The traffic assignment problem is essential for traffic flow analysis, traditionally solved using mathematical programs under the Equilibrium principle. These methods become computationally prohibitive for large-scale networks due to non-linear growth in complexity with the number of OD pairs. This study introduces a novel data-driven approach using deep neural networks, specifically leveraging the 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 drastically reduces computation time, while adapting to changes in demand and network structure without the need for recalculation. Numerical experiments are conducted on the Manhattan-like synthetic network, the Sioux Falls network, and the Eastern-Massachusetts network. The results demonstrate that the proposed model is orders of magnitude faster than conventional optimization. It efficiently estimates path-level traffic flows in multi-class networks, reducing computational costs and improving prediction accuracy by capturing detailed trip and flow information. The model also adapts flexibly to varying demand and network conditions, supporting traffic management and enabling rapid `what-if' analyses for enhanced transportation planning and policy-making.