## 1 Introduction
The continuous convex approximation algorithm was used to optimize beamforming, while the SDR algorithm was used to optimize the IRS phase shifts. For the energy efficiency maximization problem addressed in [14], an effective method based on fractional programming and alternating optimization was proposed, leading to the attainment of a local optimal solution. The authors of [15; 16] addressed the energy efficiency maximization problem, using the SDR algorithm to optimize IRS phase shifts. The distinction lies in the former study focusing on a small-cell scenario, while the latter studied a simultaneous wireless information and power transfer (SWIPT) scenario. To tackle energy efficiency optimization addressed in [17], the Dinkelbach algorithm was employed to handle the fractional function, followed by an efficient algorithm based on alternating optimization. Beamforming was solved using the SDR, MM, and Dinkelbach algorithms, while IRS phase shifts were solved using manifold optimization. The authors of [18] investigated resource allocation in IRS-assisted uplink systems, which is an NP-hard non-convex problem. A block-coordinate descent-based iterative method was proposed for solving the problem. The simulation results indicated that increasing the number of IRS elements could better improve the system's energy efficiency compared to increasing the number of BS antennas. For the BS beamforming subproblem addressed in [19], a maximum ratio transmission strategy was adopted, and, for the IRS phase-shift optimization subproblem, the optimal phase shift for the IRS was obtained using the sine-cosine algorithm. The authors of [20] employed the block-coordinate descent method to jointly optimize the BS beamforming vector, power, and phase shift matrix. The simulation results showed that the block-coordinate descent method exhibited rapid convergence, and the proposed scheme achieved a 70.7% gain in energy efficiency compared to the non-IRS scheme. Although the aforementioned works have significantly improved the energy efficiency of IRS-assisted communication systems,they mainly rely on optimization methods such as the SDR algorithm and the penalty function, which have high computational complexity [21]. Moreover, these works overlook the impact of the Doppler effect on the system's energy efficiency, reducing the applicability of the algorithms to IRS-assisted HST communication scenarios. For IRS-assisted HST communication systems, ref. [22] aims to address communication blockages, enhance spectrum efficiency, and reduce outage probability by utilizing statistical channel state information (CSI) to investigate the outage performance of the MIMO system. A closed-form expression for the outage probability is derived. Furthermore, under the constraint of transmit power, the joint design of transmit-and-receive beamforming and an IRS phase shift is conducted to minimize the outage probability. In [23], a deep reinforcement learning framework combining LSTM and DDPG is proposed to design the BS transmit beamforming and IRS phase shift for millimeter-wave HST networks, thereby improving the system's spectrum efficiency. In order to enhance the anti-interference capability of the high-speed train network, ref. [24] addresses the capacity maximization problem for an IRS-assisted high-speed train network in the presence of interference. An IRS phase-shift optimization scheme based on deep deterministic policy gradients (DDPG) is proposed. Additionally, the issue of external interference threatening train safety is considered, and the interference suppression problem is transformed into a maximum-received signal-to-noise ratio problem, which is solved using the SDR algorithm [25]. However, the high complexity of the SDR algorithm hinders its large-scale application in HST communication systems. In [26], an optimization algorithm for joint transmit-and-receive beamforming and IRS phase shifts is proposed to enhance the capacity of IRS-assisted millimeter-wave HST communication systems.