## 1 Introduction  
Space information networks consist of satellite platforms, ground stations, and other communication nodes that provide global coverage and real-time data transmission services. These networks play an important role in disaster response, emergency communications, and military operations by decreasing transmission delays and expanding coverage, ensuring seamless data exchange and thus significantly improving communication efficiency [1, 2]. In contrast to traditional terrestrial networks and localized communication systems, space information networks utilize space platforms to provide continuous communication support across geographic regions [3], and their multi-level resource management and task scheduling capabilities satisfy communication needs in complex environments [4, 5], thereby improving information transmission efficiency and system reliability [6]. Despite the advantages of space information networks in providing global coverage and real-time data transmission, it is still a significant challenge to efficiently and optimally allocate network resources under limited conditions [7]. Several specific challenges arise from the heterogeneity among LEO satellite constellations and ground stations. The differences in computational capabilities, bandwidth, and storage resources among various satellite platforms result in load imbalances during task offloading. Additionally, the hardware configurations, network connectivity, and geographical locations of ground stations further complicate resource allocation and task scheduling. In dynamic topologies and under resource fluctuations, these heterogeneities make it difficult for existing task offloading strategies to adapt to rapidly changing environments [8]. In particular, the space-terrestrial integrated network (STIN) faces specific challenges such as seamless connectivity between satellites and ground stations, cross-domain resource management, and load balancing in dynamic environments [9; 10]. Variations in satellite and ground station delays, bandwidth, and computational capabilities, coupled with external factors like orbital changes and weather disruptions, increase the complexity of task offloading decisions. Moreover, the inherent security challenges of STIN arise from its system heterogeneity, which expands potential attack surfaces and complicates privacy protection during data transmission. To cope with these challenges, existing research has mainly addressed the complexity of node heterogeneity and inter-node collaboration by introducing distributed optimization algorithms and collaborative computing strategies [11]. Researchers have proposed optimization frameworks based on Lagrangian duality and gradient descent methods to improve the efficiency of task offloading decisions and address the resource imbalance between platforms, achieving good local optimization results in static environments [12; 13]. However, these methods usually focus on a single optimization objective or local problem, which fail to consider inter-task dependencies and coordination globally and lack a systematic and holistic solution, resulting in poor performance when dealing with heterogeneous platforms and real-time scheduling in large-scale dynamic environments, and failing to adequately solve these multi-level, multi-objective and complex problems. This paper proposes a dynamic gradient descent-based task offloading method (D-GTOM), a distributed framework designed to address dynamic resource allocation and task offloading issues in STIN. This method introduces delay optimization models, resource efficiency models, and collaborative task scheduling optimization to tackle the challenges in dynamic environments. To address dynamic optimization challenges such as resource conflicts and task delays in satellite networks, our approach integrates delay optimization and resource efficiency models into a unified framework.