Multi training set/enhance (x,y) symmetry

[SimonYYS]

We would like to derive PDE equations from a particle simulation. Our 2D system (PDEs, an active nematic system like liquid crystal) is sensitive to (x,y) coordinate, and in principal, the output equations should be symmetric in x and y. For instance, the velocity is an vector in the form of (vx, vy), and I separate it into 2 equations in a bid to reduce to scaler form (not sure if this is the best way actually). Thus, the two equations of vx and vy should have the same (number of) terms with symmetry in x and y, which is not enforced now.

However, when generating training set from a particle simulation, there may be bias in x or y direction due to random initial condition and the output is not symmetric for vx and vy. I would appreciate it so much if anyone could give suggestions! Is it possible to fed the algorithm with multi training set to avoid this bias?

[Jacob-Stevens-Haas]

Cool use case!

the output equations should be symmetric in x and y.

Sounds like a good case for ConstrainedSR3 optimizer! You'll have to write constraints for the symmetric terms

Is it possible to fed the algorithm with multi training set to avoid this bias?

Yes, just pass a list of arrays for x and they will all be used for training. For PDEs they all need to be on the same spatial domain, currently.

[SimonYYS]

Thanks for the reply! I will check ConstrainedSR3 tutorial carefully.

Yes, just pass a list of arrays for x and they will all be used for training. For PDEs they all need to be on the same spatial domain, currently.

A direct linking of two training sets may cause trouble in calculating time derivative ( t = t_max followed by a novel t = 1 ). So I guess I need to link them after calculating of time derivative separately. Basically, I am not sure whether the time derivative between the ending of one training set and the begining of another could directly be connected into a larger array or not...

[Jacob-Stevens-Haas]

We calculate the derivatives before concatenating the arrays. So the time derivative in one sample is not affected by the data in another sample.

Also note that unless you're using the pysindy master branch, you need to pass multiple_trajectories=True to SINDy.fit() when passing multiple trajectories.