Discovery of interaction and diffusion kernels in particle-to-mean-field multi-agent systems

TL;DR

This paper introduces a data-driven method to identify interaction and diffusion kernels in stochastic multi-agent systems directly from trajectory data, without prior knowledge of the interaction structure, using sparse regression and mean-field approximations.

Contribution

The paper presents a novel framework combining sparse regression and mean-field approximation to learn interaction kernels from limited and partially observed data in multi-agent systems.

Findings

Accurately reconstructs interaction and diffusion kernels from limited data.

Effective for benchmark models like attraction-repulsion and bounded-confidence.

Two strategies (random-batch and mean-field) achieve comparable accuracy.

Abstract

We propose a data-driven framework to learn interaction kernels in stochastic multi-agent systems. Our approach aims at identifying the functional form of nonlocal interaction and diffusion terms directly from trajectory data, without any a priori knowledge of the underlying interaction structure. Starting from a discrete stochastic binary-interaction model, we formulate the inverse problem as a sequence of sparse regression tasks in structured finite-dimensional spaces spanned by compactly supported basis functions, such as piecewise linear polynomials. In particular, we assume that pairwise interactions between agents are not directly observed and that only limited trajectory data are available. To address these challenges, we propose two complementary identification strategies. The first based on random-batch sampling, which compensates for latent interactions while preserving the statistical structure of the full dynamics in expectation. The second based on a mean-field approximation, where the empirical particle density reconstructed from the data defines a continuous nonlocal regression problem. Numerical experiments demonstrate the effectiveness and robustness of the proposed framework, showing accurate reconstruction of both interaction and diffusion kernels even from partially observed. The method is validated on benchmark models, including bounded-confidence and attraction-repulsion dynamics, where the two proposed strategies achieve comparable levels of accuracy.