In the present study, a novel combined system consisting of solid oxide fuel cell (SOFC), organic Rankine cycle (ORC), and compressed air energy storage (CAES) is proposed, investigated, and optimized. The SOFC and CAES models are validated individually to ensure the accuracy of the results. Here, the grey wolf multi-objective optimization (MOGWO) approach is applied to find the optimal system design and performance. For this, a trained neural network is provided to the MOGWO algorithm as a fitted function, and multi-objective optimization is carried out on it. The most significant benefit of the suggested method is time-saving. The proposed system's thermodynamic performance is investigated from the energy, exergy, economic, and environmental (4E) points of view at three periods, including full-time, charging, and discharging periods. The results indicate that the Levenberg-Marquardt training algorithm has the best performance among all of the algorithms. The value of exergetic round trip efficiency (ERTE), total cost rate, and CO2 emission at the best optimum point are obtained as 45.7%, 34.2 $/h, and 0.22 kg/kWh, respectively.