The nonequilibrium vibrational kinetics of carbon dioxide is modeled taking into account complex mechanisms of relaxation and energy exchanges between modes. The possibilities of using machine-learning methods to enhance the performance of the numerical simulation of nonequilibrium carbon-dioxide flows are studied. Various strategies for increasing the efficiency of the hybrid four-temperature model of CO2 kinetics are considered. The neural-network-based approach proposed by us to calculate the rate of vibrational relaxation in each mode turns out to be the most promising. For the problem of spatially homogeneous relaxation, the error and computational costs of the developed algorithm are estimated, and its high accuracy and efficiency are demonstrated. For the first time, the carbon-dioxide flow behind a planar shock wave is simulated in a complete state-to-state approximation. The results obtained are compared with those in the hybrid four-temperature approach, and the equivalence of the approaches is shown. Based on this, the developed multi-temperature approximations may be recommended as the main tool for solving problems of nonequilibrium kinetics and gas dynamics. The hybrid four-temperature approach that uses a neural network for calculating relaxation terms reduces the numerical-simulation time by more than an order of magnitude while maintaining its accuracy. This technique can be recommended for solving complex multidimensional problems of nonequilibrium gas dynamics, including state-to-state chemical reactions.