Exploring the usage of Probabilistic Neural Networks for Ionospheric electron density estimation

A fundamental limitation of traditional Neural Networks (NN) in predictive modelling is their inability to quantify uncertainty in their outputs. In critical applications like positioning systems, understanding the reliability of predictions is paramount for constructing confidence intervals, early warning systems, and effectively propagating results. For instance, Precise Point Positioning (PPP, see Zumberge et al (1997)) in satellite navigation heavily relies on accurate error models for ancillary data (orbits, clocks, ionosphere, and troposphere) to compute precise error estimates and establish robust protection levels. As an example, one of the main objectives of the Galileo High Accuracy Service (HAS) Service Level 2 will be to provide the necessary regional atmospheric delay corrections (and associated uncertainty) in order to improve user positioning based on PPP strategies, most notably the convergence time of the solution (see for instance Juan et al (2025)). To address this challenge, the main objectives of this paper aims at exploring a potential framework capable of providing both point estimates and associated uncertainty measures of ionospheric Vertical Total Electron Content (VTEC). Probabilistic Neural Networks (PNNs) offer a promising approach to achieve this goal. However, constructing an effective PNN requires meticulous design of hidden and output layers, as well as careful definition of prior and posterior probability distributions for network weights and biases. This introduction provides a review in terms of state-of-the-art in PNN as well as the application of NN in ionospheric estimation of VTEC.