script to solve linear sde

[hpieper14]

This is a script that solves a linear SDE using the KKL expansion for the Wiener process. A couple things --

1. This script creates individual variables for the modes. Ideally, the number of modes would be n, where n is specified by the user, something like

order = n
@parameters t z[1:n]
eq = Dt(u(t,z)) ~ f(u, t) + g(u,t)*dW(t,z)
I tried this and ModelingToolkit doesn't support differentiation of array variables. I'm not sure if there's a workaround or if that functionality needs to be extended in ModelingToolkit.

2. I couldn't figure out how to write this script using an ODEProblem for the SDE. With the expansion we are using, we have time as a variable as well as n other variables from the expansion. I used a PDEProblem instead.

[ChrisRackauckas]

This looks very much on track.

I tried this and ModelingToolkit doesn't support differentiation of array variables. I'm not sure if there's a workaround or if that functionality needs to be extended in ModelingToolkit.

Turn it into an array of symbolic variables via z = collect(z).

I couldn't figure out how to write this script using an ODEProblem for the SDE. With the expansion we are using, we have time as a variable as well as n other variables from the expansion. I used a PDEProblem instead.

I meant SDEProblem (sorry if I wrote ODEProblem somewhere). I meant create a version of https://github.com/SciML/NeuralPDE.jl/blob/master/src/ode_solve.jl which takes in an SDEProblem and does this solve on it. It should look very similar to the ODEProblem solver NNODE, except with the bits for handling the Wiener expansions.