauckas]

Simple local parameter sensitivities have been implemented. They solve the sensitivity equation using an Euler scheme at the same time as the ODE. This turns out to be terrible... I need to think of a better way to do this. Likely, ParameterizedFunctions will provide a separate SensitivityFunction which holds the ODE for a sensitivity analysis, and that will be what's solved. It might also be implemented with a MultiScaleModel....

[ChrisRackauckas]

Both of these are implemented. Well, PyCall needs to fix

JuliaPy/PyCall.jl#327
JuliaPy/PyCall.jl#328

and then DiffEqBifurcate.jl is pretty much complete.