State estimations and noise identifications with intermittent corrupted observations via Bayesian variational inference

Accurate state estimation of nonlinear dynamical systems is fundamental to modern aerospace operations across air, sea, and space domains. Online tracking of adversarial unmanned aerial vehicles (UAVs) is especially challenging due to agile nonlinear motion, noisy and sparse sensor measurements, and unknown control inputs; conditions that violate key assumptions of classical Kalman filter variants and degrade estimation performance. Neural networks (NNs) can learn complex nonlinear relationships from data, but lack principled uncertainty quantification, which is critical for state estimation tasks where confidence bounds drive downstream decisions. We address this with Bayesian Neural Networks (BNNs), which model uncertainty through distributions over network weights and produce predictive means and uncertainties via Monte Carlo sampling. Building on this, we propose the Bayesian Neural Kalman Filter (BNKF): a hybrid framework coupling a trained BNN with a Kalman correction step for robust online UAV state estimation. Unlike related neural Kalman approaches, BNKF produces full state predictions and incorporates Bayesian uncertainty directly into covariance propagation, improving robustness under high noise conditions. We evaluate BNKF under varying radar noise levels and sampling rates using synthetic nonlinear UAV flight data. Five fold cross validation demonstrates that BNKF outperforms Extended and Unscented Kalman Filters in accuracy, precision, and truth containment under degraded sensing. An ensemble variant (BNKFe) further improves precision in high-noise edge cases at a slight accuracy tradeoff. Runtime analysis confirms minimal inference overhead, supporting real-time deployment feasibility.

Online latent state estimation constitutes a fundamental challenge within the artificial intelligence field, serving as a foundational tool for diverse applications, including sequential decision making, anomaly and change-point detection. In this paper, a novel online distributed sensing framework, where agents collaborate and exchange information to perform latent state estimation, is presented. The proposed estimator combines available partial domain knowledge with the representation capabilities of deep neural networks. In particular, the designed sensing framework incorporates prior estimates, optimized consensus weights, and Kalman-like recursive updates to perform decentralized inference, without relying on knowledge of noise statistics. Extensive experiments on linear, chaotic (Lorenz), and practical wireless tracking environments reveal that the proposed Covariance-Agnostic Neural Kalman Consensus Filter (CA-NKCF) outperforms traditional distributed Kalman and particle filters as well as purely model-free deep neural networks, exhibiting robustness even when the underlying motion and observation models are misspecified. It is also demonstrated that CA-NKCF's performance advantage remains stable across varying noise levels, random communication topologies, latent state dimensions, and observation clutter densities induced by scattering objects in wireless systems.

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.