论文标题
在相互作用粒子一阶系统的平均场方程中学习相互作用内核
Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles
论文作者
论文摘要
我们引入了一种非参数算法，以学习一阶相互作用粒子的一阶系统的平均场方程相互作用核。数据由解决方案的离散时空观察组成。至少与正则化的正方形，该算法有效地了解了数据自适应假设空间的内核。一个关键成分是概率误差功能，该功能来自均值场方程扩散过程的可能性。估算器在繁殖的内核希尔伯特空间和在可识别性条件下的L2空间中收敛，其速率是最佳的，因为它等于数值集成商的顺序。我们在三个典型示例上演示了算法：带有分段线性内核的意见动力学，带有二次核的颗粒状媒体模型以及带有令人震惊的吸引性核的聚集 - 扩散。
We introduce a nonparametric algorithm to learn interaction kernels of mean-field equations for 1st-order systems of interacting particles. The data consist of discrete space-time observations of the solution. By least squares with regularization, the algorithm learns the kernel on data-adaptive hypothesis spaces efficiently. A key ingredient is a probabilistic error functional derived from the likelihood of the mean-field equation's diffusion process. The estimator converges, in a reproducing kernel Hilbert space and an L2 space under an identifiability condition, at a rate optimal in the sense that it equals the numerical integrator's order. We demonstrate our algorithm on three typical examples: the opinion dynamics with a piecewise linear kernel, the granular media model with a quadratic kernel, and the aggregation-diffusion with a repulsive-attractive kernel.
