Learning interacting particle systems from unlabeled data

TL;DR

This paper introduces a new trajectory-free method for learning potentials in interacting particle systems from unlabeled data, enabling scalable and robust estimation without trajectory information.

Contribution

It proposes a novel weak-form loss function leveraging stochastic evolution equations, supporting scalable, high-dimensional, and nonparametric regression methods with theoretical convergence guarantees.

Findings

Outperforms baseline methods in numerical tests

Tolerates large observation time steps

Supports both parametric and nonparametric regression

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.