## 4 Results and Discussion
### Results of Learning  
Here, we present the experimental results of learning and show that the proposed network successfully learned the optimizer \(\Lambda\). Figure 8 shows the values of the loss function at each epoch. At the final epoch (the 6000 th), the loss function for the training data shows a decreasing trend, while the one for the validation data converged, indicating that the learning was sufficient. Figure 8: Learning curve. Blue and red points are results for training data and validation data, respectively. Figure 7: Structure of optimizer \(\Lambda\). Dense\((n,f)\) indicates a dense layer with \(n\) nodes and activation function \(f\ :\ R^{n}\to R^{n}\). DropOut\((r)\) indicates the dropout layer with dropout rate \(r\). ReLU is short for rectified linear unit function. To evaluate the performance of the learned optimizer \(\Lambda\), we validated it on a test dataset consisting of data not included in the training and validation datasets. The properties described below are common to all of the test datasets. We will discuss the results for five of the test data pairs shown in Figure 9. The first test data item (\(\ell=1\)) was a pair of a projection \(y^{1}\) with an SNR of 20 dB and the expected reconstructed image \(e^{1}\). In the second and third test data pairs (\(\ell=2\) and 3), the expected reconstructed images \(e^{2}\) and \(e^{3}\) equaled \(e^{1}\) but with different Sn Rs of the projection. On the other hand, in the fourth and fifth test data pairs (\(\ell=4\) and 5), the expected reconstructed images \(\epsilon^{4}\) and \(\tilde{e^{5}}\) were different from \(e^{1}\) and from each other, but the noise levels of the projections were 20 dB. Figure 10 plots the evaluation function values  
\[D(\ell,z):=\|e^{\ell}-z\|_{2} \tag{12}\]  
defined by the L2 norm of the difference between the \(\ell\)th true image \(e^{\ell}\) and reconstructed image \(z\) when using the parameter \(\hat{\lambda}^{\ell}=(\hat{\gamma}^{\ell},\hat{\alpha}^{\ell})\) estimated by the learned model \(\Lambda\) from the projection \(y^{\ell}\) with \(\ell=1,2,\ldots,5\). The results for the MLEM algorithm, i.e., the PDEM algorithm with \(\lambda=(1,1)\), are displayed for comparison. For the first test data pair (\(\ell=1\)), due to the high noise level, the evaluation functions for both the MLEM and the proposed algorithm, which is the PDEM using the parameter values \(\hat{\lambda}^{1}=(0.40,1.05)\) estimated by the optimizer \(\Lambda\), do not decrease monotonically but rather start to increase at different iteration numbers, around \(k=50\) and \(140\), respectively. The proposed method decreases the evaluation function significantly in the last iteration  
Figure 9: Dataset for testing. compared with that of the MLEM algorithm. This result indicates that the optimizer \(\Lambda\) can estimate parameter values that, while not optimal, significantly improve image quality. The results for the second test data pair (\(\ell=2\)), which has a different noise level than the first test data pair (\(\ell=1\)), shows that the proposed method also reduces the evaluation function compared with the MLEM algorithm, but the difference is slight. We discuss the reason in the following subsection. The proposed method can significantly reduce the evaluation function when using the third test data pair (\(\ell=3\)), which has weak noise. The reason for this is presumably that the power exponent parameter \(\gamma\) is a value that affects the step size in the discretization of the continuous dynamical system and that, when the noise is weak, a relatively large parameter is chosen for fast convergence to a value close to the true one. In fact, the estimated parameter \(\hat{\gamma}^{3}\) is \(1.64\), which is larger than the others. The results for the fourth and fifth test data pairs (\(\ell=4\) and \(5\)), where the projection contains a lot of noise, show a slight decrease in the value of the evaluation function, contrary to the results for the first test data pair (\(\ell=1\)), where the SNRs of the projections are comparable.