## 1 Introduction  
Clinical time-of-flight positron emission tomography (TOF-PET) scanners with 200-600 picosecond (ps) coincidence time resolution (CTR) have been developed [1; 2; 3]. In laboratory studies, CTR has approached \(\sim\)30 ps at the detector level [4; 5]. TOF information determines the annihilation position on the coincidence line connecting the two interaction positions of \(\gamma\)-rays with detectors, with the position uncertainty \(\Delta x\approx\varepsilon\Delta t/2\), where \(c\) is the speed of light and \(\Delta t\) is the CTR. For example, if \(\Delta t=200\) ps, \(\Delta x=30\) mm. Although the position uncertainty of 30 mm is not comparable to the spatial resolution of clinical PET scanners (4-6 mm), the signal-to-noise ratio of the reconstructed image can be improved depending on the CTR and patient size using the TOF information in the reconstruction process [3]. If we could develop a TOF-PET detector with 26-40 ps CTR, corresponding to a position uncertainty of 4-6 mm, annihilation points could be determined event-by-event. Therefore, the image reconstruction process, which can amplify the noise of PET images, could be removed from the clinical PET. Recently, Cherenkov radiation-based TOF-PET detectors have been developed [6; 7; 8]. Because the emission of the Cherenkov photon is as fast as a few ps, the detection of Cherenkov photons could potentially approach 10 ps CTR [9]. However, to stop the \(\gamma\)-ray using the Cherenkov radiator, such as a monolithic lead fluoride (PbF2) crystal with the same detection efficiency as a lutetium-basedscintillator, thicker crystal is needed, for example, 20 mm. As the radiator thickness increases, the depth of interaction (DOI) of \(\gamma\)-rays causes timing errors because of the propagation speed difference between \(\gamma\)-rays and photons in crystals [10; 11; 12]. Thus, to achieve an extremely good CTR on the order of 10 ps, both TOF and DOI information will be needed. However, as fewer than 30 Cherenkov photons are produced by the 511 keV \(\gamma\)-ray [9; 13], DOI estimation becomes challenging. Hence, DOI estimation methods based on machine learning have been developed for Cherenkov detectors [8; 14]. Although the machine learning approach enables the Cherenkov detector to estimate DOI, it requires a dataset equipped with correct DOI. The measurement of the correct DOI for the Cherenkov detector requires substantial time and effort. Some methods that can be used to avoid this problem have been developed in detectors based on a monolithic scintillator [15; 16; 17; 18]. However, these methods could not be applied to the monolithic Cherenkov detector. From a different point of view, the cause of the problem could be that we try to indirectly estimate the annihilation position from the TOF and DOI. If we can estimate the annihilation position directly from the spatial and temporal photon distribution measured by a pair of Cherenkov detectors, there would be no need to estimate the DOI in the first place. In this study, we propose a direct annihilation position classification based on deep learning using paired Cherenkov detectors. We estimate the annihilation position directly from the photon information measured by paired Cherenkov detectors using deep learning instead of intermediate estimation of TOF and DOI. Deep learning is a powerful tool that has attracted the interest of researchers in the medical imaging field [19; 20; 21]. Recently, deep learning has been used to process raw data bypassing image reconstruction, such as ghost cytometry [22] or sinogram-space machine learning [23]. In the same spirit, we use deep learning to bypass the intermediate estimation of DOI and TOF and estimate the annihilation position directly from the raw data of photon information measured by paired Cherenkov detectors.