## 1 Introduction
However, the SEC task offloading problem presents a particularly challenging scenario: the action space combines discrete decisions (task offloading choices) with continuous variables (power allocation, UAV trajectories) [17]. Traditional DRL methods either discretize the entire action space, sacrificing precision and creating computational bottlenecks, or struggle to handle hybrid action spaces effectively. Given the aforementioned challenges, this work addresses three key research questions. How can MLLMs be efficiently deployed in UAV-assisted SEC networks to provide ubiquitous services to IoT devices in remote areas? How can we jointly optimize discrete offloading decisions and continuous resource allocation variables to minimize system latency and energy consumption while ensuring MLLM service quality in dynamic SEC environments? How can we develop a DRL that effectively handles the hybrid discrete-continuous action space inherent in SEC task offloading problems? To address these research questions, this paper aims to achieve comprehensive performance optimization for MLLM inference in UAV-assisted SEC networks through intelligent MLLM task offloading, bandwidth allocation, and UAV trajectory. The primary objective is to develop a framework that minimizes system-wide latency and energy consumption while ensuring reliable MLLM service delivery to remote IoTDs. This involves formulating the joint optimization problem as a mixed-integer nonlinear programming (MINLP) problem that simultaneously handles discrete offloading decisions and continuous resource allocation variables. To solve this challenging optimization problem, we propose an action-decoupled soft actor-critic (AD-SAC) algorithm specifically designed for hybrid discrete-continuous action spaces. This approach maintains the precision of continuous control while efficiently handling discrete decisions, enabling real-time adaptation to dynamic SEC environments. The framework aims to minimize system-wide latency and energy consumption while ensuring sustainable MLLM service delivery to IoTDs in remote environments. The main contributions of this work are summarized as follows. * We develop a novel MLLM inference framework that jointly optimizes task offloading and resource allocation in UAV-assisted SEC networks, enabling ubiquitous MLLM services for IoT devices in remote environments through coordinated utilization of UAVs and LEO satellites. * We formulate the joint optimization problem as an MINLP problem that minimizes the weighted sum of MLLM offloading latency and energy consumption. The formulation simultaneously optimizes discrete offloading decisions and continuous variables, including power allocation and UAV trajectories, while satisfying accuracy requirements and accounting for dynamic LEO satellite coverage. * We propose an AD-SAC algorithm to solve the MINLP problem with hybrid discrete-continuous action spaces. By delegating discrete and continuous actions to specialized agents within a cooperative training framework, AD-SAC effectively leverages off-policy data to develop robust hybrid policies for real-time adaptation. * Extensive simulation results demonstrate that our proposed method achieves a more effectively converged policy and exhibits enhanced performance in MLLM inference when compared to prevalent DRL baseline algorithms. The organization of this manuscript is structured as follows. Section 2 reviews the related work. The system model and problem formulation are delineated in Section 3. A comprehensive description of the hybrid space SAC algorithm is provided in Section 4. The performance evaluation of the proposed algorithm is presented in Section 5, and the study is summarized in Section 6.