Learning interaction kernels in heterogeneous systems of agents from multiple trajectories
Learning interaction kernels in heterogeneous systems of agents from multiple trajectories
复制标题
DOI：
--
复制
发表时间：
2019-10
期刊：
影响因子：
--
通讯作者：
F. Lu;M. Maggioni;Sui Tang
中科院分区：
文献类型：
--
作者：
F. Lu;M. Maggioni;Sui Tang
相互作用的粒子或试剂系统在物理、化学、生物学和经济学等许多学科中具有广泛的应用。这些系统受交互定律的支配，而交互定律通常是未知的：根据观察数据估计它们是一项基本任务，可以提供有意义的见解和对代理行为的准确预测。在本文中，我们考虑了当交互核取决于成对距离时，以非参数方式学习给定来自多个轨迹的数据的交互定律的逆问题。我们建立了交互核的可学习性条件，并构造了保证以一维非参数回归的最佳最小-最大速率收敛在合适的 $L^2$ 空间中的估计器。我们提出了一种基于最小二乘的有效学习算法，该算法可以并行实现多个轨迹，因此非常适合高维大数据体系。对各种例子（包括意见动力学、捕食者群体动力学和异质粒子动力学）的数值模拟表明，实践中使用的模型满足可学习性条件，并且我们的估计器的收敛速度与理论一致。这些模拟还表明，我们的估计器对观测中的噪声具有鲁棒性，并且可以在相对较大的时间间隔内产生准确的动态预测，即使它们是从短时间间隔内收集的数据中学习的。
Systems of interacting particles or agents have wide applications in many disciplines such as Physics, Chemistry, Biology and Economics. These systems are governed by interaction laws, which are often unknown: estimating them from observation data is a fundamental task that can provide meaningful insights and accurate predictions of the behaviour of the agents. In this paper, we consider the inverse problem of learning interaction laws given data from multiple trajectories, in a nonparametric fashion, when the interaction kernels depend on pairwise distances. We establish a condition for learnability of interaction kernels, and construct estimators that are guaranteed to converge in a suitable $L^2$ space, at the optimal min-max rate for 1-dimensional nonparametric regression. We propose an efficient learning algorithm based on least squares, which can be implemented in parallel for multiple trajectories and is therefore well-suited for the high dimensional, big data regime. Numerical simulations on a variety examples, including opinion dynamics, predator-swarm dynamics and heterogeneous particle dynamics, suggest that the learnability condition is satisfied in models used in practice, and the rate of convergence of our estimator is consistent with the theory. These simulations also suggest that our estimators are robust to noise in the observations, and produce accurate predictions of dynamics in relative large time intervals, even when they are learned from data collected in short time intervals.