# Learning interaction kernels in heterogeneous systems of agents from multiple trajectories

arXiv: https://arxiv.org/abs/1910.04832 · HTML: https://mlsanity.com/paper/1910.04832/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Oct 2019.

## Built on by (most cited first, 25 of 25)

- [Learning Mean-Field Equations from Particle Data Using WSINDy](https://mlsanity.com/paper/2110.07756/) (2021)
- [Learning Stochastic Dynamics with Statistics-Informed Neural Network](https://mlsanity.com/paper/2202.12278/) (2022)
- [Neural parameter calibration for large-scale multi-agent models](https://mlsanity.com/paper/2209.13565/) (2022)
- [Identifiability of interaction kernels in mean-field equations of interacting particles](https://mlsanity.com/paper/2106.05565/) (2021)
- [Learning Anisotropic Interaction Rules from Individual Trajectories in a Heterogeneous Cellular Population](https://mlsanity.com/paper/2204.14141/) (2022)
- [Random Feature Models for Learning Interacting Dynamical Systems](https://mlsanity.com/paper/2212.05591/) (2022)
- [Learning Interaction Kernels for Agent Systems on Riemannian Manifolds](https://mlsanity.com/paper/2102.00327/) (2021)
- [Self-test loss functions for learning weak-form operators and gradient flows](https://mlsanity.com/paper/2412.03506/) (2024)
- [Learning particle swarming models from data with Gaussian processes](https://mlsanity.com/paper/2106.02735/) (2021)
- [Reproducing kernel Hilbert spaces in the mean field limit](https://mlsanity.com/paper/2302.14446/) (2023)
- [Learning Collective Behaviors from Observation](https://mlsanity.com/paper/2311.00875/) (2023)
- [Data-Driven Model Selections of Second-Order Particle Dynamics via Integrating Gaussian Processes with Low-Dimensional Interacting Structures](https://mlsanity.com/paper/2311.00902/) (2023)
- [Interacting Particle Systems on Networks: joint inference of the network and the interaction kernel](https://mlsanity.com/paper/2402.08412/) (2024)
- [Learning interacting particle systems from unlabeled data](https://mlsanity.com/paper/2604.02581/) (2026)
- [A Data-Adaptive Prior for Bayesian Learning of Kernels in Operators](https://mlsanity.com/paper/2212.14163/) (2022)
- [Learning Multi-type heterogeneous interacting particle systems](https://mlsanity.com/paper/2602.03954/) (2026)
- [Nonlocal Mean Field Schr"odinger Bridge with Learned Interactions](https://mlsanity.com/paper/2606.04265/) (2026)
- [Learning Nonlinear Couplings in Network of Agents from a Single Sample Trajectory](https://mlsanity.com/paper/2211.11069/) (2022)
- [Learning Stochastic Dynamical Systems with Structured Noise](https://mlsanity.com/paper/2503.01077/) (2025)
- [A Sparse Bayesian Learning Algorithm for Estimation of Interaction Kernels in Motsch-Tadmor Model](https://mlsanity.com/paper/2505.07068/) (2025)
- [Minimax Rates for Learning Pairwise Interactions in Attention-Style Models](https://mlsanity.com/paper/2510.11789/) (2025)
- [Data-driven Learning of Interaction Laws in Multispecies Particle Systems with Gaussian Processes: Convergence Theory and Applications](https://mlsanity.com/paper/2511.02053/) (2025)
- [Discovery of interaction and diffusion kernels in particle-to-mean-field multi-agent systems](https://mlsanity.com/paper/2603.15927/) (2026)
- [Recursive Maximum Likelihood Estimation for Interacting Particle Systems using Virtual Particles](https://mlsanity.com/paper/2605.00786/) (2026)
- [Multi-Agent System Identification with Nonlinear Sheaf Diffusion](https://mlsanity.com/paper/2605.11204/) (2026)

## Abstract (arXiv)

Systems of interacting particles or agents have wide applications in many disciplines such as Physics, Chemistry, Biology and Economics. These systems are governed by interaction laws, which are often unknown: estimating them from observation data is a fundamental task that can provide meaningful insights and accurate predictions of the behaviour of the agents. In this paper, we consider the inverse problem of learning interaction laws given data from multiple trajectories, in a nonparametric fashion, when the interaction kernels depend on pairwise distances. We establish a condition for learnability of interaction kernels, and construct estimators that are guaranteed to converge in a suitable L2 space, at the optimal min-max rate for 1-dimensional nonparametric regression. We propose an efficient learning algorithm based on least squares, which can be implemented in parallel for multiple trajectories and is therefore well-suited for the high dimensional, big data regime. Numerical simulations on a variety examples, including opinion dynamics, predator-swarm dynamics and heterogeneous particle dynamics, suggest that the learnability condition is satisfied in models used in practice, and the rate of convergence of our estimator is consistent with the theory. These simulations also suggest that our estimators are robust to noise in the observations, and produce accurate predictions of dynamics in relative large time intervals, even when they are learned from data collected in short time intervals.