[Paper Review] Learning interaction kernels in heterogeneous systems of agents from multiple trajectories

Fei Lu, Mauro Maggioni+1 more|arXiv (Cornell University)|Jan 1, 2021

Opinion Dynamics and Social Influence30 citations

TL;DR

This paper proposes a nonparametric method to learn interaction kernels in heterogeneous agent systems from multiple observed trajectories, using least squares estimation that converges at the optimal min-max rate in $L^2$ space. The approach is scalable, parallelizable, and demonstrates robust performance on opinion dynamics, predator-swarm, and particle systems, even with noisy or short-time data.

ABSTRACT

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 $L^2$ 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.

Motivation & Objective

• Address the inverse problem of learning unknown interaction laws in multi-agent systems from observed trajectories.
• Develop a nonparametric estimation framework for interaction kernels that depend on pairwise distances.
• Ensure theoretical convergence guarantees by establishing a learnability condition in $L^2$ space.
• Design a scalable, parallelizable algorithm suitable for high-dimensional, big data settings.
• Demonstrate robustness and predictive accuracy of the estimator under noisy observations and limited time data.

Proposed method

• Formulate the inverse problem as a nonparametric regression task in $L^2$ space, where the interaction kernel is estimated from pairwise distances.
• Introduce a learnability condition that ensures the existence of consistent estimators for the interaction kernel.
• Construct least squares estimators that converge at the optimal min-max rate for 1D nonparametric regression.
• Implement the algorithm in a parallelizable manner across multiple trajectories to enable scalability in big data regimes.
• Use numerical simulations to validate theoretical convergence rates and robustness to noise and short observation windows.

Experimental results

Research questions

• RQ1 Under what conditions is the interaction kernel in a heterogeneous agent system learnable from multiple trajectories?
• RQ2 Can a nonparametric estimator achieve the optimal min-max convergence rate in $L^2$ space for such systems?
• RQ3 How does the proposed least squares estimator perform in practice across diverse agent dynamics like opinion formation and swarming?
• RQ4 To what extent is the estimator robust to noise and short observation intervals in real-world data?
• RQ5 Can the method be efficiently parallelized to scale to high-dimensional, large-scale trajectory data?

Key findings

• The proposed learnability condition is satisfied in practical models such as opinion dynamics, predator-swarm systems, and heterogeneous particle dynamics.
• The least squares estimator achieves the optimal min-max convergence rate for 1D nonparametric regression in the $L^2$ space.
• Numerical simulations confirm that the estimator's convergence rate aligns with theoretical predictions.
• The method remains robust to observational noise, maintaining accurate predictions over long time intervals.
• Even when trained on short time intervals, the estimator produces reliable long-term dynamic predictions.