标题： 通过两阶段方法提取多粒子系统的相互作用核 显示英文标题

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

摘要： 本文提出了一种两阶段方法，用于学习随机多粒子系统的相互作用核。 通过使用核密度估计方法将每个粒子的随机轨迹转换为粒子密度函数后，我们方法的第一阶段结合重要性采样与自适应阈值策略，以识别核函数中的关键项，而第二阶段则利用整个数据集来优化系数。 在我们方法的实现过程中，平均场方程在将提取相互作用核的任务重新表述为可学习的回归问题中起着关键作用。 我们通过大量数值例子展示了我们方法的卓越性能，包括具有立方势的相互作用粒子系统、幂律排斥-吸引势、分段线性势以及二维径向对称势。 显示英文摘要

摘要： 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.