从无轨迹的离散数据中学习粒子系统相互作用势能
Learning interacting particle systems from unlabeled data
- 用自测试损失函数在无轨迹数据上直接估计势能
- 大时间步下仍优于传统轨迹恢复方法，支持高维大数据
- 理论证明参数估计收敛性，适合物理建模与数据驱动研究
学习相互作用粒子系统的势能是多个科学领域的基础任务。主要挑战在于，离散时间点采集的无标签数据因数据采集限制或隐私约束而缺乏轨迹信息。我们提出一种无需轨迹的自测试损失函数，利用经验分布的弱形式随机演化方程。该损失函数在势能上为二次型，支持参数与非参数回归算法，可稳健估计大规模高维系统，适用于大数据场景。系统性数值实验表明，该方法优于基于标签匹配恢复轨迹的基线方法，且能容忍较大的观测时间步长。我们建立了参数估计器随样本量增加的收敛性，为所提方法提供了理论基础。
原文摘要 · Abstract (English)
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.
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