A continuous-time average consensus system is a linear dynamical system defined over a graph, where each node has its own state value that evolves according to a simultaneous linear differential equation. A node is allowed to interact with neighboring nodes. Average consensus is a phenomenon that the all the state values converge to the average of the initial state values. In this paper, we assume that a node can communicate with neighboring nodes through an additive white Gaussian noise channel. We first formulate the noisy average consensus system by using a stochastic differential equation (SDE), which allows us to use the Euler-Maruyama method, a numerical technique for solving SDEs. By studying the stochastic behavior of the residual error of the Euler-Maruyama method, we arrive at the covariance evolution equation. The analysis of the residual error leads to a compact formula for mean squared error (MSE), which shows that the sum of the inverse eigenvalues of the Laplacian matrix is the most dominant factor influencing the MSE. Furthermore, we propose optimization problems aimed at minimizing the MSE at a given target time, and introduce a deep unfolding-based optimization method to solve these problems. The quality of the solution is validated by numerical experiments.

翻译：连续时间平均一致性系统是一种定义在图上的线性动力系统，其中每个节点拥有自身状态值，该值根据同时的线性微分方程演化。节点允许与相邻节点进行交互。平均一致性是指所有状态值收敛至初始状态值平均值的现象。本文假设节点通过加性白高斯噪声信道与相邻节点通信。我们首先利用随机微分方程（SDE）形式化含噪平均一致性系统，从而能够使用欧拉-丸山方法（一种求解SDE的数值技术）进行研究。通过分析欧拉-丸山方法残差误差的随机行为，推导出协方差演化方程。残差误差分析给出了均方误差（MSE）的紧凑公式，表明拉普拉斯矩阵逆特征值之和是影响MSE的最主要因素。此外，我们提出了旨在最小化给定目标时刻MSE的优化问题，并引入基于深度展开的优化方法求解这些问题。通过数值实验验证了解的质量。