This work presents a novel approach for designing preliminary fuel-optimal low-thrust spacecraft trajectories with gravity assists (GAs) by employing Physics-Constrained Neural Networks (PCNNs). The study introduces the Multiple-Leg Trajectory PCNN (MLT-PCNN) which embeds physical dynamics and constraints directly into the neural network architecture and computes planar fuel-optimal low-thrust trajectories with a gravity assist assuming restricted two-body motion. It models a GA as a discontinuous change in the spacecraft's heliocentric velocity vectors, arising from an instantaneous turning of the spacecraft’s hyperbolic excess velocity vectors with respect to the GA body. The model’s performance is demonstrated using Earth–Mars–Ceres and Earth-Venus-Earth-Mars-Jupiter case studies, where MLT-PCNNs are trained with various training schedules using both first-order (Adam) and second-order (L-BFGS) optimizers. Physical accuracy of MLT-PCNN-generated solutions is evaluated by error metrics that measure discrepancies in final position, velocity, and mass, obtained through numerical integration of the MLT-PCNN generated control profile. A control profile correction method based on the state transition matrix validates that only a few grams of additional fuel are required to eliminate residual errors in the final states. The results show that the MLT-PCNN can generate fuel-optimal solutions comparable or superior to state-of-the-art techniques.