Algorithm 2 FNO

$\mathbf{Require}\mathbf{:} \mathrm{Dataset} {\left\{\left(v^{(i)}\left(\mathbf{x}\right),u^{(i)}(\mathbf{x})\right)\right\}}_{i=1}^N$

$\mathbf{Require}\mathbf{:} \mathrm{Lifting} \mathrm{operator} P:{\mathbb{R}}^m\to {\mathbb{R}}^d, \mathrm{Fourier} \mathrm{layers} {\mathcal{F}}_j \mathrm{for} j=1,\dots ,L, \mathrm{decoder} Q$

1: $\mathrm{Initialize} \mathrm{weights} {\theta }_P,{\left\{W_j,R_j,b_j\right\}}_{j=1}^L,{\theta }_Q$

2: $\mathbf{for} \mathrm{epoch}=1 \mathrm{to} E \mathrm{do}$

3:            $\mathbf{for} i=1 \mathrm{to} N \mathrm{do}$

4:                         $h_0^{(i)}(\mathbf{x})\leftarrow P\left(v^{(i)}(\mathbf{x})\right) ▹ \mathrm{Lift} \mathrm{to} \mathrm{latent} \mathrm{space}$

5:                         $\mathbf{for} j=0 \mathrm{to} L-1 \mathrm{do}$

6:                                    ${\hat{h}}_{j+1}^{(i)}(\mathbf{x})\leftarrow {\mathcal{F}}^{-1}\left[R_j\cdot \mathcal{F}\left[h_j^{(i)}\right]\right](\mathbf{x}) ▹ \mathrm{Fourier} \mathrm{transform} \mathrm{and} \mathrm{filtering}$

7:                                    $h_{j+1}^{(i)}(\mathbf{x})\leftarrow \sigma \left(W_j{h}_j^{(i)}(\mathbf{x})+{\hat{h}}_{j+1}^{(i)}(\mathbf{x})+b_j\right) ▹ \mathrm{Nonlinear} \mathrm{activation}$

8:                         end for

9:                         ${\hat{u}}^{(i)}(\mathbf{x})\leftarrow Q\left(h_L^{(i)}(\mathbf{x})\right) ▹ \mathrm{Decode} \mathrm{output} \mathrm{wavefield}$

10:                   $\mathcal{L}\leftarrow \mathrm{M}\mathrm{S}\mathrm{E}\left({\hat{u}}^{\left(i\right)},u^{\left(i\right)}\right)$

11:          $\mathrm{Update} \mathrm{all} \mathrm{parameters} \mathrm{via} \mathrm{backprop} \mathrm{on} \mathcal{L}$

12:             end for

13: end for