We provide a Lyapunov convergence analysis for time-inhomogeneous variable coefficient stochastic differential equations (SDEs). Three typical examples include overdamped, irreversible drift, and underdamped Langevin dynamics. We first formulate the probability transition equation of Langevin dynamics as a modified gradient flow of the Kullback-Leibler divergence in the probability space with respect to time-dependent optimal transport metrics. This formulation contains both gradient and non-gradient directions depending on a class of time-dependent target distribution. We then select a time-dependent relative Fisher information functional as a Lyapunov functional. We develop a time-dependent Hessian matrix condition, which guarantees the convergence of the probability density function of the SDE. We verify the proposed conditions for several time-inhomogeneous Langevin dynamics. For the overdamped Langevin dynamics, we prove the $O(t^{-1/2})$ convergence in $L^1$ distance for the simulated annealing dynamics with a strongly convex potential function. For the irreversible drift Langevin dynamics, we prove an improved convergence towards the target distribution in an asymptotic regime. We also verify the convergence condition for the underdamped Langevin dynamics. Numerical examples demonstrate the convergence results for the time-dependent Langevin dynamics.

Belief and opinion evolution in social networks (SNs) can aid in understanding how people influence others’ decisions through social relationships as well as provide a solid foundation for many valuable social applications. As large numbers of users are involved in SNs, the complexity of traditional optimization techniques is high as they deal with the interactions between users separately. Moreover, the state variable (opinion) is high-dimensional because a person usually has opinions about many different social issues. To overcome those challenges, we formulate the opinion evolution in SNs as a high-dimensional stochastic mean field game (MFG). Numerical methods for high-dimensional MFGs are practically non-existent because of the need for grid-based spatial discretization. Thus, we propose a machine-learning based method, where we use an alternating population and agent control neural network (APAC-net), to tractably solve high-dimensional stochastic MFGs. Through APAC-net, solving MFGs can be regarded as a special case of training a generative adversarial network (GAN). To the best of our knowledge, the APAC-Net is the first model that can solve high-dimensional stochastic MFGs. The simulation results affirm the efficiency of the APAC-net.

With the increasing popularity of mobile vehicles, such as unmanned aerial vehicles (UAVs) and mobile robots, it is foreseen that they will play an important role in Internet-of-Things (IoT) networks due to their high mobility and rapid deployment. Specifically, mobile vehicles equipped with sensors act as IoT devices and can be dispatched to several sensing regions to perform sensing tasks. In this article, we consider mobile vehicles for sensing applications and investigate the corresponding joint task assignment and collision-free trajectory optimization problem. This problem is challenging as the number of involved vehicles can be very large, and to tackle the problem efficiently, we reformulate the original optimization problem into a mean-field-game (MFG) problem by simplifying the interaction between vehicles as a distribution over their state space, known as the mean-field term. To solve the MFG problem efficiently, we propose a G-prox primal-dual hybrid gradient (PDHG) algorithm that transforms the MFG problem into a saddle-point problem by defining a Lagrangian functional with a proximal operator. The complexity of this algorithm is shown to be linear with the total number of grid points in the proposed MFG problem. We provide a comprehensive theoretical analysis of the proposed model and algorithm. Numerical results together with the practical implementation on real mobile robots show that our proposed system model and algorithm are of significant effectiveness and efficiency.

When a disaster happens in a metropolitan area, wireless communication systems in the area are highly affected, degrading the efficiency of the search and rescue (SAR) mission. An emergency wireless network must be deployed quickly and efficiently to preserve human lives. Teams of low-altitude rotarywing unmanned aerial vehicles (UAVs) are useful as on-demand temporal wireless networks because they are generally faster to deploy, flexible to reconfigure, and able to provide good communication services with short line-of-sight links. However, rotary-wing UAVs' limited on-board batteries require that they need to recharge and reconFigure frequently during a mission. Therefore, we formulate the velocity control problem for massive numbers of rotary-wing UAVs as a Schrödinger bridge problem which can describe the frequent reconfiguration of UAVs. Then we transform it into a mean field game and solve it with the Gprox primal dual hybrid gradient (PDHG) method. Finally, we show the efficiency of our algorithm and analyze the influence of wind dynamics with numerical results.

In recent years, there has been a growing interest in utilizing mobile robot swarm to execute several tasks at the same time. However, how to assign tasks to the swarm and optimize the trajectory of the robots scientifically to minimize energy consumption is still a big challenge. In this paper, we consider a mobile robot swarm system where a large number of robots are deployed by a centralized controller to execute a series of tasks, such as target detection tasks, cooperatively. The controller controls the velocity strategy of each robot, and makes corresponding task assignment decisions to minimize the overall cost of the robot swarm. Since the number of involving robots is large, it will be extremely difficult to consider the interaction between them. In this regard, we adopt the concept of mean-field term to approximate the behaviors and states of the robots, and formulate the joint task assignment and trajectory optimization problem as a mean-field game. To solve the problem efficiently, a primal-dual hybrid gradient algorithm is proposed to find the optimal trajectory and corresponding task assignment decisions for each robot. The numerical simulation results show the effectiveness of the proposed algorithm.

Multi-access edge computing (MEC) can use the distributed computing resources to serve the large numbers of mobile users in the next generation of communication systems. In this new architecture, a limited number of mobile edge servers will serve a relatively large number of mobile users. Heterogeneous servers can provide either single resource or multiple different resources to the massive number of selfish mobile users. To achieve high quality of service (QoS) and low latency under these two cases, we construct two system models and formulate our problems as two non-cooperative population games. Then we apply our proposed mean field evolutionary approach with two different strategy graphs to solve the load balancing problems under those two cases. Finally, to evaluate the performance of our algorithms, we employ the following performance indicators: overall response time (average response time of the whole system), individual response time (response time of each server), and fairness index (equality of users' response time).

In mobile edge computing(MEC), the computing resources used to be centralized on the core cloud are extended to the mobile edge hosts (i.e. cloudlets) dispersedly deployed near the mobile users. In this new architecture, myriad of mobile terminals including vehicles, smart phones and different kinds of computers will form an ultra dense network. This motivates us to consider an ultra dense MEC system in which limited mobile edge hosts serve a relatively huge amount of mobile users. Considering the substantial amount and selfish manner of the users, we formulate the problem as a non-cooperative dynamic population game. The main contribution of this paper is as following. First, we consider the channel interference, average response time, load balance among servers and fairness (the variance of individual response time). Second, we propose an innovative mean field evolutionary approach which is robust to the channel interpolation as shown by the simulation results. Eventually, we consider a more challenging situation when the computing resources are rather limited.

The Wasserstein distance serves as a loss function for unsupervised learning which depends on the choice of a ground metric on sample space. We propose to use the Wasserstein distance itself as the ground metric on the sample space of images. This ground metric is known as an effective distance for image retrieval, that correlates with human perception. We derive the Wasserstein ground metric on pixel space and define a Riemannian Wasserstein gradient penalty to be used in the Wasserstein Generative Adversarial Network (WGAN) framework. The new gradient penalty is computed efficiently via convolutions on the $L^2$ gradients with negligible additional computational cost. The new formulation is more robust to the natural variability of the data and provides for a more continuous discriminator in sample space.