An anisotropic interaction model with collision avoidance

Abstract

In this article an anisotropic interaction model avoiding collisions is proposed. Starting point is a general isotropic interacting particle system, as used for swarming or follower-leader dynamics. An anisotropy is induced by rotation of the force vector resulting from the interaction of two agents. In this way the anisotropy is leading to a smooth evasion behaviour. In fact, the proposed model generalizes the standard models, and compensates their drawback of not being able to avoid collisions. Moreover, the model allows for formal passage to the limit 'number of particles to infinity', leading to a mesoscopic description in the mean-field sense. Possible applications are autonomous traffic, swarming or pedestrian motion. Here, we focus on the latter, as the model is validated numerically using two scenarios in pedestrian dynamics. The first one investigates the pattern formation in a channel, where two groups of pedestrians are walking in opposite directions. The second experiment considers a crossing with one group walking from left to right and the other one from bottom to top. The well-known pattern of lanes in the channel and travelling waves at the crossing can be reproduced with the help of this anisotropic model at both, the microscopic and the mesoscopic level. In addition, the 'right-before-left' and 'left-before-right' rule appear intrinsically for different anisotropy parameters.

Parameter calibration with stochastic gradient descent for interacting particle systems driven by neural networks

“…Nevertheless, in combination with the relaxation term that drives the pedestrian towards their desired destination, we expect to have a reasonable model. This problem for radially symmetric interactions is well-known and reported for example in [27]. Therein, an approach to solve this issue is presented as well.…”

“…In the recent years, many models for interaction dynamics with various applications such as swarming, sheep and dogs, crowd motion, traffic and opinion dynamics have been proposed, see, e.g. [1,[3][4][5]8,9,21,26,27] for an overview. Typically, the models are based on ordinary differential equations (ODEs) which describe the dynamics of each particle (or agent) in the system by interaction forces.…”

Space mapping-based optimization with the macroscopic limit of interacting particle systems

“…In the recent decades interacting particle systems attracted a lot of attention from researchers of various fields such as swarming, pedestrian dynamics and opinion formation (cf. Albi and Pareschi 2013;Helbing and Molnár 1995;Toscani 2006;Totzeck 2020 and the references therein). In particular, a model hierarchy was established Carrillo et al 2010;Golse 2003.…”

“…For applications with many particles involved, this microscopic modelling leads to a huge amount of computational effort and storage needed. Especially, when it comes to the optimization of problems with many particles There is also an intermediate level of accuracy given by the mesoscopic description, see Albi and Pareschi 2013;Carrillo et al 2010;Totzeck 2020. We do not want to give its details here, instead, we directly pass to the macroscopic level, where the velocities are averaged and a position-dependent density describes the probability of finding a particle of the dynamics at a given position.…”

Mean-field optimal control for biological pattern formation

“…In the past years, interacting particle or agent systems have been widely used to model collective behavior in biology, sociology and economics. Among the many examples of applications are biological phenomena such as animal herding or flocking [7,8,13], cell movement [19], as well as sociological and economical processes like opinion formation [15], pedestrian flow dynamics [6,28], price formation [4], robotics [27] and data science [23].…”