Evaluation on any point

[douhongkun]

Hello, thanks for your nice work!
I have some question when reading your paper and codes. As is shown following,

how to evaluate at any point by the GANO model in practice, and tackle with the irregular input? Thanks.

[kazizzad]

Hi, great questions,
By irregular input, I assume you mean, input on an irregular mesh. And also, I assume you are asking about FNO-based UNO. In that case, the point-wise operators carry as they are, and for the Fourier transform, i.e., \int v(x)exp(i2\pi x omega), you deploy Riemann sum to approximate it for different omega and the rest follow.
For the output, you can invert the Fourier transform and query the function at any point, and for the pointwise part, you need either to follow kernel integration or interpolation. If you do GNO-based UNO, then you can input directly at any point, and for the output, query the kernel integral operators at any point you desire.

[douhongkun]

Thanks for your reply, I still have some doubts about the implementation. Suppose I have trained the FNO-based UNO and only have the weight matices in the fourier space, how can I query the function when conduct the FFT and IFFT operator, and how to perform the kernel integration in practice in this case?

[kazizzad]

Good question,
In FNO, we project the input function on a few Fourier bases functions {\phi_i}_i^m (these are continues functions). After applying the matrix R on the coefficients, and computing the "m" coefficient, {\alpha_i}_i^m , the output of this part of the layer will be sum_i alpha_i phi_i. This is a function that you can query at any point in practice.