Swarm intelligence and mathematics
Text written by Bojan Crnković and Vedrana Mikulić Crnković, Faculty of Mathematics, University of Rijeka
Mathematical optimisation is a branch of mathematics that studies methods for determining the maximum or minimum of real functions. Many practical problems in science and in everyday life can be reduced to determining the minima of a function, that is a consequence of the mathematical model of the observed problem. It is precisely for this reason that optimisation methods are often used in applied mathematics. Problems that are solved using these methods can be found in various fields: technical sciences, economics, computer science, etc.
One of the optimisation methods is based on behaviours that we observe in nature, i.e. swarm intelligence.
Swarm intelligence is the collective behaviour of decentralised, self-organised systems that arises from the simple rules of behaviour of individuals. The term was introduced in 1989 by Gerardo Beni and Jing Wang in connection with cellular robotic systems. Such systems typically consist of a population of simple agents or individuals whose collective task is to discover a minimum of a function in their virtual or real environment. The inspiration for such procedures comes from nature, especially from biological systems or swarms that can perform very complex procedures. The agents follow very simple rules, and although there is no centralized control to dictate how the individual agents should behave, the rules of behaviour lead to a collective global behaviour that looks like a collective intelligence of the swarm.
The individual agents have no insight into the overall picture, but only communicate with their neighbours and behave randomly to a certain extent. Examples of swarm intelligence in natural systems are ant colonies, bee colonies, flocks of birds and schools of fish. Each of these swarms has its own peculiarities, which scientists have translated into algorithms that run on a computer and successfully solve many problems that scientists and engineers face every day in their work. The most common application is problems with the organisation of simple robots.
In addition to scientific research, such algorithms are also used to simulate the behaviour of large groups of people, which is particularly useful in filmmaking, where it is often a problem to create convincing computer-generated scenes. Some of the films that have used these methods are Lord of the Rings, Batman etc.