A Coordinate-Conditioned Multiscale Framework for Short-Horizon Trajectory Forecasting: Mathematical Analysis and Numerical Experiments

This paper studies short-horizon trajectory forecasting through a coordinate-conditioned multiscale architecture, termed AeroMixer. The framework combines segment-wise local Cartesian re-representation, multiscale decomposition, bidirectional trend mixing, scale-specific prediction heads, and direct scale aggregation for the one-step prediction of highly dynamic aerial trajectories. The mathematical part of the paper establishes a local tangent-plane approximation bound for the geodetic-to-local map, a horizontal metric-scale characterization related to latitude-dependent distortion, and smoothness as well as spectral perturbation characterizations for the weighted and kinematically regularized objective. The analysis quantifies the local approximation error and objective regularity for the proposed representation and loss. Numerical experiments on 72,000 simulated trajectory samples compare five deep learning models under trajectory-level splits, three seeds, and batch size 256. AeroMixer achieves the lowest batch-wise and pooled global position RMSE in this matched set. Regime-wise and ablation analyses further identify the conditions under which the observed differences arise.