Data-Driven Model Selections of Second-Order Particle Dynamics via Integrating Gaussian Processes with Low-Dimensional Interacting Structures

TL;DR

This paper introduces a Gaussian Process-based method for data-driven discovery of second-order particle models, effectively capturing complex collective behaviors and interactions in high-dimensional systems with uncertainty quantification.

Contribution

It develops a nonparametric, scalable framework that integrates Gaussian Processes with low-dimensional interaction structures for modeling collective agent dynamics.

Findings

Successfully models fish flocking and milling behaviors in up to 248 dimensions.

Outperforms existing methods on real-world datasets with small sample sizes.

Provides theoretical insights into kernel recovery conditions.

Abstract

In this paper, we focus on the data-driven discovery of a general second-order particle-based model that contains many state-of-the-art models for modeling the aggregation and collective behavior of interacting agents of similar size and body type. This model takes the form of a high-dimensional system of ordinary differential equations parameterized by two interaction kernels that appraise the alignment of positions and velocities. We propose a Gaussian Process-based approach to this problem, where the unknown model parameters are marginalized by using two independent Gaussian Process (GP) priors on latent interaction kernels constrained to dynamics and observational data. This results in a nonparametric model for interacting dynamical systems that accounts for uncertainty quantification. We also develop acceleration techniques to improve scalability. Moreover, we perform a theoretical analysis to interpret the methodology and investigate the conditions under which the kernels can be recovered. We demonstrate the effectiveness of the proposed approach on various prototype systems, including the selection of the order of the systems and the types of interactions. In particular, we present applications to modeling two real-world fish motion datasets that display flocking and milling patterns up to 248 dimensions. Despite the use of small data sets, the GP-based approach learns an effective representation of the nonlinear dynamics in these spaces and outperforms competitor methods.