相互作用粒子系统及其平均场极限的矩方法估计器
A method of moments estimator for interacting particle systems and their mean field limit
AI总结：
本文提出一种基于矩方法和平均场极限的估计器，通过单粒子轨迹求解低维线性系统来推断相互作用粒子系统的未知参数，并证明其渐近无偏性。
摘要：
我们研究从系统中单个粒子的轨迹学习具有多项式漂移、相互作用和扩散函数的随机相互作用粒子系统中未知参数的问题。我们的估计器通过求解一个线性系统获得，该系统通过对平均场极限的不变分布的矩以及过程的二次变差施加适当条件来构造。我们的方法易于实现，因为它只需要通过遍历定理近似矩以及求解一个低维线性系统。此外，我们证明了在无限数据和无限粒子数（平均场极限）的极限下，我们的估计器是渐近无偏的。另外，我们给出了若干数值实验，验证了理论分析，并展示了我们的方法在相互作用粒子系统中准确推断参数的有效性。
英文摘要：
We study the problem of learning unknown parameters in stochastic interacting particle systems with polynomial drift, interaction and diffusion functions from the path of one single particle in the system. Our estimator is obtained by solving a linear system which is constructed by imposing appropriate conditions on the moments of the invariant distribution of the mean field limit and on the quadratic variation of the process. Our approach is easy to implement as it only requires the approximation of the moments via the ergodic theorem and the solution of a low-dimensional linear system. Moreover, we prove that our estimator is asymptotically unbiased in the limits of infinite data and infinite number of particles (mean field limit). In addition, we present several numerical experiments that validate the theoretical analysis and show the effectiveness of our methodology to accurately infer parameters in systems of interacting particles.
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