Benefiting from the growth of the bandwidth, Terahertz (THz) communication can support the new application with explosive requirements of the ultra-high-speed rates for future 6G wireless systems. In order to compensate for the path loss of high frequency, massive multiple-input multiple-output (MIMO) can be utilized for high array gains by beamforming. However, since a large number of analog phase shifters should be used to realize the analog beamforming, the existing THz communication with massive MIMO has very high energy consumption. To solve this problem, a reconfigurable intelligent surface (RIS)-based hybrid precoding architecture for THz communication is developed in this paper, where the energy-hungry phased array is replaced by the energy-efficient RIS to realize the analog beamforming of the hybrid precoding. Then, based on the proposed RIS-based architecture, a sum-rate maximization problem for hybrid precoding is investigated. Since the phase shifts implemented by RIS in practice are often discrete, this sum-rate maximization problem with a non-convex constraint is challenging. Next, the sum-rate maximization problem is reformulated as a parallel deep neural network (DNN)-based classification problem, which can be solved by the proposed low-complexity deep learning-based multiple discrete classification (DL-MDC) hybrid precoding scheme. Finally, we provide numerous simulation results to show that the proposed DL-MDC scheme works well both in the theoretical Saleh-Valenzuela channel model and practical 3GPP channel model. Compared with existing iterative search algorithms, the can proposed DL-MDC scheme reduces the runtime significantly with a negligible performance loss.