arXiv:2610.24588v2 Announce Type: replace Abstract: Understanding how a particular automated vehicle reacts to nearby traffic can improve motion prediction, control development, traffic simulation, and safety evaluation. However, commercial automated driving systems are proprietary, and single encounters may provide only a few seconds of noisy data. Standard physical models are interpretable but inflexible, whereas neural models are often data-hungry. We propose a physics-informed Koopman framework that combines a calibrated car-following law with a correction learned from the observations. A weak-form estimator learns this correction from velocity measurements integrated over time, avoiding noise-sensitive numerical differentiation. The resulting continuous-time model predicts the follower's response and provides interpretable measures of driving behavior. Experiments on OpenACC and CATS-CF evaluate 5-, 7-, and 11-second histories. Averaged over these histories, the constant-time-headway branch reduces speed prediction error by 16.0% and 15.4% on OpenACC and CATS-CF, respectively, relative to the best neural baselines. The PI-Koopman branches achieve the lowest mean speed and gap errors among the physical and neural comparators. Weak-form estimation reduces acceleration-field reconstruction error by 74.5% in noisy known-law simulations and noise-induced field perturbation by 91.6% in paired real-vehicle measurements. The same PI-Koopman system identification supports following-mode recognition with an approximately eleven-fold difference in median residual variance between human drivers and automated vehicles, and a single residual threshold achieves 85.4% balanced accuracy. The branch informed by the intelligent driver model takes a median 17 to 26 ms for identification and forecasting on the tested central processing units, supporting frequent target adaptation.