Extracting Interaction Kernels for Many-Particle Systems by a Two-Phase Approach

TL;DR

This paper introduces a two-phase approach for learning interaction kernels in stochastic many-particle systems, transforming trajectories into density functions and refining kernel coefficients through importance sampling and regression, validated by extensive numerical examples.

Contribution

The paper proposes a novel two-phase method combining importance sampling and regression to accurately extract interaction kernels from particle system data.

Findings

Effective in identifying key kernel terms.

Accurate kernel coefficient refinement demonstrated.

Works well across various potential types.

Abstract

This paper presents a two-phase method for learning interaction kernels of stochastic many-particle systems. After transforming stochastic trajectories of every particle into the particle density function by the kernel density estimation method, the first phase of our approach combines importance sampling with an adaptive threshold strategy to identify key terms in the kernel function, while the second phase uses the whole dataset to refine the coefficients. During the implementation of our method, the mean-field equation plays a key role in reformulating the task of extracting the interaction kernels into a learnable regression problem. We demonstrate the outstanding performance of our approach through extensive numerical examples, including interacting particle systems with a cubic potential, power-law repulsion-attraction potential, piecewise linear potential, as well as a two-dimensional radially symmetric potential.