## 1 Introduction
To combat this problem, regularization techniques such as constraining neuron weights or random neuron dropout are used, to name a few. The problems of stochastic inference interpreted as ML tasks have attracted considerable attention in the field of physical layer communication [7; 8; 9; 10]. The classical mathematical formulations of detection, estimation, and signal processing algorithms usually require a precise analytical knowledge of the system and observation model and usually lead to provable optimal closed-form results. Alternative ML-based approaches provide in such situations only numerically solved approximations to those solution. Although this has a limited applicability in classical scenarios, there are situations where the system and observation model are not fully known and/or the resulting closed-form solutions are too complex or unknown. In that situation, the ML approach provides an answer. As a demonstrative example, a particularly obvious area where this applies in the context of stochastic graphs is WPLNC (wireless physical layer network coding). Recently, many interesting applications of ML were proposed for usage in wireless communications. To provide several motivating examples, in [11], the authors used ANN to perform joint OFDM channel estimation and symbol detection, where the learning phase was performed offline based on known channel statistics. In [8; 9], a deep learning approach to jointly optimize the whole wireless communication chain was proposed, and this approach was successfully tested in an over-the-air transmission. Similar to this, in the context of WPLNC, the authors in [10] addressed parts of the network chain as individual deep neural networks, divided into the source modulator, a relay node, and the demodulator. Attention attracted also the approach of modulation classification in [12], where it was interpreted as an image classification problem. These papers largely motivated our work. So far, utilization of the ML in WPLNC is not deeply explored. The concept of WPLNC recently attracted much attention. A comprehensive tutorial of WPLNC techniques might be found in [13], where fundamental concepts, advantages, and challenges were covered. This paper investigates the applicability of the ML concept in the parametric, slowly varying two-way relay channel (2 WRC) WPLNC communication. This scenario is on the edge which still can be handled by the traditional approaches, see [14]. This allows us to find the required reference scenarios to which we compare the ML solution and gives a hint about how the ML solution could be applicable in more complex scenarios, which are beyond the capability of classical analytical solutions. As was stated in [7], it is suitable to consider ML, when (1) the physical system model is not known, (2) the training dataset is available, (3) a clear metric of the task can be defined, and (4) a detailed explanation of the obtained result is not required. These general conditions are easily met in our context of wireless communication networks. Our contribution in this paper is an exploration of a novel approach to compensate the effects of the variable phase of the wireless channel in a fundamental WPLNC scenario without the need for explicit channel state estimation (which would generally require orthogonal pilots). Instead, we use an ML approach that inherently tracks (learns) the varying hierarchical channel relative phase with tractable computational complexity. This is achieved with a hypothesis class of the ANN, and the task is interpreted as a classification problem. We explore methods to re-use previously trained coefficients of the ANN, thus saving computational resources. Validation of the results was performed by means of a numerical analysis. The rest of the paper is organized as follows. Section 3 introduces the notation and describes the system model. Section 4 outlines a generic approach of ANN utilization for the classification of hierarchical symbols.