Poly chaos

[FHoltorf]

Hey,

I need #8 going forward and so I prepared a first draft. Specifically, this implementation addresses the generation of moment equations for polynomial models via Galerkin projection.
Extending it for non-polynomial terms is left for future work as non-polynomial terms need to be handled approximately and accordingly there are many different approaches to do that approximation. Any polynomial basis as supported by PolyChaos.jl is supported. That means that a large subset of polynomials from the Askey scheme and some additional useful ones are available (including polynomials wrt user-defined weighting/density functions). That said, it also means that currently only dense polynomial bases with equal degree in all dimensions are supported. While that wouldn't be strictly necessary and this implementation could be adjusted for more liberty in that regard, it is not natural with the way PolyChaos.jl is constructed and I think we should change PolyChaos.jl itself to accommodate sparse polynomial bases. That change would probably be less involved. The generalization would then carry over automatically.

I will add a tutorial style example to showcase potential usage soon.

Flemming

[FHoltorf]

Hey, any thoughts on this @ChrisRackauckas @bowenszhu? The basic functionality of generating PCE moment equations works quite nicely and robustly for polynomial explicit ODE models (see example above). In the future, it would be good to explore the following extensions (importance vaguely in descending order):

• Support for PDE models (integration with MethodOfLines.jl) -> PCE moment equations generated from discretized models vs. discretization of PCE moment equations for continuous model
• Support for general nonlinearities via collocation.
• Support for general sparse polynomial bases (perhaps better built into PolyChaos.jl, very awkward first attempt here)
• Support for algebraic models
• Support for stochastic optimal control problems (ControlSystems.jl)
• Support for multi-element PCE

@bowenszhu already contacted me and would like to help with some of the PCE efforts going forward as well, so it would be nice to get a pair of eyes on this as soon as possible.

Aside: Along the way of creating this, I found that working with PolyChaos.jl is extremely cumbersome. Among many things, the type structure underpinning PolyChaos.jl makes it particularly difficult to extend. For now I decided to work around that but we may wanna consider making changes to PolyChaos itself (I can go into more detail if we have any interest in that).

Flemming

[bowenszhu]

With this interface, users can set the PCE of their own choice, which is fantastic!

I think we can make it more powerful by doing this step for our users.

My idea is as follows

1. asking users to add the probability distributions of stochastic parameters as symbolic metadata. See ModelingToolkit.jl - Symbolic metadata - Probability distributions.
2. we internally set the corresponding appropriate orthogonal polynomials according to the given probability distributions.

I would be happy to adapt this PR if you think this is feasible.

[FHoltorf]

Sounds great to me. And at least for canonical distributions it not only should be feasible but even very easy to do. That said, I am unsure how easy it would be to support more obscure distributions. With the current implementation it is very easy. So for now, I would keep this and add the interface you are proposing for canonical distributions as well. There seems to be no harm in keeping both and they both address valid use cases.