Underground reservoir information can be obtained through well-log interpretation. However, some logs might be missing due to various reasons, such as instrument failure. A deep-learning-based method that combines a convolutional layer and a long short-term memory (LSTM) layer is proposed to estimate the missing logs without the expensive relogging. The convolutional layer is used to extract the depth-series features initially, which are then input into the LSTM layer. To improve the feature memory and extraction capabilities of the LSTM layer, we construct two LSTM-based components: the first component uses an attention mechanism to optimize the LSTM units by adaptively adjusting network weights, and the second component uses a period-skip mechanism, which enhances the sensitivity of aperiodic changes in the depth series by learning the information of the shallow sequence. In addition, we add an autoregressive component to enhance the linear feature extraction capability while learning the nonlinear relationship between different logs. A total of 13 wells from two different regions are used for experiments, including 11 training and two test wells. We use one well to calculate the uncertainties of four time-series networks, i.e., our proposed network and three benchmark models (recurrent neural network, gated recurrent unit, and LSTM), to demonstrate the stability and robustness of the proposed method. Furthermore, we evaluate the performance of our proposed method in several crossover experiments, e.g., different logging intervals, depths, and input logs. Compared to a state-of-the-art deep learning method and a classic LSTM network, the proposed network has higher reliability, which is reflected in the feature extraction of depth series with a larger span. The experimental results demonstrate that our proposed network can accurately generate sonic and other unknown logs.