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In this paper, we propose a local discontinuous Galerkin (LDG) method for the Novikov equation that contains cubic nonlinear high-order derivatives. Flux correction techniques are used to ensure the stability of the numerical scheme. The $H^1$-norm stability of the general solution and the error estimate for smooth solutions without using any priori assumptions are presented. Numerical examples demonstrate the accuracy and capability of the LDG method for solving the Novikov equation. 

Federated Learning (FL) trains a shared model using data and computation power on distributed agents coordinated by a central server. Decentralized FL (DFL) utilizes local model exchange and aggregation between agents to reduce the communication and computation overheads on the central server. However, when agents are mobile, the communication opportunity between agents can be sporadic, largely hindering the convergence and accuracy of DFL. In this paper, we propose Cached Decentralized Federated Learning (Cached-DFL) to investigate delay-tolerant model spreading and aggregation enabled by model caching on mobile agents. Each agent stores not only its own model, but also models of agents encountered in the recent past. When two agents meet, they exchange their own models as well as the cached models. Local model aggregation utilizes all models stored in the cache. We theoretically analyze the convergence of Cached-DFL,explicitly taking into account the model staleness introduced by caching. We design and compare different model caching algorithms for different DFL and mobility scenarios. We conduct detailed case studies in a vehicular network to systematically investigate the interplay between agent mobility, cache staleness, and model convergence. In our experiments, Cached-DFL converges quickly, and significantly outperforms DFL without caching. 

Federated Learning (FL) aims to train a shared model using data and computation power on distributed agents coordinated by a central server. Decentralized FL (DFL) utilizes local model exchange and aggregation between agents to reduce the communication and computation overheads on the central server. However, when agents are mobile, the communication opportunity between agents can be sporadic, largely hindering the convergence and accuracy of DFL. In this paper, we study delay-tolerant model spreading and aggregation enabled by model caching on mobile agents. Each agent stores not only its own model, but also models of agents encountered in the recent past. When two agents meet, they exchange their own models as well as the cached models. Local model aggregation works on all models in the cache. We theoretically analyze the convergence of DFL with cached models, explicitly taking into account the model staleness introduced by caching. We design and compare different model caching algorithms for different DFL and mobility scenarios. We conduct detailed case studies in a vehicular network to systematically investigate the interplay between agent mobility, cache staleness, and model convergence. In our experiments, cached DFL converges quickly, and significantly outperforms DFL without caching. 

With the development of space-air-ground integrated networks, Low Earth Orbit (LEO) satellite networks are envisioned to play a crucial role in providing data transmission services in the 6G era. However, the increasing number of connected devices leads to a surge in data volume and bursty traffic patterns. Ensuring the communication stability of LEO networks has thus become essential. While Lyapunov optimization has been applied to network optimization for decades and can guarantee stability when traffic rates remain within the capacity region, its applicability in LEO satellite networks is limited due to the bursty and dynamic nature of LEO network traffic. To address this issue, we propose a robust Lyapunov optimization method to ensure stability in LEO satellite networks. We theoretically show that for a stabilizable network system, traffic rates do not have to always stay within the capacity region at every time slot. Instead, the network can accommodate temporary capacity region violations, while ensuring the long-term network stability. Extensive simulations under various traffic conditions validate the effectiveness of the robust Lyapunov optimization method, demonstrating that LEO satellite networks can maintain stability under finite violations of the capacity region.