some questions about Geo-FNO

[gokhalen]

@zongyi-li

I'm reading your paper on learned deformations.

Could you please check if my following understanding is correct?

In Geo-FNO, the input mesh is regarded as coming from some probability distribution. By sampling this probability distribution, we generate training data on different meshes. The neural network $\phi^{-1}_a$ learns to map these sampled meshes into a latent uniform space. When we encounter a new mesh, we use the learned neural network $\phi^{-1}_{a}$ to approximately map the new mesh into a uniform grid in latent space, where standard FNO operates and then we map the solution back into the physical domain. Since the mapping to the latent space $\phi^{-1}_{a}$ is not perfect, this may be a source of (small) error.

Also, I have the following questions:

1. In equation (12) is $|\mathcal{T}^i|$ the volume/area of the mesh? Why is it in the denominator? Why is it necessary while going from (11) to (12) by approximating the integral? A simple approximation of the integral wouldn't have it in the denominator...

2. What exactly is $\rho_a(x)$?

3. I'm looking at the definition of $\phi^{-1}_a$ here and it doesn't seem that anything special is done to make sure that the output of $\phi^{-1}_a$ is uniform. It seems to learn to produce uniform output as a result of training. Is this correct?

Thanks,

-Nachiket

[zongyi-li]

Hi Nachiket, Thank you for the question and sorry for the delayed response. Yes your understanding is correct. For your questions

1. T^i is the input meshes and |T^i| is the number of points |{x}| in the meshes. m(x)/|T^i| corresponds to dx in the Riemann sum.
2. \rho is the density of the input meshes. For example, if the input mesh is uniform, \rho(x) = 1, Otherwise, consider a cosine mesh (Chebshev node) on [-1, 1], x_i = cos(2pi i/N) and rho(x) ~ cos'(2pi x) = 2pi sin(2pi x).
3. Right, so far we don't have any techniques to make sure the output is uniform, but we will use a uniform FFT that implicit assume the output is uniform. It will be interesting to add other techniques, for example adding the laplacian as a loss.

[gokhalen]

@zongyi-li

Thank you very much for your response. Still not clear about point 2, but let me think about it. Perhaps it will be useful for me to take a look at the code.