ReviewCollective learning modeling based on the kinetic theory of active particles
Section snippets
From five key questions to the plan of the paper
Since the first half of the last century, some pioneering studies were devoted to the mathematical formulation and representation of learning phenomena [54], [55], [56], [61], [74], followed over the years by a growing interest toward the derivation of suitable theories and models. In particular, [29] is a survey of several studies developed in the '50s where two main schools of thinking are pointed out. The models related to the first one are known in the literature as “stimulus sampling
Perception, learning and classification
This section focuses on the first key question:
The aim is to understand how the learning phenomena can develop within a system composed of many interacting individuals, who “learn” by interactions. This can be viewed as a first step toward the derivation of mathematical tools for modeling purpose.How can the collective perception and learning dynamics in systems composed of many living entities be interpreted and classified by a mathematician?
We do not naively claim that this paper will cover
Representation of complex learning systems
Let us now consider the first step of the modeling approach, which is related to the second key question:
The hierarchical structure of the system is represented in Fig. 1(a) along with the interactions that will be discussed in the next section. More in detail, individuals are regarded as active particles and are assumed to be distributed in a network composed by nodes further subdivided in a number ofHow can a large system of individuals who learn be represented by mathematical variables?
Modeling interactions
This section focuses on the third key question:
At least two scales can have an influence on the modeling of interactions, namely the microscopic scale and the macroscopic scale. The microscopic scale refers to interactions between active particles and involves their microstate. In such interactions three types of particles can be distinguished:How can the interactions leading to a learning dynamics be accounted for and which are the metrics appropriate to model their quantitative role?
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Test
From mathematical structure to a collective learning process
This section focuses on the fourth key question:
How can the learning dynamics be inserted into a mathematical structure?
The mathematical approach proposed in [19] leads straightforwardly to the derivation of a mathematical structure which includes the various interactions previously presented. This mathematical structure describes the evolution in time of the probability distribution over the learning variable for each functional system and consists of a system of integro-differential
Hallmarks of the modeling approach and application
The hallmarks of the approach presented in the previous sections can be summarized as follows:
- 1.
Assessment of the FSs that can have an important role in the learning dynamics;
- 2.
Selection of the type of interactions that have an influence on the aforementioned dynamics and their modeling;
- 3.
Derivation of the system of equations which describe the time evolution of the distribution functions linked to the FSs.
Critical analysis and research perspectives
This section focuses on the last key question:
Which are the challenging research perspectives in the modeling of collective learning dynamics?
Rather than presenting a list of open problems, which in the authors opinion can be very many, we here reconsider in some more depth the two examples of complex learning dynamics presented in Subsection 2.1 and, for each of them, we propose some hints toward possible research perspectives.
Acknowledgements
L.G. acknowledges the financial support received from the European Union's Seventh Framework Programme (FP7/2014–2016) under Grant Agreement Number 607626 (Safeciti). Project title: “Simulation Platform for the Analysis of Crowd Turmoil in Urban Environments with Training and Predictive Capabilities”. This publication reflects the views only of the authors and the Commission cannot be held responsible for any use which may be made of the information here contained.
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