## 4 Discussion  
### Impacts of Training Data on Reconstruction Quality and Generalization Performance  
We discuss the performance difference between the BP-DNN and NI-DNN. First, we discuss the performance of the BP-DNN. From the visual evaluation, the BP-DNN was able to reconstruct structures down to \(1/10\) of a wavelength. This is finer than \(1/2\) wavelength, which is the minimum resolution (diffraction limit) for imaging using the nature of wave interference. The reasons and limitations of this phenomenon are discussed from the perspective of the constraints imposed on the task. Model-driven methods, such as FWI, utilise physical models to perform sound speed reconstruction. In this case, the constraints (prior information) of the problem are given in a formulated form, such as total variation regularisation [39] and Tikhonov regularisation [40]. On the other hand, DNNs are data-driven methods and utilise the constraints estimated from the training data to perform sound speed reconstruction. In this case, there is no need to provide additional constraints by regularisation. The BP-DNN in this study is considered to have estimated the constraints of the problem from the training data and restricted its solution space to a smaller subspace. Therefore, it is assumed that the BP-DNN is able to provide detailed texture from the measured signal even at a large wavelength relative to the structure. Figure 6: Generalization performance against tumours in the breast. Each row shows a ground truth and the DNNs predictions of the breast model, according to their tumour sizes. Left column: ground truth. Centre column: prediction of the BP-DNN. Right column: prediction of the NI-DNN. The NI-DNN could capture the tumours with a resolution of about half a wavelength, while the BP-DNN overlooked the tumours. This approach is effective when there is no domain shift, i.e., the training data and the real-world data are sampled from the same distribution [41]. However, as discussed in the introduction, the true sound speed distributions of breasts are unknown. Even if the true sound speed distribution could be obtained by other methods, it would be difficult to collect a large amount of data on abnormal structures such as tumours, which are important in medical examinations. This means that it is difficult to avoid a domain shift. Therefore, there is concern that the DNN may not work properly if there is a difference between the artificially produced breast phantom and the actual breast in pattern of sound speed distribution, or if there is a tumour that appears infrequently. Actually, our experimental results showed the BP-DNN had poor generalization performance, as it was unable to reconstruct structures and produced false structures for out-of-domain data (tumour models and natural image data). This suggests that the training on the breast phantom data led to an insufficient model in generalization performance. On the other hand, the NI-DNN showed high generalization performance. The NI-DNN was able to reconstruct structures not only on the test data of natural image data, but also on the breast phantom and tumour models data which have patterns not included in the training data. Although its resolution was inferior to the BP-DNN, it was able to reconstruct structures down to half a wavelength, which is fine enough for practical use. These results can be explained as follows. The BP-DNN has an excessively restricted solution space, which means that more accurate solutions can be reached for the data inside the training data domain, resulting in very fine resolution. Simultaneously, the generalization performance is significantly degraded for out-of-domain data. On the other hand, the NI-DNN with a wide variety of patterns is assumed to have a large solution space, so the range of possible solutions is wide, resulting in a resolution that is not extremely small. Simultaneously, the generalization performance is improved for data with patterns not included in the training data.