Networked dynamical systems are widely used as formal models of real-world
cascading phenomena, such as the spread of diseases and information. Prior
research has addressed the problem of learning the behavior of an unknown
dynamical system when the underlying network has a single layer. In this work,
we study the learnability of dynamical systems over multilayer networks, which
are more realistic and challenging. First, we present an efficient PAC learning
algorithm with provable guarantees to show that the learner only requires a
small number of training examples to infer an unknown system. We further
provide a tight analysis of the Natarajan dimension which measures the model
complexity. Asymptotically, our bound on the Nararajan dimension is tight for
almost all multilayer graphs. The techniques and insights from our work provide
the theoretical foundations for future investigations of learning problems for
multilayer dynamical systems.

研究网络动力系统在多层网络上的可学习性，并提出了基于 PAC 学习算法的有效算法及对 Natarajan 维度的严格分析，为多层动力系统的学习问题提供了理论基础。

多层网络上动力系统的高效 PAC 可学习性

Efficient PAC Learnability of Dynamical Systems Over Multilayer Networks

This paper analyzes distributed control protocols for first- and second-order
networked dynamical systems. We propose a class of nonlinear consensus
controllers where the input of each agent can be written as a product of a
nonlinear gain, and a sum of nonlinear interaction functions. By using integral
Lyapunov functions, we prove the stability of the proposed control protocols,
and explicitly characterize the equilibrium set. We also propose a distributed
proportional-integral (PI) controller for networked dynamical systems. The PI
controllers successfully attenuate constant disturbances in the network. We
prove that agents with single-integrator dynamics are stable for any integral
gain, and give an explicit tight upper bound on the integral gain for when the
system is stable for agents with double-integrator dynamics. Throughout the
paper we highlight some possible applications of the proposed controllers by
realistic simulations of autonomous satellites, power systems and building
temperature control.

分析了用于网络动力系统的分布式控制协议，提出了一类非线性共识控制器；并通过使用积分李雅普诺夫函数证明了所提出的控制协议的稳定性，同时明确表述了均衡集；之后又提出了分布式比例积分控制器，用于网络动力系统的稳定；通过在自主卫星，电源系统和建筑温度控制的实际模拟中突出了所提出的控制器的一些可能的应用。