Energy-Efficient Predictive RIS Beamforming in Multi-Satellite LEO Networks Using Twin-Delayed Deep Reinforcement Learning

This paper presents an energy-efficient downlink control framework for multi-satellite low-Earth-orbit (LEO) networks assisted by terrestrial reconfigurable intelligent surfaces (RIS). The proposed method integrates three key components: (i) an extended Kalman filter (EKF) for mobility and channel prediction; (ii) a single-step affine calibration process that constrains the predicted signal-to-noise ratio (SNR) to a prescribed normalized meansquare error (NMSE); and (iii) a twin-delayed deep deterministic policy gradient (TD3) controller that dynamically adjusts the RIS phase configurations in real time. An analytical model of energy efficiency (EE), accounting for satellite transmit power, RIS circuit power, and processing overhead, has been developed and employed both as a benchmarking reference and as the optimization target for the learning process. Simulation results across varied mobility conditions, satellite multiplicities, and RIS configurations show that the EKF-calibration loop maintains the SNR-prediction NMSE at or below 8 × 10 − 3 , enabling stable predictive control. Increasing the RIS aperture produces the expected ≈ 6 dB SNR improvement with diminishing marginal EE gains. for example, doubling the aperture from 64 to 128 elements improves EE by approximately 59%; from 128 to 256 elements by 29%; and from 256 to 512 elements by 4.5%. With TD3-optimized phases, the achieved EE attains 93 − 97% of the analytical upper bound (0.5 − 3.4 bit J − 1), while per-satellite transmit-power savings exceed 50% in typical regimes, and network-wide savings reach up to 12% compared with non-predictive baselines. For moderate satellite diversity with large RIS apertures, the incremental gain from learning-based control is up to 0.38 bits J − 1. These findings demonstrate that the integration of EKF-based motion prediction, NMSE-constrained SNR calibration, and learning-based RIS optimization yields a scalable and near-optimal solution for energy-efficient downlink operations in LEO satellite systems.