Switches-based hybrid architecture has attracted much attention, especially in directional-of-arrival (DOA) sensing, due to its ability of significantly reducing the hardware cost by compressing massive multiple-input multiple-output (MIMO) arrays with switching networks. However, this structure will lead to a degradation in the degrees of freedom (DOF) and accuracy of DOA estimation. To address these two issues, we first propose a switches-based sparse hybrid array (SW-SHA). In this method, we design a dynamic switching network to form a synthesized sparse array, i.e., SW-SHA, that can enlarge the virtual aperture obtained by the difference co-array, thereby significantly enhancing the DOF. Second, in order to improve the DOA estimation accuracy of switches-based hybrid arrays, a deep neural network (DNN)-based method called ASN-DNN is proposed. It includes an antenna selection network (ASN) for optimizing the switch connections based on the criterion of minimizing the Cramer-Rao lower bound (CRLB) under the peak sidelobe level (PSL) constraint and a DNN for DOA estimation. Then by integrating ASN and DNN into an iterative process, the ASN-DNN is obtained. Furthermore, the closed-form expression of CRLB for DOA estimation is derived to evaluate the performance lower bound of switches-based hybrid arrays and provide a benchmark for ASN-DNN. The simulation results show the proposed ASN-DNN can achieve a greater performance than traditional methods, especially in the low signal-to-noise ratio (SNR) regions.