The Traffic Assignment Problem (TAP) is essential for traffic flow analysis, traditionally solved using mathematical programs under the User Equilibrium (UE) principle. These methods become computationally prohibitive for large-scale networks due to non-linear growth in complexity with the number of origin-destination (OD) pairs. This study introduces a novel data-driven approach using Deep Neural Networks (DNNs), specifically leveraging Transformer architecture, to predict equilibrium path flows directly. By focusing on path-level traffic distribution, the proposed model captures intricate correlations between OD pairs, offering a more detailed and flexible analysis compared to traditional link-level approaches. The Transformer-based model significantly reduces computation time and adapts to changes in both demand and network structure, predicting optimal solutions without the need for recalculating from scratch. Numerical experiments are conducted on the Manhattan-like synthetic network and the Sioux Falls network. The results demonstrate that the proposed model efficiently estimates optimal traffic flow distribution at the path level, saving computational resources, enhancing traffic flow prediction accuracy by capturing detailed trip usage and flow information, and offering flexibility to adapt to varying demand and network changes, thus supporting traffic management, and enabling "what-if" analyses for improved transportation planning and policy-making.