# Separable Gaussian Neural Networks: Structure, Analysis, and Function Approximations
## 6 Comparison of SGNN and GRBFNN  
The performances of SGNN and GRBFNN in approximating two-dimensional and three-dimensional candidate functions are presented in Tables 4 and 5, respectively. For comparison, the centers and width of Gaussian neurons of GRBFNN are set to be trainable variables as well. We focus on the differences of total epochs, training time per epoch, and losses for comparison. The results are obtained by averaging the results of 30 runs.  
As shown in Table 4, when approximating two-dimensional functions, SGNN can achieve comparable accuracy as GRBFNN, with differences of less than one-order-of magnitude in most cases. The worst case occurs with approximating \(f_{1}\). However, the absolute difference is around \(1.0\times 10^{-3}\), and SGNN can still give a reasonably good approximation. On the other hand, the training time per epoch of SGNN is roughly one-tenth of that of GRBFNN.  
The advantage of SGNN becomes more evident in three-dimensional function approximations. SGNN can gain a one-hundred-time speedup over GRBFNN but still maintain a similar level of accuracy. Surprisingly, SGNN can also yield more accurate results when approximating \(f_{3}\) to \(f_{6}\).  
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**SGNN**} & \multicolumn{5}{c}{**GRBFNN**} \\  & **Epoch** & **Seconds/Epoch** & **Ave Loss** & **Min Loss** & **Epoch** & **Seconds/Epoch** & **Ave Loss** & **Min Loss** \\ \hline \(f_{1}\) & 483 & 0.027 & \(1.86\times 10^{-3}\) & \(1.32\times 10^{-3}\) & 568 & 0.232 & \(1.98\times 10^{-4}\) & \(1.25\times 10^{-4}\) \\ \(f_{2}\) & 333 & 0.028 & \(6.51\times 10^{-5}\) & \(3.65\times 10^{-5}\) & 351 & 0.199 & \(2.17\times 10^{-5}\) & \(1.35\times 10^{-5}\) \\ \(f_{3}\) & 334 & 0.027 & \(1.32\times 10^{-5}\) & \(6.44\times 10^{-6}\) & 349 & 0.193 & \(6.21\times 10^{-6}\) & \(2.30\times 10^{-6}\) \\ \(f_{4}\) & 223 & 0.028 & \(7.12\times 10^{-6}\) & \(5.53\times 10^{-6}\) & 209 & 0.194 & \(4.62\times 10^{-6}\) & \(2.68\times 10^{-6}\) \\ \(f_{5}\) & 123 & 0.030 & \(3.72\times 10^{-6}\) & \(1.82\times 10^{-6}\) & 126 & 0.198 & \(1.99\times 10^{-6}\) & \(1.15\times 10^{-6}\) \\ \(f_{6}\) & 595 & 0.027 & \(2.41\times 10^{-4}\) & \(1.50\times 10^{-4}\) & 629 & 0.209 & \(7.41\times 10^{-5}\) & \(3.20\times 10^{-5}\) \\ \(f_{7}\) & 602 & 0.027 & \(6.17\times 10^{-4}\) & \(4.93\times 10^{-4}\) & 636 & 0.192 & \(1.77\times 10^{-4}\) & \(8.24\times 10^{-5}\) \\ \(f_{8}\) & 822 & 0.026 & \(3.62\times 10^{-4}\) & \(2.70\times 10^{-4}\) & 756 & 0.191 & \(3.48\times 10^{-4}\) & \(1.56\times 10^{-4}\) \\ \(f_{9}\) & 625 & 0.026 & \(8.56\times 10^{-4}\) & \(4.38\times 10^{-4}\) & 591 & 0.190 & \(4.84\times 10^{-4}\) & \(1.45\times 10^{-4}\) \\ \(f_{10}\) & 437 & 0.026 & \(2.97\times 10^{-5}\) & \(2.26\times 10^{-5}\) & 444 & 0.199 & \(1.09\times 10^{-5}\) & \(6.15\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Two-dimensional function approximations using SGNN and GRBFNN. Data are generated by averaging the results of 30 runs. Sample points: 1024; mini-batch size: 64; neurons per layer: 10.  
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**SGNN**} & \multicolumn{5}{c}{**GRBFNN**} \\  & **Epoch** & **Seconds/Epoch** & **Ave Loss** & **Min Loss** & **Epoch** & **Seconds/Epoch** & **Ave Loss** & **Min Loss** \\ \hline \(f_{1}\) & 269 & 0.038 & \(1.42\times 10^{-3}\) & \(9.33\times 10^{-4}\) & 270 & 4.049 & \(1.65\times 10^{-4}\) & \(9.60\times 10^{-5}\) \\ \(f_{2}\) & 253 & 0.039 & \(2.47\times 10^{-4}\) & \(1.02\times 10^{-4}\) & 157 & 4.055 & \(4.75\times 10^{-5}\) & \(