## 1 Introduction  
Robot formations, as a part of swarm intelligence, consist of a group of robots showing a collective behaviour, which is usually achieved from emerging collaborations with the objective of performing some specific global tasks. Unmanned Aerial Vehicles (UAVs), as swarm members in a formation, can be arranged in a specific three-dimensional shape to perform different types of missions, such as surveillance [1], synchronisation of space-crafts [2], salvage missions [3], and localisation and mapping [4], as well as representing dynamic deforming figures [5]. These types of missions usually present problems such as the unknown initial positions of the swarm members, as well as the need for path planning from these positions to the final locations. There is also a challenging adaptability to real situations, e.g., asteroid observation or secorting a rogue drone (RD) out of a restricted area, especially when there are collisions, communication losses, or robot failures. This problem cannot be accurately represented using a mathematical model where UAV iterations, such as collision avoidance manoeuvres, have to be taken into account. Therefore, a simulator is frequently used to model these real-world problems. Simulating a UAV swarm in a 3D space often uses a high amount of computing resources (and time) to achieve high levels of accuracy, especially when using a multi-physics robot simulator, e.g., ARGoS [6]. The use of a communication layer plus an inertial model for calculating each simulation step demands from seconds to minutes, depending on the number of robots modelled [7]. Optimising these problems requiring simulations rapidly becomes unaffordable, usually due to the high number of evaluations needed to successfully obtain an optimal solution. Hence, an alternative technique, e.g., surrogate models, has been proposed to successfully complete such studies. Bayesian optimisation, as a surrogate model [8], can be used to estimate the fitness value for the objective function during an optimisation process. It leads to an efficient reduction in the computation times required to evaluate expensive optimisation problems. Evolutionary algorithms (EAs) and gradient methods have been used in combination with surrogate models [9]. They have been applied to different problems such as modelling circuits and systems [10], forecasting wildfires [11], predicting noise emission and aerodynamic performance of propellers [12], sustainable building design [13], and modelling groundwater [14]. In previous research works [7], we have found evidence that the optimisation of UAV formations consisting of swarms of more that ten robots requires a high number of computationally expensive simulations, which makes it unaffordable in most cases. In general, the required number of UAVs to efficiently surround a rogue drone is greater than ten in such 3D formations where the virtual sphere's radius could be several metres. We study in this article the viability of using surrogate models based on Gaussian processes (GPs) and artificial neural networks (ANNs) as predictors of the UAV swarm behaviour when arranged in formation. By doing so, we will be able to address the optimisation of a greater number of UAVs, increasing the efficiency and utility of the robot formation. The main contributions of this paper are:  
1. The study, training, and testing of six surrogate models to predict the behaviour of a UAV swarm in formation. 2. A hybrid EA (HEA) to be used as the optimisation algorithm for the parameters of our formation system, which combines simulations and predictions to balance efficiency and accuracy. 3. The evaluation of the optimised formation swarm in 150 unseen scenarios comprising up to 30 UAVs in terms of accuracy and stability. The rest of this paper is organised as follows. A review of the state-of-the-art research related to our proposal can be found in the next section. In Section 3, we describe our UAV formation system, the simulation model, and the six proposed surrogate models.