Data-Driven Discovery of Dynamics in Interacting Agent Systems and Linear Diffusion Processes

这个项目的目标是为动力系统开发数据驱动的方法，特别是关于相互作用的试剂/粒子系统和线性扩散过程，这些过程出现在不同学科中，如社会影响下的意见动力学、猎物-捕食者系统、动物群体的成群和成群、网络上的谣言/威胁传播和道路网络上的交通流量。该项目将侧重于从统计学习中发现支配规律的想法，并将观测数据转化为可用于预测的方程。虽然机器学习技术在这项任务中特别有希望，但它们在学习动态系统方面的应用仍处于初级阶段。该项目将开发有效的算法，从各种类型的观测轨道数据中学习系统的未知结构和参数，并将制定一个严格的量化框架，以指导选择能够很好地概括未知数据的模型。学生将参与并接受跨学科方面的培训。该项目的第一部分解决了基于回归的学习方法，以从各种类型的轨迹数据中发现代理之间的相互作用规律，并将其应用于物理、生物、生态和社会科学的系统，使用机器学习和反问题的接口方法。将开发系统的学习理论来研究适定性和模型选择，以实现统计上的最优性能。该项目的第二部分将开发稳健的方法，从演化状态的部分观测中恢复图上的线性扩散过程，并将其应用于图信号处理。特别是，该项目将开发用于收集时空样本的采样定理以及稳健的重建算法。抽样定理将阐明如何利用图上的动力学和图的结构来补偿空间信息的损失。将研究不完美数据造成的影响的理论和算法分支，以在各种图表的合成和真实数据集上测试建议的算法。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。

The goal of this project is to develop data-driven methods for dynamical systems and specifically on interacting agent/particle systems and linear diffusion processes that arise in various disciplines such as opinion dynamics under social influence, prey-predator systems, flocking and swarming of animal groups, rumor/threat propagations over networks, and traffic flow over road networks. The project will focus on ideas from statistical learning for the discovery of governing laws and turning the observational data into equations that can be used for predictions. While machine learning techniques are particularly promising for this task their application to learning dynamical systems is still in its infancy. This project will develop efficient algorithms to learn unknown structures and parameters of the systems from various types of observational trajectory data, together with a rigorous quantitative framework to guide the selection of models that generalize well on unseen data. Students will be involved and trained in interdisciplinary aspects. The first part of the project addresses regression-based learning approaches to discover interaction laws between agents from various types of trajectory data, with applications to systems arising from physics, biology, ecology, and social sciences, using methods at the interface of machine learning and inverse problems. Systematic learning theories will be developed to study the well-posedness and model selections to achieve statistically optimal performance. The second part of the project will develop robust methods to recover linear diffusion processes over graphs from partial observations of evolving states, with applications to graph signal processing. In particular the project will develop sampling theorems to collect space-time samples as well as robust reconstruction algorithms. The sampling theorems will shed light on how to utilize dynamics over graphs and the structure of graphs to compensate for the loss of spatial information. Theoretical and algorithmic ramifications of the effects caused by imperfect data will be studied to test the proposed algorithms on synthetic and real data sets over a wide variety of graphs.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.