Abstract: Low Earth Orbit (LEO) satellite networks, characterized by their ubiquity, low communication latency and long-range propagation capabilities, have emerged as a key development trend for next-generation networks. The integration of LEO satellite networks with Mobile Edge Computing (MEC) technology enables the provisioning of high-speed, ubiquitous computation offloading services to terrestrial end-users, which effectively mitigates the resource constraints of insufficient network bandwidth and computing power at the user side. Such integration is also a pivotal enabler for the construction of integrated space-air-ground computing network. This study investigates the satellite-terrestrial collaborative computing problem in LEO satellite networks, and establish a system model consisting of terrestrial users, LEO satellites with their adjacent satellites nodes, and ground cloud service centers. The computation task offloading problem in this network is formulated as a Mixed-Integer Programming (MIP) problem, with the dual objectives of minimizing the total system energy consumption and task processing delay. Given the NP-hard (Non-deterministic Polynomial time hard) nature of the formulated problem, obtaining exact solutions through conventional convex optimization methods entails prohibitively high computational complexity and requires global information, making it impractical for real-world implementation.To address this challenge, this study employs Deep Reinforcement Learning (DRL) techniques and propose a GCN-AC based task offloading scheme for satellite-terrestrial collaborative computing, which integrates Graph Convolutional Networks (GCN) and the Actor-Critic (AC) algorithm. Simulation results demonstrate that the proposed scheme can effectively reduce system’s end-to-end latency and energy consumption. Furthermore, compared with traditional state-of-the-art solutions, it achieves evident improvement in convergence speed, overall performance, and cost efficiency.