How to define a custom loss for an inversion problem when the observed data is not directly the solution of the PDE?

[zjuliu]

Thanks a lot for sharing such a powerful method.
I'm using DeepXDE to invert the parameters in the following PDE equation:

where u = u(a, t), k = k(u), σ' = σ'(u); other variables are constant. The constant parameters in k(u) and σ'(u) need to be inverted.

My observed data is not u but the a function of u, which is an array with shape of (n, 2) like
[t, f(u(a,t))], t = t0, t1, t2, ....
In my custom loss function, I want to extract the predicted 𝑢(𝑎,𝑡) around the observed times, apply the function 𝑓(𝑢(𝑎,𝑡)), and then compare the result with the observed data. But some errors always occurs. The time index always exceeds the dimension of y_pred in my custom loss function.
How can I solve this problem, and is there a more efficient way to achieve the same goal? Thanks a lot!
My code is as follows:

[lululxvi]

Let v = f(u(a,t)) be another PDE equation, although this is not really PDE. Then data is for v.