While Artificial Neural Networks (ANN) representational capabilities have long been considered to benefit guidance and control tasks, a knowledge gap exists in the literature for their application to augment real-time navigation algorithms. Deep learning has excelled in various tasks, but its computational complexity limits its use in onboard spacecraft navigation. The study focuses on Single Layer Feedforward Networks, simpler ANN structures, to design shallow Physics-Informed Neural Networks (PINNs). The research aims to merge physics-based models with shallow neural networks to improve computational efficiency and enhance autonomy of conventional navigation algorithms. The proposed methodology involves designing a novel architecture combining an Extended Kalman Filter (EKF) with shallow PINNs, while also developing a novel incremental learning technique. Numerical simulations are performed for a relative navigation LEO scenario and results are presented. The PINN-EKF architecture emerges as a robust system, capable of taking advantage of physics-based knowledge, as well as of automatically predicting the dynamics from observations. In particular, it provides substantial computational advantage over standard nonlinear sequential estimators at a limited accuracy and autonomy performance degradation.