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Abstract Global Navigation Satellite System (GNSS) is pervasively used in position, navigation, and timing (PNT) applications. As a consequence, important assets have become vulnerable to intentional attacks on GNSS, where of particular relevance is spoofing transmissions that aim at superseding legitimate signals with forged ones in order to control a receiver’s PNT computations. Detecting such attacks is therefore crucial, and this article proposes to employ an algorithm based on deep learning to achieve the task. A data-driven classifier is considered that has two components: a deep learning model that leverages parallelization to reduce its computational complexity and a clustering algorithm that estimates the number and parameters of the spoofing signals. Based on the experimental results, it can be concluded that the proposed scheme exhibits superior performance compared to the existing solutions, especially under moderate-to-high signal-to-noise ratios. 

Hybrid physics-based data-driven models, namely, augmented physics-based models (APBMs), are capable of learning complex state dynamics while maintaining some level of model interpretability that can be controlled through appropriate regularizations of the data-driven component. In this article, we extend the APBM formulation for high-order Markov models, where the state space is further augmented with past states (AG-APBM). Typically, state augmentation is a powerful method for state estimation for a high-order Markov model, but it requires the exact knowledge of the system dynamics. The proposed approach, however, does not require full knowledge of dynamics, especially the Markovity order. To mitigate the extra computational burden of such augmentation we propose an approximated-state APBM (AP-APBM) implementation leveraging summaries from past time steps. We demonstrate the performance of AG- and AP-APBMs in an autoregressive model and a target-tracking scenario based on the trajectory of a controlled aircraft with delay-feedback control. The experiments showed that both proposed strategies outperformed the standard APBM approach in terms of estimation error and that the AP-APBM only degraded slightly when compared to AG-APBM. For example, the autoregressive (AR) model simulation in our settings showed that AG-APBM and AP-APBM reduced the estimate error by 31.1% and 26.7%. The time cost and memory usage were reduced by 37.5% and 20% by AP-APBM compared to AG-APBM.