Elsevier

Physics of Life Reviews

Volume 16, March 2016, Pages 123-139
Physics of Life Reviews

Review
Collective learning modeling based on the kinetic theory of active particles

https://doi.org/10.1016/j.plrev.2015.10.008Get rights and content

Highlights

  • Collective perception and learning are interpreted and classified.

  • Heterogeneity and non-linearity are expected to play a central role.

  • The Kinetic Theory of active particles provides a unified framework for the modeling.

  • The dynamics of probability distributions offers deep insights into learning processes.

Abstract

This paper proposes a systems approach to the theory of perception and learning in populations composed of many living entities. Starting from a phenomenological description of these processes, a mathematical structure is derived which is deemed to incorporate their complexity features. The modeling is based on a generalization of kinetic theory methods where interactions are described by theoretical tools of game theory. As an application, the proposed approach is used to model the learning processes that take place in a classroom.

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:

How can the collective perception and learning dynamics in systems composed of many living entities be interpreted and classified by a mathematician?

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.

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:

How can a large system of individuals who learn be represented by mathematical variables?

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 of

Modeling interactions

This section focuses on the third key question:

How can the interactions leading to a learning dynamics be accounted for and which are the metrics appropriate to model their quantitative role?

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:
  • 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.

As an example, we consider the learning process that takes place in a classroom as described

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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