Deep video compression has made significant progress in recent years, achieving rate-distortion performance that surpasses that of traditional video compression methods. However, rate control schemes tailored for deep video compression have not been well studied. In this paper, we propose a neural network-based $λ$-domain rate control scheme for deep video compression, which determines the coding parameter $λ$ for each to-be-coded frame based on the rate-distortion-$λ$ (R-D-$λ$) relationships directly learned from uncompressed frames, achieving high rate control accuracy efficiently without the need for pre-encoding. Moreover, this content-aware scheme is able to mitigate inter-frame quality fluctuations and adapt to abrupt changes in video content. Specifically, we introduce two neural network-based predictors to estimate the relationship between bitrate and $λ$, as well as the relationship between distortion and $λ$ for each frame. Then we determine the coding parameter $λ$ for each frame to achieve the target bitrate. Experimental results demonstrate that our approach achieves high rate control accuracy at the mini-GOP level with low time overhead and mitigates inter-frame quality fluctuations across video content of varying resolutions.