Learning Dynamics from Data: Discovering Interaction Laws of Particle and Agent Systems

基于粒子和智能体的系统在科学中是普遍存在的，例如，基础物理学中的粒子系统，在社会影响下模拟意见动力学的基于智能体的系统，猎物-捕食者动力学，集群和蜂群，以及细胞动力学中的趋光性。要了解这些系统的机制，一个根本的挑战是从观测数据中推断出粒子和作用剂之间的相互作用规律。这个项目的目的是开发数学和统计理论以及计算效率高的算法，以系统轨迹的形式从观测中学习这些相互作用规律。该理论在估计中提供了性能保证和不确定性量化，从而为模型选择和最优数据收集提供了基础。这些算法可扩展到大数据集，避免了维度灾难，适用于物理、生物、生态和社会科学的各种系统，不同系统的相互作用规律差别很大，一般没有解析形式。PIs提出了学习相互作用定律的非参数统计推断方法，而没有关于其分析形式的参考或假设。这项研究将为相互作用核的非参数回归发展一种系统的学习理论，其值不能从系统中粒子或代理人的轨迹数据中观察到也不能计算。该理论将研究相互作用核的可辨识性，估计器的一致性，以及假设空间的最佳选择，以实现估计器的最优收敛速度。在学习理论的指导下，我们将设计计算效率高的算法，具有以下特点：(I)通过关注交互核心的内在维度来避免维度诅咒；(Ii)具有理论保证和不确定性量化，可用于模型选择和最佳数据收集；(Iii)可通过并行实施扩展到大型数据集。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Particle and agent-based systems are ubiquitous in science, for example, particle systems in fundamental physics, agent-based systems that model opinion dynamics under the social influence, prey-predator dynamics, flocking and swarming, and phototaxis in cell dynamics. To understand the mechanism of these systems, a fundamental challenge is to infer the laws of interaction between the particles and agents from observational data. This project aims to develop mathematical and statistical theory and computationally efficient algorithms for learning these interaction laws from observations in the form of trajectories of the systems. The theory provides performance guarantees and uncertainty quantification in the estimations, therefore providing foundations for model selection and for optimal data collection. The algorithms are scalable to large data sets, avoiding the curse of dimensionality, and are applicable to a wide variety of systems from Physics, Biology, Ecology and Social Sciences.The interaction laws vary largely for different systems, and there is no analytical form in general. The PIs propose non-parametric statistical inference approaches for learning the interaction laws, with no reference or assumption on their analytical form. The research will develop a systematical learning theory for the non-parametric regression of the interaction kernels, whose values are not observed and can not be computed from the data of trajectories of the particles or agents in the system. The theory will study the identifiability of the interaction kernels, the consistency of the estimators, and the optimal choice of hypothesis spaces to achieve optimal rate of convergence of the estimators. With the guidance from the learning theory, we will design computationally efficient algorithms with the following features: (i) avoiding the curse of dimensionality by focusing on the intrinsic dimension of the interaction kernels; (ii) with theoretical guarantee and uncertainty quantification which can be used for model selection and optimal data collection; (iii) scalable to large data sets by implementing in parallel.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.