PC-Transformer: Probabilistic flight trajectory prediction with multi-stage predictive coding

Reliable Flight Trajectory Prediction (FTP) is fundamental to critical Air Traffic Management (ATM) applications. Existing FTP methods that employ deterministic point estimation fail to model multi-source uncertainties such as route adjustments and atmospheric turbulence in large-scale data. This limitation restricts the probabilistic outputs needed for risk-aware airspace decisions. To address this, we propose the Predictive Coding Transformer (PC-Transformer). Our approach integrates physical interpretability with high-order probabilistic reasoning through a three-stage framework. First, baseline predictions are generated using classical kinematic models. Second, residual encoding maps prediction errors to discrete index spaces, mitigating high-dimensional computational bottlenecks. Finally, probabilistic mapping is achieved via a multi-scale routing encoder and Mamba-Attention decoder. This captures non-Gaussian features and spatial-topological semantics during trajectory evolution. Extensive experiments on ADS-B datasetsfrom the OpenSky Network and ATC systems demonstrate several key findings: (1) Predictive coding significantly enhances baseline methods, yielding interpretable and highly accurate predictions. (2) PC-Transformer achieves substantial improvements in trajectory prediction accuracy over baseline models. It exhibits particularly strong long-term prediction performance, reducing the mean deviation error (MDE) by 24 %-35 %. (3) Across three representative airspace scenarios and three typical flight states, PC-Transformer accurately predicts basic flight modes from short-term historical trajectories. It also precisely estimates future waypoint motion states and outputs their corresponding probability distributions.Furthermore, the probabilistic outputs provide direct support for critical air traffic control decisions, significantly enhancing conflict detection reliability and slot allocation robustness compared to deterministic baselines.