On Covariance Estimation in Physics Informed Neural Networks for Orbit Determination

Artificial intelligence (AI), particularly machine learning (ML), is widely applied in fields such as medicine, autonomous driving, and manufacturing. Over time, ML has also seen increasing use in space and geosciences, where its algorithms hold the potential to enhance orbit prediction and orbit determination (OD) by utilizing measurement data. However, ML models like Artificial Neural Networks (ANNs) are limited to problems with abundant data and are often considered "black boxes", as their predictions lack interpretability in a scientifically meaningful way. To address these challenges, Raissi et al. 2018 introduced Physics Informed Neural Networks (PINNs), a specialized type of ANN. PINNs integrate the governing differential equations of a system into the learning process, imposing a physical constraint on the network's training and predictions. This approach allows effective training with small datasets, removing the reliance on large amounts of measurements. Additionally, PINNs can estimate unknown or poorly defined parameters within the differential equations, making them conceptually similar to classical OD algorithms like the Weighted Least Squares method. Building on this, Scorsoglio et al. 2023 successfully applied a variant of PINNs, called Physics Informed Extreme Learning Machines (PIELMs), for OD. In this study, a similar approach is employed for OD within the AI4POD (Artificial Intelligence for Precise Orbit Determination) software tool, focusing on resident space objects (RSOs) in low Earth orbit. Following this, we explore various methods, such as output perturbation, to determine the covariance matrix for the PINN-based OD approach. The covariance matrix provides an assessment of uncertainty in the predicted orbit and therefore being an essential tool in real space missions and collision avoidance. These methods are compared for their realism and effectiveness, both against each other and against the covariance matrix results from classical approaches. This study aims to evaluate whether the proposed methods can replicate and potentially improve upon traditional covariance estimation techniques.