Low Earth orbit (LEO) satellite mega-constellation networks aim to address the high connectivity demands with a projected 50,000 satellites in less than a decade. To fully utilize such a large-scale dynamic network, an air network composed of stratospheric nodes, specifically high altitude platform station (HAPS), can help significantly with a number of aspects including mobility management. HAPS-LEO network will be subject to time-varying conditions, and in this paper, we introduce an artificial intelligence (AI)-based approach for the unique channel estimation and synchronization problems. First, channel equalization and carrier frequency offset with residual Doppler effects are minimized by using the proposed convolutional neural networks based estimator. Then, the data rate is compounded by increasing spectral efficiency using non-orthogonal multiple access method. We observed that the proposed AI-empowered HAPS-LEO network provides not only a high data throughput per second but also higher service quality thanks to the agile signal reconstruction process.