Investigation of the Robustness of Neural Density Fields
Authors:   Advanced Concepts Team, European Space Agency, European Space Research and Technology Centre (ESTEC), Noordwijk, Netherlands, Schuhmacher Jonas , Gratl Fabio , Izzo Dario , Gómez Pablo 
Published: January 2024

Recent advances in modeling density distributions, so-called neural density fields [1], can accurately describe the density distribution of even irregular celestial bodies, such as asteroids and comets. This representation has several advantages as it relies on no prior information, converges even inside the Brillouin sphere, and is extensible to heterogeneous density distributions of the celestial bodies [1,2]. The accurate knowledge about the density representation of a body is of crucial interest in several fields. For example, spacecraft operations in close proximity to these bodies require this knowledge to be capable of designing safe and efficient trajectories [3]. However, there are open questions in terms of the robustness of this approach with regard to several factors. Previous work utilized a synthetic gravity signal generated with a mascon model as ground truth [1] or measured data from the OSIRIS-REx mission for the asteroid Bennu [4] to train these neural density fields. Further, the influence of other perturbations (e.g. solar radiation pressure), noise and a potentially weak gravitational signal depending on distance warrant further study.
Here, we perform a detailed study of these factors, further investigating the capabilities and robustness of neural density fields. As mascon models discretize the body into point masses, their application is accompanied by the appearance of inaccuracy, especially close to the surface inside the Brillouin sphere. The present work resolves this by using the polyhedral gravity model using the line integral approach capable of analytically computing the gravity tensor given a polyhedral mesh assuming homogeneous density [5, 6]. Herewith, this work presents the obtainable precision of whether a mascon or polyhedral ground truth is used for training the neural network and if the polyhedral model can eliminate errors in the surface region, given its smoother analytical form.
Further, we explore the robustness against noise when training the neural network. Noise can appear due to multiple sources like numerical errors of the numerical integration of the gravitational triple integral, measurement errors or due to limited sampling of the volume given spacecraft trajectories. Second, it can result from non-gravitational accelerations like the Yarkovsky effect or solar radiation pressure (suspected to have affected the measurements of [4]). Hence, we investigate the effects of noise in the measurements, adding systematic and random noise to the training data in order to present its impact when compared to training without it. Results are presented in terms of absolute and relative errors but also visually explored in representations of the neural density field and acceleration errors.
Another central question is how sparse or weak the gravitational signal can be. This is especially a problem during the approach to the body where – due to distance – the gravitational signal is close to a spherical/ point mass, and any irregularities are more challenging to find. Here, we present different sampling strategies given different distance thresholds and we study the resulting precision while also taking propagated orbits for the sampling process into account. This provides an essential insight into whether neural density fields could be employed in a specific mission scenario or whether the gravitational signal is insufficient.
For detailed and robust results, we compare these aspects on several celestial bodies, including 67P Churyumov-Gerasimenko, 433 Eros, 101955 Bennu and 25143 Itokawa.
Overall, we thus take the next step on bringing neural density fields to an onboard mission scenario, where they can be a useful and potent tool complementing existing approaches such as polyhedral or mascon models. All data and code used for the study are available online.
References:
[1] Izzo, D. and Gómez, P., 2021. Geodesy of irregular small bodies via neural density fields: geodesyNets. arXiv preprint arXiv:2105.13031.
[2] Cui, Pingyuan, and Dong Qiao. "The present status and prospects in the research of orbital dynamics and control near small celestial bodies." Theoretical and Applied Mechanics Letters 4.1 (2014): 013013.
[3] Leonard, J.M., Geeraert, J.L., Page, B.R., French, A.S., Antreasian, P.G., Adam, C.D., Wibben, D.R., Moreau, M.C. and Lauretta, D.S., 2019, August. OSIRIS-REx orbit determination performance during the navigation campaign. In 2019 AAS/AIAA Astrodynamics Specialist Conference (pp. 1-20).
[4] von Looz, M., Gómez, P. and Izzo, D., 2021. Study of the asteroid Bennu using geodesyANNs and Osiris-Rex data. arXiv preprint arXiv:2109.14427.
[5] Tsoulis, D., 2012. Analytical computation of the full gravity tensor of a homogeneous arbitrarily shaped polyhedral source using line integrals. Geophysics, 77(2), pp.F1-F11.
[6] Tsoulis, D. and Gavriilidou, G., 2021. A computational review of the line integral analytical formulation of the polyhedral gravity signal. Geophysical Prospecting, vol. 69, no. 8-9, pp. 1745–1760.