Algorithm 2 Online phase

Require: choose a query parameter ${\mathit{\theta }}_q$, choose a time step $\delta t^{ROM}>0$.

Initialization: $t=t^{0,ROM}=0$, ${\mathbf{\alpha }}^0=0$, ${\mathbf{\xi }}^0={\mathbf{\xi }}^D\left(0\right)$;

Compute ${\mathrm{\Psi }}_u\left({\mathit{\theta }}_q\right)$ and ${\mathrm{\Psi }}_v\left({\mathit{\theta }}_q\right)$ from the diffuse approximation approach;

for $i=1\dots ,N_t$ do

$\left\{\mathit{u}\right\}(\left\{\mathit{x}\right\},t^i,{\mathit{\theta }}_q)\leftarrow {\mathsf{\Phi }}_u^{r}A\left(t^i\right){\mathrm{\Psi }}_u\left({\mathit{\theta }}_q\right)$;

$\left\{\mathit{v}\right\}(\left\{\mathit{x}\right\},t^i,{\mathit{\theta }}_q)\leftarrow {\mathsf{\Phi }}_v^{r}B\left(t^i\right){\mathrm{\Psi }}_v\left({\mathit{\theta }}_q\right)$;

end for

$\mathcal{U}\left({\mathit{\theta }}_q\right)\leftarrow [\left\{\mathit{u}\right\}(\left\{\mathit{x}\right\},t^1,{\mathit{\theta }}_q),\dots ,\left\{\mathit{u}\right\}(\left\{\mathit{x}\right\},t^{N_t},{\mathit{\theta }}_q)]$;

$\mathcal{V}\left({\mathit{\theta }}_q\right)\leftarrow [\left\{\mathit{v}\right\}(\left\{\mathit{x}\right\},t^1,{\mathit{\theta }}_q),\dots ,\left\{\mathit{v}\right\}(\left\{\mathit{x}\right\},t^{N_t},{\mathit{\theta }}_q)]$;

Compute $\mathsf{\Phi }\left({\mathit{\theta }}_q\right)$, $\mathrm{\Gamma }\left({\mathit{\theta }}_q\right)$, $Q\left({\mathit{\theta }}_q\right)$, ${\mathbf{\alpha }}^D\left(t^i\right)$ and ${\mathbf{\xi }}^D\left({\mathit{\theta }}_i\right)$, $i=1,\dots ,N_t$;

while $t<T_f$ do

$t\leftarrow t+\delta t^{ROM}$; $t^{i+1,ROM}=t^{i,ROM}+\delta t^{ROM}$;

${\mathbf{\alpha }}^{i+1}={\mathbf{\alpha }}^i+\delta t^{ROM}Q\left({\mathit{\theta }}_q\right){\mathbf{\xi }}^i$;

Compute ${\mathit{a}}^k\left(t^{i+1,ROM}\right)$, $k=1,\dots ,m_v$ from the diffuse approximation approach;

${\xi }_k^{i+1}=\mathit{p}{\left({\mathbf{\alpha }}^{i+1}\right)}^T{\mathit{a}}^k\left(t^{i+1,ROM}\right)$;

If needed, reconstruct the high-dimensional displacements/velocity fields:

$\left\{\mathit{u}\right\}\left(\left\{\mathit{X}\right\},t^{i+1,ROM},{\mathit{\theta }}_q\right)=\mathsf{\Phi }\left({\mathit{\theta }}_q\right){\mathbf{\alpha }}^{i+1}$;

$\left\{\mathit{v}\right\}\left(\left\{\mathit{X}\right\},t^{i+1,ROM},{\mathit{\theta }}_q\right)=\mathrm{\Gamma }\left({\mathit{\theta }}_q\right){\mathbf{\xi }}^{i+1}$;

end while