Extracting interaction kernels for many-particle systems by a two-phase approach
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This study proposes a two-phase machine learning approach to extract interaction kernels from trajectory data of many-particle systems. The research is motivated by understanding interaction laws governing collective behaviors such as flocking birds, schooling fish, and opinion dynamics. Based on mean-field theory, as particle number N approaches infinity, the system can be described by the McKean-Vlasov equation. The method consists of two phases: Phase I employs importance sampling with adaptive sparsification for initial estimation; Phase II refines parameter estimation using the complete dataset. Results are evaluated through free energy comparison, kernel function errors, and Wasserstein distance metrics. Numerical experiments cover diverse scenarios including cubic potential, power-law repulsion-attraction, double-well potential, opinion dynamics (polarization and consensus models), and 2D radial evolution patterns. The method successfully reconstructs interaction kernels across all cases. The approach combines kernel density estimation with symbolic regression techniques, enabling identification of both smooth and discontinuous interaction functions.
