\mathbb{R}^{N_\text{FOM}}
\bar{u}(0)
\bar{u}(T)
\text{FOM}
\text{Linear ROM}
\text{Nonlinear ROM}
Full order model (FOM)
\boldsymbol{u}(\boldsymbol{x}, t; \boldsymbol{\mu})
\approx
g_\text{FOM}(\boldsymbol{x}, \textcolor{red}{\bar{u}(t; \boldsymbol{\mu}}))
=
\mathbf{\Phi} \cdot \textcolor{red}{\bar{u}(t; \boldsymbol{\mu}})
Linear POD-ROM
Nonlinear ROM
\textcolor{red}{\bar{u}(t; \boldsymbol{\mu})}
\approx
g'_\text{ROM}(\textcolor{orange}{\bar{u}(t; \boldsymbol{\mu}})
=
\bar{u}_0 + \mathbf{P} \cdot
\textcolor{orange}{\tilde{u}(t; \mathbf{\mu})}
\boldsymbol{u}(\boldsymbol{x}, t; \boldsymbol{\mu})
\approx
g_\text{ROM}(\boldsymbol{x}, \textcolor{blue}{\tilde{u}(t; \boldsymbol{\mu}}))
=
\mathrm{NN}_\theta\left(\boldsymbol{x}, \textcolor{blue}{\tilde{u}(t; \boldsymbol{\mu}}) \right)
\frac{\partial \boldsymbol{u}}{\partial t}
=
\mathcal{L}(\boldsymbol{x}, t, \boldsymbol{u}; \boldsymbol{\mu})