Bug in the Bayesian inference for PINNs

[affans]

Describe the bug 🐞

The method getlogpdf dispatched on GridTraining constructs the grid of points to evaluate the PINN at.

ts = collect(eltype(strategy.dx), tspan[1]:(strategy.dx):tspan[2])

To this time vector, it appends dataset[end - 1] (in the case of an inverse problem)

  t = isempty(ltd.dataset) ? ts : vcat(ts, ltd.dataset[end - 1])

The problem is the dataset has the form [x, y, time] (in the 2d case) and so it appends tot he time vector ltd.dataset[end - 1] which is not time but the independent variable y.

I believe this is a bug. Should it not concatenate the last element of dataset? I have dataset defined as

# let's add some noise 
x = u[1, :] + (u[1, :]) .* (0.3 .* randn(length(u[1, :])))
y = u[2, :] + (u[2, :]) .* (0.3 .* randn(length(u[2, :])))
dataset = [x, y, time] # noisy data [x, y, time]

[affans]

False alarm. Reading through the documentation in advancedHMC_MCMC.jl, it states:

 `dataset`: Is either an empty Vector or a nested Vector of the form `[x̂, t, W]` where `x̂` are dependant variable observations, `t` are time points and `W` are quadrature weights for domain. The dataset is used to compute the L2 loss against the data and also for the Data Duadrature loss function.  For multiple dependant variables, there will be multiple vectors with the last two vectors in dataset still being for `t`, `W`.
 Is empty by default assuming a forward problem is being solved.

so seems like my code dataset = [x, y, time] is not correct and I need to add another element for the weights. However, I got this from the example posted here: https://docs.sciml.ai/NeuralPDE/stable/tutorials/Lotka_Volterra_BPINNs/ so seems like the example itself is wrong. Both the documentation and the example page should be updated. Happy to submit PR if that's the way to go.