## 2 Related Works  
In this section, we analyse some recent works related to robot simulations using surrogate models. A review addressing computational time, accuracy, and problem size in surrogate models is available in [15], while in [16], the authors surveyed the use of surrogate models in optimisation algorithms. In [17], a surrogate-based method is used to set up a parameter of the Rossler chaotic system to improve coverage of the CACOC (Chaotic Ant Colony Optimisation for Coverage). The authors proposed Bayesian optimisation to efficiently explore the parameter space, avoiding using costly simulations. Their results show that this method permitted efficient exploration of a bifurcation diagram bypassing periodic regions, providing two groups of points with excellent results in terms of coverage for the swarm. In [18], the authors present a control system for a quadcopter using several machine learning techniques. Time series, Gaussian processes, and neural networks are proposed to calculate optimum control gains for a specific mission and overcome environmental uncertainties. These predictors are used in an optimisation process and tested using simulations. Their results show performance improvements when compared to nominal control gains due to a better exploration of the search space. In [19], a surrogate model based on gene expression programming is proposed for the optimisation of an autonomous underwater vehicle's shape using computational fluid dynamics. This surrogate model of resistance and surrounded volume is also compared with the response surface model. The results obtained using a multi-objective particle swarm optimisation are compared with hydrodynamic calculations. It shows that the reduced computational cost when using the surrogate model and the model's accuracy improved the optimal shape design. In [20], a mathematical-computational model for the control and navigation of robots is proposed. The authors use a combination of a 2D cellular automata, Tabu search, ant colonies, and greedy approaches for selecting elitist cells. Then, a genetic algorithm is used to optimise the parameters for two proposed surrogate models. The main objective of this system is the maximisation of area coverage by using a pheromone-based approach. The validation of the models was performed using Webots simulator and E-Puck robots. In [21], the authors present a surrogate approach using the Kriging method to optimise the design of the delta wing and the canard wing of a tube-fan hybrid UAV. Moreover, a multi-objective genetic algorithm is proposed with the objective of maximising the UAVs lift and minimising the energy consumption. Computational fluid dynamics simulations were used to validate the calculated solutions. In [22], a distributed Bayesian optimisation framework for deep neuroevolution is presented. An acquisition function is defined to mimic the actual model according to a set of input parameters. The actual model is a neural network with its training dataset and the proposed optimisation strategy, i.e., distributed swarm-based neuroevolution. The authors use the proposed method for training various feed-forward neural networks for pattern classification problems. Their results show a promising performance of the proposed method, which has a reduced computational time for large deep learning problems. In our present work, we analyse Gaussian processes as well as other methods to calculate an accurate surrogate model for our problem, as some of the aforementioned articles also do. Conversely, we use the best performing predictor to approximate the results from simulations and speed up the optimisation of our 3D formation problem. We have proposed an optimisation algorithm, i.e., a hybrid EA, different to those used in related works, which allowed us to address bigger swarms with affordable execution times. Summing up, in this article, we propose six surrogate models for the robot formation problem, then train them and analyse the results in terms of accuracy and execution times.