In remote regions lacking terrestrial infrastructure, aviation users increasingly demand Internet services comparable to ground cellular networks. This paper proposes a space-air-ground integrated network (SAGIN) architecture that combines ground networks, civil aviation networks, and low Earth orbit (LEO) satellites. Based on real North Atlantic flight data, a heterogeneous aviation Internet is constructed to support diverse aircraft service requirements. A composite Quality of Experience (QoE) metric, characterized by transmission delay, jitter, and content similarity, guides efficient resource utilization. The dynamic connectivity of SAGIN is modeled using a reconfigurable time-expanded graph (RTEG). With Network Function Virtualization (NFV), data delivery is formulated as a Service Function Chain (SFC) deployment across ground stations, relay aircraft, and satellites, resulting in a long-term QoE maximization problem. A tailored Multi-Agent Deep Deterministic Policy Gradient (MADDPG) algorithm incorporating LCSS-based path filtering and softmax-based action selection is developed. Simulation results demonstrate that the proposed approach enables seamless connectivity for heterogeneous aircraft and consistently outperforms game-theoretic and MATD3 methods in terms of QoE.

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