We consider the problem of inferring the interaction kernel of stochastic interacting particle systems from observations of a single particle. We adopt a semi-parametric approach and represent the interaction kernel in terms of a generalized Fourier series. The basis functions in this expansion are tailored to the problem at hand and are chosen to be orthogonal polynomials with respect to the invariant measure of the mean-field dynamics. The generalized Fourier coefficients are obtained as the solution of an appropriate linear system whose coefficients depend on the moments of the invariant measure, and which are approximated from the particle trajectory that we observe. We quantify the approximation error in the Lebesgue space weighted by the invariant measure and study the asymptotic properties of the estimator in the joint limit as the observation interval and the number of particles tend to infinity, i.e. the joint large time-mean field limit. We also explore the regime where an increasing number of generalized Fourier coefficients is needed to represent the interaction kernel. Our theoretical results are supported by extensive numerical simulations.

翻译：我们考虑从单个粒子的观测数据推断随机交互粒子系统交互核的问题。采用半参数化方法，将交互核表示为广义傅里叶级数形式。该展开中的基函数根据问题特性专门设计，选择为正交多项式，其正交性相对于平均场动力学的不变测度。广义傅里叶系数通过求解适当的线性系统获得，该系统系数依赖于不变测度的矩，这些矩通过观测到的粒子轨迹进行近似计算。我们在由不变测度加权的Lebesgue空间中量化近似误差，并研究估计量在观测区间与粒子数趋于无穷大联合极限（即联合大时间-平均场极限）下的渐近性质。同时探讨了需要递增的广义傅里叶系数来表示交互核的数学机制。理论结果得到了大量数值模拟的验证。