Over the past forty years, air traffic flow management (ATFM) has garnered significant attention since the initial approach was introduced to address single-airport ground delay issues. Traditional methods for solving both single- and multi-airport ground delay problems primarily rely on operations research techniques and are typically formulated as mixed-integer problems (MIPs), with solvers employed to approximate optimal solutions. Despite their effectiveness in smaller-scale problems, these approaches struggle with the complexity and scalability required for large-scale, multi-sector ATFM, leading to suboptimal performance in real-time scenarios. To overcome these limitations, we propose a novel neural network-based demand and capacity balancing (NN-DCB) method that leverages neural branching and neural diving to efficiently solve the ATFM problem. Using data from 15,927 flight trajectories across 287 airspace sectors on a typical day in February 2024, our method re-allocates trajectory entry and exit times in each sector. The results demonstrate that large-scale ATFM problems can be solved within 15 min, offering a significant performance improvement over the state-of-the-art methods. This study confirms that neural network-based approaches are more effective for large-scale ATFM problem-solving.