This paper proposes a new digital signal recovery (DSR) technique for next-generation power efficient low Earth orbit (LEO) satellite-to-ground communication systems, which feature additive white Gaussian noise (AWGN) channel and significant power variation. This technique utilizes the prior knowledge (i.e., nonlinearities of radio frequency power amplifiers (RF-PAs)) of space-borne transmitters to improve the quality of the signal received at ground stations by modeling and mitigating the imperfection using deep neural networks (DNNs). Benefiting from its robustness against noise and power variation, the proposed DNN based DSR technique (DNN-DSR), can correct high signal distortions caused by the nonlinearities and hence allows RF-PAs to work close to their saturation region, leading to a high power efficiency of the LEO satellites. This work has been validated by both simulations and experiments, in comparison with the power back-off technique as well as memory polynomial based DSR solutions. Experimental results show that the DNN-DSR technique can increase the drain efficiency of the space-borne RF-PA from 32.6% to 45% while maintaining the same error vector magnitude as the power back-off technique. It has also been demonstrated that the proposed DNN-DSR technique can handle a signal power variation of 12 dB, which is challenging for conventional solutions.