Multi-beam low earth orbit (LEO) satellites, with their wide coverage, high communication rates, low latency, and flexibility, are essential components in the 5G and 6G eras. However, challenges such as uneven ground user distribution, multi-dimensional resource coupling, inter-beam interference, and dynamic network topology complicate efficient resource management. This paper presents a joint beam hopping and resource allocation (BHRA) scheme within a Digital Twin (DT)empowered multi-beam LEO satellites network. The complex resource management problem is divided into two sub-problems: traffic-satellite allocation and joint multi-satellite BHRA. First, cell traffic is allocated among satellites to balance load using deep reinforcement learning (DRL). Then, Hierarchical Proximal Policy Optimization (HPPO) optimizes multi-satellite beam hopping and resource allocation to meet demand. Extensive simulation results demonstrate that the proposed method reduces satellite load imbalance by approximately 83%, achieves the highest throughput compared to other algorithms, and generalizes well as traffic demand increases.

Low-Earth orbit (LEO) satellites utilizing beam hopping (BH) technology offer extensive coverage, low latency, high bandwidth, and significant flexibility. However, the uneven geographical distribution and temporal variability of ground traffic demands, combined with the high mobility of LEO satellites, present significant challenges for efficient beam resource utilization. Traditional BH methods based on GEO satellites fail to address issues such as satellite interference, overlapping coverage, and mobility. This paper explores a Digital Twin (DT)-based collaborative resource allocation network for multiple LEO satellites with overlapping coverage areas. A two-tier optimization problem, focusing on load balancing and cell service fairness, is proposed to maximize throughput and minimize inter-cell service delay. The DT layer optimizes the allocation of overlapping coverage cells by designing BH patterns for each satellite, while the LEO layer optimizes power allocation for each selected service cell. At the DT layer, an Actor-Critic network is deployed on each agent, with a global critic network in the cloud center. The A3C algorithm is employed to optimize the DT layer. Concurrently, the LEO layer optimization is performed using a Multi-Agent Reinforcement Learning algorithm, where each beam functions as an independent agent. The simulation results show that this method reduces satellite load disparity by about 72.5% and decreases the average delay to 12ms. Additionally, our approach outperforms other benchmarks in terms of throughput, ensuring a better alignment between offered and requested data.

Numerical Partial Differential Equation (PDE) solvers often require discretizing the physical domain by using a mesh. Mesh movement methods provide the capability to improve the accuracy of the numerical solution without introducing extra computational burden to the PDE solver, by increasing mesh resolution where the solution is not well-resolved, whilst reducing unnecessary resolution elsewhere. However, sophisticated mesh movement methods, such as the Monge-Ampère method, generally require the solution of auxiliary equations. These solutions can be extremely expensive to compute when the mesh needs to be adapted frequently. In this paper, we propose to the best of our knowledge the first learning-based end-to-end mesh movement framework for PDE solvers. Key requirements of learning-based mesh movement methods are: alleviating mesh tangling, boundary consistency, and generalization to mesh with different resolutions. To achieve these goals, we introduce the neural spline model and the graph attention network (GAT) into our models respectively. While the Neural-Spline based model provides more flexibility for large mesh deformation, the GAT based model can handle domains with more complicated shapes and is better at performing delicate local deformation. We validate our methods on stationary and time-dependent, linear and non-linear equations, as well as regularly and irregularly shaped domains. Compared to the traditional Monge-Ampère method, our approach can greatly accelerate the mesh adaptation process by three to four orders of magnitude, whilst achieving comparable numerical error reduction.