Non code related questions

[Jose-Bastos]

Hello!

I'm really interested in this topic and I got some questions, I do not know if this is the right place to post them( if you prefer I can send an email to any of you) but here they go:

• If I understood it right, when the training is done we get a coef matrix and combining it with our Library Function(theta) we can build the underlying PDE of the data. But if I want to reconstruct the original underlying equation I still need to solve the PDE using some numerical method right?
• Do you think this could be used to find the PDE of data we get little to no knowledge of the underlying PDE? If we build a Library function with big enough components this method should be able to work? And if the data shows periodicity it gets harder because we have to insert sinusoidal components with the right period for it to get the right solution no? And then the Library with all terms, derivatives and cross terms gets huge. So do you think this is feasible or not?

Thank you!

[GJBoth]

Hi Jose,

This seems good enough to discuss - maybe it can be useful to someone in the future!

If I understood it right, when the training is done we get a coef matrix and combining it with our Library Function(theta) we can build the underlying PDE of the data.

Yes, that's correct. If a term is zero, it doesn't feature in the equation, and if it is non-zero, it's coefficient of that term in the equation.

But if I want to reconstruct the original underlying equation I still need to solve the PDE using some numerical method right?

You don't need to solve the PDE using numerical methods, since the neural network learned the denoised output, which is the solution of he found PDE! Ofcourse, you still can solve the discovered PDE and that will probably give you more accurate results, but it's not strictly necessary.

Do you think this could be used to find the PDE of data we get little to no knowledge of the underlying PDE? If we build a Library function with big enough components this method should be able to work?

We have done some internal tests where we used 30-40 terms and it still seemed to work pretty well, but I can't give you any hard guarantees.

And if the data shows periodicity it gets harder because we have to insert sinusoidal components with the right period for it to get the right solution no?

You should be able to learn the periodicity, see the ODE examples for mechanical models.

And then the Library with all terms, derivatives and cross terms gets huge. So do you think this is feasible or not?

A lot depends on the size and quality of your data, how correlated your terms are etc, so there's no straight answer. It certainly isn't impossible, but the only way to know is to try :-)

Does this answer your questions?

[Jose-Bastos]

Hello, thank you for your answer, it helped me in most of my doubts! And thank you for this framework, it's really nice and easy to get started!

You should be able to learn the periodicity, see the ODE examples for mechanical models.

But in situations that you want to model data that shows some periodicity over time, for example let's say we have data that should behave as something like this: f(t) = k1sin(w1t)+ k2sin(w2t). If I do not insert sinusoidal terms with w1 and w2 in my Library function the model won't be able to get the PDE right?

You don't need to solve the PDE using numerical methods, since the neural network learned the denoised output, which is the solution of he found PDE! Of course, you still can solve the discovered PDE and that will probably give you more accurate results, but it's not strictly necessary.

After I made this post I re-read your papers and guessed it would be this way. But then I used the trained Neural Net(a SIREN one) to make predictions and even if the coef matrix that comes out has all entries as 0 the model is still able to fit the data and make,not that bad, predictions. Is it just over-fitting by SIREN? Or is it something else(like low threshold on the estimator or maybe a mistake on my side)

Once again thank you for your answers and for your time!

[GJBoth]

If I do not insert sinusoidal terms with w1 and w2 in my Library function the model won't be able to get the PDE right?

Yes, but the value of w1 and w2 can be learned from the data.

But then I used the trained Neural Net(a SIREN one) to make predictions and even if the coef matrix that comes out has all entries as 0 the model is still able to fit the data and make,not that bad, predictions.

Not sure I fully understand what you mean, but it does sound like a mistake; I don't remember seeing something like this before.