This paper introduces a novel numerical approach to achieve smooth lane-change trajectories in autonomous driving scenarios. Our trajectory generation approach leverages particle swarm optimization (PSO) techniques, incorporating Neural Network predictions for trajectory refinement. The generation of smooth and dynamically feasible trajectories for lane change is facilitated by combining polynomial curve fitting with particle propagation, which accounts for vehicle dynamics. The proposed planning algorithm is capable of determining feasible trajectories with real-time computation capability. To validate our approach, we conduct comparative analyses with two baseline methods, involving analytic solutions and heuristic techniques in numerical simulations. The simulation results support the efficacy and effectiveness of our proposed approach.