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

arXiv: https://arxiv.org/abs/2603.15927 · HTML: https://mlsanity.com/paper/2603.15927/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 39 references that are in the map.
- First posted on arXiv 16 Mar 2026.

## Builds on (most cited first, 10 of 10)

- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Nonparametric inference of interaction laws in systems of agents from trajectory data](https://mlsanity.com/paper/1812.06003/) (2018)
- [Learning interaction kernels in heterogeneous systems of agents from multiple trajectories](https://mlsanity.com/paper/1910.04832/) (2019)
- [Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories](https://mlsanity.com/paper/2007.15174/) (2020)
- [Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles](https://mlsanity.com/paper/2010.15694/) (2020)
- [Learning Theory for Inferring Interaction Kernels in Second-Order Interacting Agent Systems](https://mlsanity.com/paper/2010.03729/) (2020)
- [Boltzmann type control of opinion consensus through leaders](https://mlsanity.com/paper/1405.0736/) (2014)
- [Computation and Control of Unstable Steady States for Mean Field Multiagent Systems](https://mlsanity.com/paper/2406.11725/) (2024)
- [Data-driven entropic spatially inhomogeneous evolutionary games](https://mlsanity.com/paper/2103.05429/) (2021)
- [Augmented data and neural networks for robust epidemic forecasting: application to COVID-19 in Italy](https://mlsanity.com/paper/2510.09192/) (2025)

## Abstract (arXiv)

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.