Learning Interaction Kernels in Stochastic Systems of Interacting Particles from Multiple Trajectories
Learning Interaction Kernels in Stochastic Systems of Interacting Particles from Multiple Trajectories
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学习多轨迹相互作用粒子随机系统中的相互作用核
DOI：
10.1007/s10208-021-09521-z
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发表时间：
2021
影响因子：
3
通讯作者：
Tang, Sui
中科院分区：
文献类型：
--
作者：
Lu, Fei;Maggioni, Mauro;Tang, Sui
相互作用的粒子系统在物理学、化学、生物学和经济学等学科中有着广泛的应用。这些系统由相互作用定律支配，而这些定律通常是未知的：从观察数据中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，我们考虑的逆问题的学习相互作用的法律给定的数据从多个轨迹，在一个非参数的方式，当相互作用的内核依赖于成对的距离。我们建立了一个相互作用核的可学习性条件，并构造了一个基于最小化适当的正则化最小二乘泛函的估计器，该估计器保证在适当的L2空间中以最优的最小-最大速率收敛于1维非参数回归。我们提出了一种有效的学习算法来构建这样的估计器，它可以并行实现多个轨迹，因此非常适合于高维，大数据制度。对意见动力学、捕食者-被捕食者动力学、群体动力学和异质粒子动力学等模型的数值模拟表明，在实际应用中，模型满足可学习性条件，且估计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，并且可以在大的时间间隔内对轨迹进行准确的预测，即使它们是从短时间间隔内的观测中学习的。
Systems of interacting particles, or agents, have wide applications in many disciplines, including 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 an estimator based on the minimization of a suitably regularized least squares functional, that is guaranteed to converge, in a suitableL2space, at the optimal min-max rate for 1-dimensional nonparametric regression. We propose an efficient learning algorithm to construct such estimator, 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-prey and 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 can produce accurate predictions of trajectories in large time intervals, even when they are learned from observations in short time intervals.
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影响因子：
4.5
作者：
Richard Nickl;Kolyan Ray
通讯作者：
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影响因子：
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作者：
Zhongyan Li;F. Lu;M. Maggioni;Sui Tang;C. Zhang
通讯作者：
Zhongyan Li;F. Lu;M. Maggioni;Sui Tang;C. Zhang
DOI：
10.1142/s0218202519500015
发表时间：
2019-01-01
影响因子：
3.5
作者：
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通讯作者：
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DOI：
10.1142/s0219493723400038
发表时间：
2020
期刊：
ArXiv
影响因子：
--
作者：
Zhongyan Li;F. Lu
通讯作者：
F. Lu
DOI：
--
发表时间：
1996
期刊：
影响因子：
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作者：
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通讯作者：
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