Understanding few lines from NeuralPDE.jl PDE example (phi solution)

Specific Domains Machine Learning
1

Hi all,
first time I’m trying to use PINNs with Julia, as I’m reading the docs there are some parts that I’m missing.
In this example :

I see that the lines used for the predictions are:

phi = discretization.phi
u_predict = reshape([first(phi([x,y],res.u)) for x in xs for y in ys],(length(xs),length(ys)))

However, in this example, the predictions are calculated like this:

phi = discretization.phi
depvars = [:u,:w]
minimizers_ = [res.u.depvar[depvars[i]] for i in 1:length(chain)]
u_predict  = [[phi[i]([t,x], minimizers_[i])[1] for t in ts  for x in xs] for i in 1:2]

Can someone explain to me the difference between the two u_predict lines? I’m not sure I’m following just by reading the docs.
My other question is about the phi object, what does it contain exactly? Is it the trained network?

2

It’s a difference between whether it’s a system of equations (array of neural networks) or a single equation (thus it’s just a neural network directly).

It’s the neural network you passed but munged to directly represent the PDE solution. So phi[i]([t,x]) is equal to u[i](t,x) when trained for a multi-dimensional system, and for a single dimensional system it’s just phi(t,x).

3

Thank you so much!
Can you please share why [1] or first is used when calling phi[i]([t,x], minimizers_[i])[1]? What does the first index represent?

4

The first NN