## Nomenclature  
\begin{tabular}{l l} \hline \hline \(T\) & temperature, K \\ \(u\) & flow rate, m\(\cdot\)s\({}^{-1}\) \\ \(RH\) & relative humidity \\ GDL & gas diffusion layer \\ MPL & micro-porous layer \\ CL & catalyst layer \\ \(L\) & length of the channel, cm \\ \(W\) & width of the channel, mm \\ \(d\) & depth of the channel, mm \\ \(A\) & area, m\({}^{2}\) \\ \(K\) & permeability, mol\(\cdot\)m\({}^{-1}\cdot\)s\({}^{-1}\cdot\)Pa\({}^{-1}\) \\ \(I\) & current density, A\(\cdot\)m\({}^{-2}\) \\ \hline \hline \end{tabular}  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Mean/Maximum Relative Error, \%**} \\ \cline{2-3}  & **Train** & **Validation** & **Test** \\ \hline two-hidden-layer M-ANN (ReLU) & 0.082/1.768 & 0.128/0.768 & 0.143/0.458 \\ two-hidden-layer M-ANN (sigmoid) & 0.051/1.351 & 0.139/0.518 & 0.155/0.359 \\ \hline \hline \end{tabular}  
\end{table}
Table 5: Prediction performance (mean and maximum relative error) of different modeling methods in the train, validation and test set for the anode recirculation mode.  
\begin{tabular}{l l} \(c\) & reactant concentration, mol\(\cdot\)m\({}^{-3}\) \\ \(S\) & source term, mol\(\cdot\)m\({}^{-2}\)\(\cdot\)s\({}^{-1}\) \\ \(N\) & the number of control volumes along the channel \\ \(J\) & the convection flux along the channel, mol\(\cdot\)m\({}^{-2}\)\(\cdot\)s\({}^{-1}\) \\ \(M\) & molar weight, kg\(\cdot\)mol\({}^{-1}\) \\ \(s\) & liquid volume fraction \\ \(EW\) & equivalent weight of the membrane, kg\(\cdot\)mol\({}^{-1}\) \\ \(C_{\rm P}\) & specific heat capacity, J\(\cdot\)kg\({}^{-1}\)\(\cdot\)K\({}^{-1}\) \\ \(P\) & pressure, Pa \\ \(V\) & voltage, V \\ \(E\) & potential, V \\ \(R\) & area resistance, \(\Omega\cdot\)m\({}^{2}\); universal gas constant, 8.314 J\(\cdot\)K\({}^{-1}\cdot\)mol\({}^{-1}\) \\ \(n\) & the number of electron transfer \\ \(F\) & Faraday constant, 96,485 C\(\cdot\)mol\({}^{-1}\) \\ ReLU & rectified linear unit \\
**Greek Letter** \\ \(\delta\) & thickness, m \\ \(\varepsilon\) & porosity \\ \(\theta\) & contact angle, \(\circ\) \\ \(\omega\) & ionomer volume fraction of the catalyst layer \\ \(\alpha\) & transfer coefficient \\ \(\phi\) & diffusion flux, mol\(\cdot\)m\({}^{-2}\)\(\cdot\)s\({}^{-1}\) \\ \(\rho\) & density, kg\(\cdot\)m\({}^{-3}\) \\ \(\lambda\) & membrane water content \\
**Subscript and Superscript** \\
0 & reference value \\ a & anode \\ c & cathode \\ GDL & gas diffusion layer \\ MPL & micro-porous layer \\ CL & catalyst layer \\ m & membrane \\ act & activation \\ CH & channel \\ in & flow inlet; inlet surface of the control volume \\ out & flow outlet \\ ref & reference \\ \(k\) & time step \\ \(n\) & control volume along the channel \\ \(i\) & gas species \\ CH-GDL & interface between the channel and gas diffusion layer \\ lq & liquid water \\ w & water \\ \(p\) & components of the electrode \\ m & membrane \\ mw & membrane water \\ T & Energy \\ eff & effective coefficient \\ cro & crossover \\ rev & reversible \\
**Abbreviation** \\ DEA & dead-ended anode \\ PEM & proton exchange membrane \\ MPR & multivariate polynomial regression \\ ANN & artificial neural network \\ CFD & computational fluid dynamics \\ BO & bonobo optimizer \\ LSTM & long short-term memory \\ \end{tabular}