Kraichnan-Orszag system

[mikelevvra]

Hello,

Great work with the library (and paper)! :)
I am running some examples to familiarize with the library, and I am a little stuck with the data from the Kraichnan-Orszag system.
As I understand it, the f_train data should be the data for the derivatives of x1, x2 and x3; right?

In case it is so, I don´t understand the following:
t_u_train[1] = array([1.5140404])
t_f_train[3] = array([1.5140404])

As such, f_train[3,0] should be the derivative of x1 at timestamp 1.5140404, and it should equal to: ax2x3
However:
f_train[3,0] = -0.0759072570021982
u_train[1,1]*u_train[1,2] = -0.1654690761472197

Shouldn´t those values be the same? ( as dx1/dt=ax2x3)
Please let me know if I am mistaken,

Best :)

[ZongrenZou]

Thanks for liking our library and paper.

Regarding your first question, the f_train data are data of the equation residual, which is LHS (the derivatives) - RHS (the terms involving x1, x2, x3). They are not the data for the derivatives of x1, x2, and x3. In the deterministic and clean-data setting, f_train should be all zero. But here we consider UQ and hence assume that we have some uncertainty over the residual. That's why f_train are randomly generated from a Gaussian distribution with mean zero and some positive standard deviation.

Let me know if this is unclear or you have more questions.

Best