## 1 Introduction
Initially, the 2D SOVA was developed for holographic data storage systems; however, [PERSON] et al. proposed an iterative 2D SOVA for bit-patterned media [16]. Because 2D SOVA takes the received signal through two 1D detectors, the complexity is similar to the VA, and it fits 2D interference. Recently, [PERSON] and [PERSON] improved the 2D SOVA into a multipath ISI structure that is suitable for a staggered BPMR structure [17]. In addition to parallel detection, serial detection for BPMR was also proposed by [PERSON] and [PERSON] [18]. In serial detection, GPR is decomposed into a series of two (down-track and cross-track) 1D PR targets to estimate the 2D interference. Based on the serial form of the GPR, serial detection that includes two serial 1D Viterbi detectors is proposed. The first detector is used for inner detection, using the VA to convert the received signal into a six-level signal. This implies that the inner detection still detects the signal in the hard output, which reduces the performance of serial detection. Therefore, [PERSON] and [PERSON] [19] resolved this problem using a soft output for inner detection in serial detection. To create a soft output, the noise information is estimated from the received signal, and then added to the hard output of the inner detection. In addition to the VA, there are many detection schemes based on the BCJR algorithm. [PERSON] et al. proposed an iterative row-column soft decision feedback algorithm (IRCSDFA) to address the 2D interference [20]. Because [PERSON] et al. considers the 2D input in the trellis, IRCSDFA is similar to 2D detection; however, the complexity of IRCSDFA is extremely high. Subsequently, [PERSON] et al. proposed a less complex IRCSDFA applying Gaussian approximation (IRCSDFA-GA) [21]. Although IRCSDFA-GA has low-complexity compared to IRCSDFA, it is still more complex than the VA. In recent years, the neural network approach has been extensively studied and widely applied to symbol detection. In the field of communication, neural networks are used as nonlinear equalizers having a higher performance capacity compared to the linear equalizer [22]. Further, in data storage, the neural networks can replace the equalizer and improve the performance compared to the conventional equalizer [23]. In [24], the detection scheme using a neural network coupled with TMR predicting ability achieved better performance than partial response maximum likelihood (PRML) detection. For the modulation code, [PERSON] showed that a decoder using a neural network outperformed a decoder with a minimum Euclidean distance [25]. In this study, based on the concepts in [19] and the wide application of neural networks in data storage systems, we built a model using a neural network to predict the noise information for the soft output in serial detection. However, when the neural network is trained at a low noise level, it does not satisfactorily predict the noise because of overfitting. Therefore, we designed a noise power estimator to reduce the overfitting. The proposed model includes two parts--one predicting the distortion of the noise using the neural network, and the other estimating the noise power using a parameter that is multiplied by the predicted noise. We also compared the proposed model with the model in [19]. The results show that our model achieves a better BER performance than the soft output model in [19]. The remainder of this paper is organized as follows. The serial detection and theory of the proposed model are described in Section 2. The proposed model is explained in Section 3. In addition, we also compare our model with the soft output model in [19]. The results and discussion are presented in Section 4. Finally, in Section 5, the conclusions are presented.