Learning interacting particle systems from unlabeled data

2026-04-02

AI Summary

This work addresses the challenge of learning interaction potentials in particle systems from unlabeled, discrete-time observations without access to trajectory information. The authors propose a self-supervised learning framework grounded in the weak-form stochastic evolution equation satisfied by empirical distributions, formulating a quadratic loss functional with respect to the potential function. This approach accommodates both parametric and nonparametric regression and comes with theoretical convergence guarantees. Notably, it is the first method to dispense with explicit particle trajectory matching, achieving superior performance over existing trajectory-reconstruction-based baselines—particularly under large time steps and high-dimensional, large-scale data settings—while ensuring consistent convergence of the estimator as the sample size increases.

Abstract

Learning the potentials of interacting particle systems is a fundamental task across various scientific disciplines. A major challenge is that unlabeled data collected at discrete time points lack trajectory information due to limitations in data collection methods or privacy constraints. We address this challenge by introducing a trajectory-free self-test loss function that leverages the weak-form stochastic evolution equation of the empirical distribution. The loss function is quadratic in potentials, supporting parametric and nonparametric regression algorithms for robust estimation that scale to large, high-dimensional systems with big data. Systematic numerical tests show that our method outperforms baseline methods that regress on trajectories recovered via label matching, tolerating large observation time steps. We establish the convergence of parametric estimators as the sample size increases, providing a theoretical foundation for the proposed approach.

Problem

• interacting particle systems
• unlabeled data
• trajectory-free learning
• potential estimation
• discrete-time observations

Innovation

• trajectory-free learning
• weak-form stochastic evolution
• interacting particle systems
• self-test loss
• nonparametric regression