# Integrated Photonic Neural Networks for Equalizing Optical Communication Signals: A Review
## 3 Training Algorithms of Photonic Neural Networks for Equalizing Communication Signals  
One obstacle faced by photonic neural networks is the need to perform their training stage still in the electronic domain [89]. This training consists of an optimization process in which the neural network's parameters are adjusted so that it can perform specific tasks after this process. To date, the particle swarm optimizer (PSO) has been the most widely used training algorithm for photonic neural networks when applied for equalizing optical communication signals [76; 78; 79; 80; 81]. The PSO is an approach introduced by [PERSON] and [PERSON] in 1995 [90], initially motivated by the natural behavior of bird flocks searching for food sources. Currently, the PSO and its modifications have been successfully utilized to solve real-life problems and engineering design optimization due to their simplicity and ability to be employed in diverse applications [91]. For instance, the PSO has been applied to enhance the efficiency and the precision of support vector machine classifiers [92], to predict hydrogen yield from gasification operations [93], for cardiovascular disease  
Figure 3: Architecture of integrated photonic reservoir computing for equalizing optical communication signals. Reproduced without modification from [PERSON] et al. [73], © 2018, licensed under a Creative Commons Attribution International License 4.0 (CC BY 4.0). Figure 2: Sketch of integrated photonic reservoir computing. Reproduced without modification from [PERSON]. [PERSON] et al. [60], © 2020, licensed under a Creative Commons Attribution International License 4.0 (CC BY 4.0). prediction [94], for early detection of forest fires [95], and for gas detection [96], among other applications [97; 98]. Its fundamental principles that allow the optimization process to be carried out are described below. Specifically, in this optimizer, individuals are called particles and fly through the search space, looking for the best position that minimizes (or maximizes) a given situation [98]. For its full implementation, that is, finding the best solution, the movement of each particle towards its prior personal best position (_pbest_) and the global best position (_gbest_) in the swarm needs to be calculated, as described below [91]:  
\[ \begin{split}&\textit{pbest}(i,t)=\operatorname*{arg\,min}_{k=1, \,\ldots,\,t}[f(P_{i}(k))],\quad i\in\{1,2,\ldots,N_{p}\},\end{split}  \tag{4}\]  
\[ \begin{split}&\textit{gbest}(i,t)=\operatorname*{arg\,min}_{i=1, \,\ldots,\,N_{p}}[f(P_{i}(k))],\\ &\textit{k}=1,\,\ldots,\,t\end{split}  \tag{5}\]  
where \(i\) represents the particle index, \(N_{p}\) the number of particles, \(t\) the current iteration, \(f\) the fitness function, and \(P\) the position. The velocity \(V\) and the position \(P\) of the particles are determined, respectively, by [91]  
\[V_{i}(t+1)=\omega V_{i}(t)+c_{1}r_{1}(\textit{pbest}(i,t)-P_{i}(t))+c_{2}r_{2} (\textit{gbest}(t)-P_{i}(t)), \tag{6}\]  
\[P_{i}(t+1)=P_{i}(t)+V_{i}(t+1), \tag{7}\]  
where \(\omega\) is the inertia weight employed to balance the global and local exploitations, \(r_{1}\) and \(r_{2}\) are arbitrary variables uniformly determined, and \(c_{1}\) and \(c_{2}\) are positive constant parameters named acceleration coefficients [91]. For photonic neural networks based on reservoir computing architectures, since only the weights in the output layer are adjusted, the training process can be performed through a simple linear regression, making the training process fast and simple [75]. In terms of the training stage of photonic neural networks, which is still performed in the electrical domain, we can mention, as an example of how this process is performed, the work carried out by [PERSON] et al., in which the PSO was employed during the training phase of a photonic neural network for chromatic distortion precompensation [81]. The training is carried out by a computer that elaborates the output signal and regulates a current control following the PSO algorithm.