## 2 Previous Works  
Feedforward neural networks (FNNs) have been applied to a wide variety of problems arising from a variety of disciplines, including mathematics, computer science, and engineering [25]. However, the training algorithm has a profound impact on the learning capacity and performance of the network. Two challenges need to be resolved in training neural networks: how to avoid local minimum and how to achieve a fast convergence rate. The ANN training process is an optimization task that aims to find a set of weights and biases to minimize an error value. Since the search space is highly dimensional, ANN training needs more powerful optimization techniques; therefore, several conventional gradient descent algorithms, such as back-propagation (BP), are used to solve this issue. However, gradient-based algorithms are sometimes susceptible to being converged into local optima, because they are local search methods in which the final solution depends strongly on the initial weights.  
In order to cope with the local minimum problem, heuristic optimization algorithms, such as GA, ACO, and PSO have been adopted for training ANNs. These algorithms do not use any gradient information, and have a better chance in avoiding local optima by simultaneously sampling multiple regions of the search space. They have the advantage of being applicable to any type of ANN, feedforward or not, with any activation function [26]. They are particularly useful for dealing with large complex problems, which generate many local optima. In order to find the global optimum, a heuristic algorithm should have two main characteristics: exploration and exploitation [27]. Exploration is the ability to search whole parts of problem space, whereas exploitation is the convergence ability to reach the best solution. Several recent heuristic algorithms, including Backtracking Search Optimization (BSO), Differential Search Optimization (DSO), Artificial Bee Colony (ABC), Harmony Search Algorithm (HSA), Gravitational Search Algorithm (GSA) and Cuckoo Search (CS), all inspired by the behavior of natural phenomena, were developed for solving optimization problems. To a certain extent, through some comparison and test studies, these algorithms have been proven to be powerful in solving various problems such as real-valued numerical optimization problems and the problem of transforming the geocentric Cartesian coordinates into geodetic coordinates [28; 29; 30; 31; 32]. An algorithm is efficient when it has a balance between the ability of exploration and exploitation. Many studies have shown that merging different algorithms is a way to achieve this balance.  
PSO is a population-based heuristic global optimization algorithm, and is often referred to as a swarm-intelligence technique. Due to its merits, including simplicity, convergence rate, and its ability to search for a global optimum, PSO is one of the most commonly used algorithms in hybrid methods, such as PSOGA [33; 34], PSODE [35], and PSOACO [36]. These hybrid algorithms are aimed at reducing the probability of being trapped in local optimum. The convergence rate and the global search ability of the aforementioned works can be further improved. The idea of developing hybrid algorithm has appeared for a long time. However, as far as we are aware, not much research has been undertaken to combine PSO with CS. [PERSON] and [PERSON] [37] added swarm intelligence to the cuckoo birds, in order to increase the chance of their eggs' survival. Each cuckoo bird will record the best personal experience during its own life. The goal of this is to inform each other from their position and help each other to immigrate to a better place. By the use of swarm intelligence, it can be seen that the hybrid algorithm observes more search space and can effectively reach better solutions. In our work, another hybrid algorithm of the PSO and CS is proposed. In the proposed algorithm, the remarkable strengths of both PSO and CS algorithms are integrated to ensure finding the global optimum.