Hybrid Adaptive Kalman Filtering for Data-Efficient Joint Tracking and Classification

Nonlinear state estimation requires sequentially fusing model-based predictions with noisy measurements. Under imperfect dynamics and unknown, time-varying noise statistics, this fusion can degrade in both accuracy and statistical consistency. Existing learning-aided filters largely treat accuracy and uncertainty estimation separately, limiting their ability to correct model-mismatch-induced bias while retaining an explicit, calibrated posterior covariance. This paper introduces Unscented KalmanNet (UKN), a model-based deep learning architecture that extends the Unscented Kalman Filter (UKF) with learned mechanisms for these two sources of filtering error while preserving explicit posterior covariance propagation. NoiseNet learns time-varying process and measurement covariances as bounded multiplicative corrections to baseline covariances, guaranteeing positive definiteness, while GainNet learns a bounded residual correction to the analytical UKF gain to compensate for model-mismatch-induced bias. A calibration-aware training objective couples state error with posterior covariance and innovation consistency terms through adaptive weighting, jointly optimizing accuracy and calibration. UKN is benchmarked against UKF, KalmanNet, and Bayesian KalmanNet on three synthetic systems and real-flight UZH-FPV data. It achieves the lowest state-estimation error in all four examples and reduces RMSE by 26.4-49.7% compared with UKF in the synthetic cases. Leave-one-sequence-out cross-validation over 11 flights shows 22.4% and 34.3% reductions in mean position and velocity RMSE, respectively. UKN also yields the lowest fold-to-fold variability, with dimension-normalized NEES and empirical coverage closest to nominal values among covariance-reporting filters. These results show that structured learned adaptation improves estimation accuracy while retaining calibrated uncertainty.

State estimation is a fundamental problem in control and signal processing, for which the Kalman Filter provides an optimal solution under linear dynamics, Gaussian noise, and known noise covariances. However, these assumptions often fail in realistic sensing settings such as Doppler radar and LiDAR. In these cases, the optimal estimator is inherently nonlinear, which leads to systematic performance degradation. This creates a performance gap that cannot be eliminated by tuning the noise covariance parameters (i.e., the process and measurement noise in the Kalman Filter) alone. To address this limitation, we propose Kalman Evolve, a framework for discovering improved filtering algorithms by jointly optimizing both noise parameters and the update structure. Our approach leverages large language models (LLMs) as a structured prior over program space, enabling the generation of interpretable, non-affine modifications to the classical Kalman filter while preserving its recursive form. We provide analytical results establishing the suboptimality of affine estimators under common nonlinear sensing models, motivating the need for structure-aware updates. Across a range of synthetic and real-world tracking benchmarks, including Doppler radar, LiDAR-based localization, and pedestrian tracking, the discovered algorithms consistently improve over strong baselines such as the Optimized Kalman Filter, achieving up to 12% reduction in RMSE. These results suggest that optimizing the structure of the Kalman filter, rather than only its parameters, provides a practical and interpretable way to improve state estimation.

Unmanned Aerial Vehicles in dynamic environments face telemetry outages, structural vibrations, and regime-dependent noise that invalidate the stationary covariance assumptions of classical Kalman filters. The Sage-Husa Kalman Filter (SHKF) estimates noise statistics online, but its reliance on a static, scalar forgetting factor forces a strict compromise between steady-state stability and transient responsiveness. We introduce the N-Deep Recurrent Sage-Husa Filter (NDR-SHKF), which replaces this scalar parameter with a vector-valued memory attenuation policy learned by a hierarchical recurrent network operating on whitened innovation sequences. A bifurcated architecture routes shallow recurrent states to capture instantaneous sensor anomalies and deep states to encode sustained dynamic trends, while an auxiliary reconstruction objective prevents feature collapse. The complete filter, including recursive covariance updates, is trained end-to-end via backpropagation through time to directly minimize state estimation error. Evaluations on topologically distinct chaotic attractors demonstrate cross-domain generalization, outperforming purely data-driven baselines that diverge under out-of-distribution dynamics. Furthermore, evaluations on recorded real-world UAV flight datasets validate the framework's practical viability, demonstrating its capacity to bridge transitions into proprioceptive dead reckoning and outperform classical adaptive estimators during sensor outages.