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

2025-11-03

AI Summary

This work addresses the challenge of modeling intra- and inter-species asymmetric interactions in multi-species particle systems. We propose the first nonparametric Bayesian learning framework tailored to heterogeneous multi-species systems. Methodologically, we extend Gaussian process regression to the multi-species setting for the first time, enabling flexible non-symmetric kernel modeling—e.g., predator–prey dynamics—and rigorously establish identifiability of interaction kernels and statistical optimality of the posterior estimator. Theoretically, we derive error convergence rates; empirically, our approach significantly outperforms existing kernel learning methods in accuracy, robustness, and generalization. Our core contribution lies in unifying the treatment of species heterogeneity, interaction asymmetry, and nonparametric kernel inference—yielding a provably optimal, scalable paradigm for inferring microscopic interaction rules directly from trajectory data.

Technology Category

• Machine Learning: Kernel Methods
• Game Theory and Economic Paradigms: Adversarial Learning
• Multiagent Systems: Multiagent Learning

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

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.

Problem

• Learning interaction kernels in multi-species particle systems
• Addressing heterogeneous interactions within and across species
• Establishing statistical guarantees for kernel recoverability and optimality

Innovation

• Gaussian process framework for multi-species particle systems
• Nonparametric Bayesian learning of interaction kernels
• Statistical guarantees for kernel recoverability and optimality