Interacting Particle Systems and Their Applications

Overview

Interacting particle systems (IPSs) are ubiquitous in science and engineering. The application ranges from physical systems such as rarified gases, to biological systems such as cell pattern evolution, to artificial systems such as ensemble type algorithms and neural networks. The core of the field revolves around examining the collective dynamics of a large number of interacting agents, and quantifying the evolution of their statistical properties. Rooted in non-equilibrium statistical physics, the modeling, analysis and application angles are all integrated in the study of IPSs. The associated mathematical techniques are closely tied to the mean-field limit derivation, and long-time equilibrium analysis for partial and/or stochastic differential
equations.

Besides the traditional application domain in physical sciences, the renewed interests are largely drawn by the recent advancements in machine learning (ML) and artificial intelligence (AI). In these applications, one is typically presented a system composed of a large number of samples, and these samples advance in time interactively to achieve a desired property, either to provide samples from a equilibrium distribution, or to find a global optimizer. Compared to classical problem settings rooted from physical sciences that are process-oriented, these systems are goal-oriented: A task is given a-priori, such as Bayesian sampling or optimizing, researchers have the freedom to design and model the dynamics to achieve this goal. This perspective change brings many challenges as well as flexibility, and thus opens up a new avenue of research.

On the fundamental level, all the design and study of IPSs for these newly emerged problems translate to answering the following two questions:
• How to design the interaction between particles so to achieve a desired property efficiently? This is the analysis leg of the framework. It comes down to designing a mechanism that best makes use of the interactions between agents to speed up, in comparison to single-particle- systems, the convergence to a desired goal. Such mechanism is typically presented using a
PDE formulation.
• How to design an algorithm pipeline to solve the designed PDE?
This is the computation leg of the framework. Among many choices, the classical Monte Carlo approach is undoubtly the most popular, and can be deployed to translate the continuous-in- space and continuous-in-time PDE formulation to the discrete-in-space and continuous-in-time IPSs. ODE/SDE solvers are then employed to simulate them