Non-cooperative interactions commonly occur in multi-agent scenarios such as car racing, where an ego vehicle can choose to overtake the rival, or stay behind it until a safe overtaking "corridor" opens. While an expert human can do well at making such time-sensitive decisions, autonomous agents are incapable of rapidly reasoning about complex, potentially conflicting options, leading to suboptimal behaviors such as deadlocks. Recently, the nonlinear opinion dynamics (NOD) model has proven to exhibit fast opinion formation and avoidance of decision deadlocks. However, NOD modeling parameters are oftentimes assumed fixed, limiting their applicability in complex and dynamic environments. It remains an open challenge to determine such parameters automatically and adaptively, accounting for the ever-changing environment. In this work, we propose for the first time a learning-based and game-theoretic approach to synthesize a Neural NOD model from expert demonstrations, given as a dataset containing (possibly incomplete) state and action trajectories of interacting agents. We demonstrate Neural NOD's ability to make fast and deadlock-free decisions in a simulated autonomous racing example. We find that Neural NOD consistently outperforms the state-of-the-art data-driven inverse game baseline in terms of safety and overtaking performance.