The phenomenon of capillary rise in a tube is mainly affected by the inertia, capillary force, gravitational force, and viscous force of fluids. It has been observed that certain fluid properties and tube geometries can cause the liquid column to oscillate. In this paper, we first present a general second-order ordinary differential equation that can be used to describe the dynamic process of oscillation in terms of the tube length, tube radius, and contact angle. However, it is difficult to obtain its finding an analytical solution owing to the existence of nonlinear terms. Thus, we adopt a deep neural network (DNN) method to solve this equation for its distinct feature in function-fitting ability. Simulations are conducted to test the performance of the DNN method, and the results are found to be in good agreement with data reported in the literature. Based on a suitably designed and trained DNN, we further perform a comprehensive analysis of the rising height and moving velocity of the fluid during the oscillatory process for different combinations of tube length, tube radius, and contact angle. The results show that the DNN method provides an effective tool for studying the oscillation dynamics of capillary rise.