Algorithm 1 DeepONet

$\mathbf{Require}\mathbf{:} \mathrm{Dataset} {\left[\left(v^{(i)}(\mathbf{x}),{\mathbf{x}}^{(i)}\right),u^{(i)}\right]}_{i=1}^N$                                                                               $▹\mathbf{x}=(\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{t})$

$\mathbf{Require}\mathbf{:} \mathrm{Branch} \mathrm{Net} \mathcal{B}:{\mathbb{R}}^m\to {\mathbb{R}}^q, \mathrm{Trunk} \mathrm{Net} \mathcal{T}:{\mathbb{R}}^4\to {\mathbb{R}}^q ▹\mathrm{m}=\mathrm{dimension} \mathrm{of} v^{(i)}$

1: $\mathrm{Initialize} \mathrm{weights} {\theta }_b,{\theta }_{\tau }$

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

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

4:                          ${\mathcal{B}}^{(i)}\leftarrow \mathcal{B}\left(v^{(i)};{\theta }_b\right) ▹ \mathrm{Branch} \mathrm{input} \mathrm{from} \mathrm{discretized} v(\mathbf{x})$

5:                          ${\mathcal{T}}^{(i)}\leftarrow \mathcal{T}\left({\mathbf{x}}^{(i)};{\theta }_{\tau }\right) ▹ \mathrm{Trunk} \mathrm{input}: \mathrm{x}=(x,y,z,t)$

6:                          ${\hat{u}}^{(i)}\leftarrow {\sum }_{k=1}^q {\mathcal{B}}_k^{(i)}\cdot {\mathcal{T}}_k^{(i)} ▹ \mathrm{Operator} \mathrm{prediction}$

7:                          $\mathbf{if} \mathrm{Physics}\text{-}\mathrm{informed} \mathbf{then}$

8:                                   $R_Q^{(i)}\leftarrow \mathrm{PDE} \mathrm{residual} \mathrm{at} {\mathbf{x}}^{(i)}$

9:                                   $B_Q^{(i)}\leftarrow \mathrm{B}/ \mathrm{IC} \mathrm{residual} \mathrm{at} {\mathbf{x}}^{(i)}$

10:                             $\mathcal{L}\leftarrow {\lambda }_1{‖R_Q^{\left(i\right)}‖}^2+{\lambda }_2{‖B_Q^{\left(i\right)}‖}^2$

11:                   else

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

13:                    end if

14:                    $\mathrm{Update} {\theta }_b,{\theta }_{\tau } \mathrm{via} \mathrm{backprop} \mathrm{on} \mathcal{L}$

15:            end for

16: end for