Data-driven Learning of Interaction Laws in Multispecies Particle Systems with Gaussian Processes: Convergence Theory and Applications

TL;DR

This paper introduces a Gaussian process framework for learning interaction laws in multi-species particle systems from trajectory data, providing theoretical guarantees and demonstrating effectiveness through numerical experiments.

Contribution

It extends previous single-species models to multi-species systems, addressing heterogeneity and asymmetric interactions with rigorous statistical analysis.

Findings

The method accurately recovers interaction kernels.

Provides quantitative error bounds for estimations.

Demonstrates superior performance over existing methods.

Abstract

We develop a Gaussian process framework for learning interaction kernels in multi-species interacting particle systems from trajectory data. Such systems provide a canonical setting for multiscale modeling, where simple microscopic interaction rules generate complex macroscopic behaviors. While our earlier work established a Gaussian process approach and convergence theory for single-species systems, and later extended to second-order models with alignment and energy-type interactions, the multi-species setting introduces new challenges: heterogeneous populations interact both within and across species, the number of unknown kernels grows, and asymmetric interactions such as predator-prey dynamics must be accommodated. We formulate the learning problem in a nonparametric Bayesian setting and establish rigorous statistical guarantees. Our analysis shows recoverability of the interaction kernels, provides quantitative error bounds, and proves statistical optimality of posterior estimators, thereby unifying and generalizing previous single-species theory. Numerical experiments confirm the theoretical predictions and demonstrate the effectiveness of the proposed approach, highlighting its advantages over existing kernel-based methods. This work contributes a complete statistical framework for data-driven inference of interaction laws in multi-species systems, advancing the broader multiscale modeling program of connecting microscopic particle dynamics with emergent macroscopic behavior.