Friction-Feasible Trajectory Prediction for Vehicles on Winter Roads: A Warm-Up Physics Loss with Velocity-Space Learning

Vehicle trajectory prediction on icy and snowy roads is safety-critical because reduced tyre–road adhesion compresses the set of dynamically feasible manoeuvres and can substantially increase braking distances relative to dry pavement. The objective of this study is to develop a lightweight physics-enhanced predictor that jointly minimises positional error and friction-circle violations on low-friction winter roads, so that short-horizon forecasts remain usable by advanced driver-assistance functions. We propose PE-DNN, a physics-informed multi-layer perceptron that predicts absolute future velocities in the ego frame and activates friction-circle and jerk penalties through a linear warm-up schedule. Unlike prior physics-informed trajectory predictors that apply constraints from the first training epoch on zero-mean residual targets, PE-DNN treats target-space design, constraint scheduling, and mechanistic interpretation of the violation rate (VR) as coupled methodological choices. On the Northeastern Vehicle Trajectory Dataset (NVTD; 289 vehicles, 19,049 frames), PE-DNN achieves an average displacement error (ADE) of 2.019 m and VR=3.8%, with the lowest ADE–VR pair among retrained learning baselines under identical splits, while trading somewhat higher final displacement error (FDE) and extreme-subset ADE (ADE-Ext) for improved physical feasibility. Ablation and sensitivity analyses show that velocity-space labels and warm-up scheduling are essential design components rather than optional refinements.