Space weather phenomena occur from the Sun to the Earth with damaging impacts on ground-based and space-borne technological infrastructure. The geomagnetic auroral electrojet indices, AU, AL, and AE, have been widely used for monitoring space weather and geomagnetic activities during space storms and substorms. The time series data of solar wind monitored by upstream satellite and ground-based auroral electrojet indices form the input-output system characterizing the dynamic coupling among solar wind, Earth’s magnetosphere, and ionosphere. The data-driven predictions of auroral electrojet indices during geomagnetic storms and substorms face the challenges of capturing the variations of ionospheric electrojet current driven by multiple solar wind variables and are modeled as a coupled complex system with finite and variable memory. The recurrent neural network (RNN) based Long Short-Term Memory (LSTM) machine learning algorithm is well suited to classify, process, and make predictions of the coupled solar wind-magnetosphere-ionosphere system by preserving important information from earlier parts of the coupled time series and carrying it forward. In this study, an RNN-based LSTM model has been built to predict the time series of AE/AL indices with multi-variate solar wind inputs. Both 5-minute and hourly long-term time series data from the NASA OMNI database were used to drive the LSTM model. The coupled time series data are divided into training and testing datasets. The Root-Mean-Square-Error (RMSE) between the predicted and actual AE/AL indices of the testing sets was used to evaluate the roles of the number of layers in the LSTM, memory length of the coupled system, prediction time, and different combinations of solar wind input parameters (magnetic field, velocity, and density). The performance of the LSTM model in predicting AL/AE indices during major geomagnetic storm and substorm events is analyzed. The differences and challenges of applying LSTM to predict 5-min and hourly AE/AL indices are also discussed.