## 4 Numerical Simulations
### Comparison of the Different Methods  
The Tables 1 and 2, and Figure 6 shows the mean squared error and the mean absolute errors, which correspond to the errors calculated using \(\ell_{2}\) and \(\ell_{1}\) norms, respectively. The error is calculated as the difference between the actual and predicted conductivities on the entire test set for all the different methods where the Gaussian width is fixed. The vertical  
Figure 4: Reconstruction images by all the different models. The true conductivity is shown in a finer mesh. The images were reconstructed on a coarser mesh. The models (except the Gaussian noise model) were trained on data with 4% Gaussian noise, while the test data had 6% Gaussian noise. (**a**) True conductivity distribution. (**b**) Reconstruction with fixed Gaussian width and no regularization. (**c**) Reconstruction with variable Gaussian width and no regularization. (**d**) Reconstruction with fixed Gaussian width and Tikhonov regularization. Regularization parameter \(\lambda=10^{-5}\). (**e**) Reconstruction with variable Gaussian width and Tikhonov regularization. Regularization parameter \(\lambda=10^{-5}\). (**f**) Reconstruction with fixed Gaussian width and Lasso regularization. Regularization parameter \(\lambda=10^{-4}\). (**g**) Reconstruction with variable Gaussian width and Lasso regularization. Regularization parameter \(\lambda=10^{-4}\). (**h**) Reconstruction with fixed Gaussian width and Elastic Net regularization. Regularization parameters in Equation (22) are \(\lambda_{1}=10^{-5}\) and \(\lambda_{2}=5\times 10^{-2}\). (**j**) Reconstruction with fixed Gaussian width and the training data containing different levels of Gaussian noise. (**k**) Reconstruction with variable Gaussian width and the training data containing different levels of Gaussian noise. Figure 5: Reconstructed images and the original conductivity distribution for a triple-inclusion example with 5% Gaussian noise. All the models were trained on single- and double-inclusion examples only, and had fixed Gaussian width \(b_{j}=1\). (**a**) True conductivity distribution. (**b**) Reconstruction without regularization. (**c**) Reconstruction with Tikhonov regularization. Regularization parameter \(\lambda=10^{-5}\). (**d**) Reconstruction with Lasso regularization. Regularization parameter \(\lambda=10^{-4}\). (**e**) Reconstruction with Elastic Net regularization. Regularization parameters in Equation (22) are \(\lambda_{1}=10^{-5}\) and \(\lambda_{2}=5\times 10^{-2}\). (**f**) Reconstruction with no regularization and where the training data contained different levels of Gaussian noise. it performs poorly on new examples outside of the training set. Using too many radial basis functions in the network also causes overfitting. However, by adding some regularization terms, we can mitigate the overfitting problem even for many RBFs in the hidden layer. The existence of a regularization term in the RBF loss function optimization problem is absent primarily in the treatment of the EIT inverse problem in the literature [21; 22; 23; 24; 25; 26]. We demonstrate that regularization is necessary to obtain a proper inverse solver of EIT. The RBF network cannot reconstruct well outside the training data without regularization. Even though regularization causes the quality of the reconstructed image to deteriorate, it improves stability by correctly reconstructing images when tested on unseen data. In summary, we conclude that by looking at the \(R^{2}\) score, one can determine the optimal number of radial basis functions; the fixed-width radial basis function network with regularization results in improved performance. The fixed-width Gaussian with Tikhonov regularization performs very well. The regularization helps reconstruct the images outside of the training data. The regularization may cause the reconstruction quality to deteriorate. However, the stability is much improved. In terms of robustness, the RBF with Lasso and Elastic Net seem more robust compared to Tikhonov.