Abstract
High-fidelity representations of the gravity field underlie all applications in astrodynamics. Traditionally these gravity models are constructed analytically through a potential function represented in spherical harmonics, mascons, or polyhedrons. Such representations are often convenient for theory, but they come with unique disadvantages in application. Broadly speaking, analytic gravity models are often not compact, requiring thousands or millions of parameters to adequately model high-order features in the environment. In some cases these analytic models can also be operationally limiting—diverging near the surface of a body or requiring assumptions about its mass distribution or density profile. Moreover, these representations can be expensive to regress, requiring large amounts of carefully distributed data which may not be readily available in new environments. To combat these challenges, this paper aims to shift the discussion of gravity field modeling away from purely analytic formulations and toward machine learning representations. Within the past decade there have been substantial advances in the field of deep learning which help bypass some of the limitations inherent to the existing analytic gravity models. Specifically, this paper investigates the use of physics-informed neural networks (PINNs) to represent the gravitational potential of two planetary bodies—the Earth and Moon. PINNs combine the flexibility of deep learning models with centuries of analytic insight to learn new basis functions that are uniquely suited to represent these complex environments. The results show that the learned basis set generated by the PINN gravity model can offer advantages over its analytic counterparts in model compactness and computational efficiency.
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The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
References
Blacker, P., Bridges, CP., Hadfield, S.: Rapid Prototyping of Deep Learning Models on Radiation Hardened CPUs. IEEE, pp 25–32, https://doi.org/10.1109/AHS.2019.000-4, https://ieeexplore.ieee.org/document/8792934/ (2019)
Bottou, L.: Stochastic Gradient Descent Tricks, vol 7700. Springer (2012), https://doi.org/10.1007/978-3-642-35289-8_25, http://link.springer.com/10.1007/978-3-642-35289-8_25
Cheng, L., Wang, Z., Song, Y., Jiang, F.: Real-time optimal control for irregular asteroid landings using deep neural networks. Acta Astronaut. 170, 66–79 (2020), https://doi.org/10.1016/j.actaastro.2019.11.039
Courant, R., Hilbert, D.: Methods of Mathematical Physics. Wiley, New York. https://doi.org/10.1002/9783527617234 (1989)
Furfaro, R., Barocco, R., Linares, R., Topputo, F., Reddy, V., Simo, J., et al.: Modeling irregular small bodies gravity field via extreme learning machines and Bayesian optimization. Adv. Space Res. 67, 617–638 (2021). https://doi.org/10.1016/j.asr.2020.06.021
Gao, A., Liao, W.: Efficient gravity field modeling method for small bodies based on Gaussian process regression. Acta Astronaut. 157, 73–91 (2019). https://doi.org/10.1016/j.actaastro.2018.12.020
Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. J. Mach. Learn. Res. 9, 249–256 (2010)
Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. MIT Press, Cambridge (2016)
Goossens, S., Lemoine, F.G., Sabaka, T.J., Nicholas, J.B., Mazarico, E., Rowlands, D.D., et al.: A Global Degree and Order 1200 Model of the Lunar Gravity Field using GRAIL Mission Data. https://ui.adsabs.harvard.edu/abs/2016LPI....47.1484G (2016)
Gottlieb, D., Shu, C.W.: On the Gibbs phenomenon and its resolution. SIAM Rev. 39, 644–668 (1997). https://doi.org/10.1137/S0036144596301390
Hewitt, E., Hewitt, R.E.: The Gibbs–Wilbraham phenomenon: an episode in Fourier analysis. Arch. Hist. Exact Sci. 21, 129–160 (1979). https://doi.org/10.1007/BF00330404
Huang, G.B., Zhu, Q.Y., Siew, C.K.: Extreme learning machine: theory and applications. Neurocomputing 70, 489–501 (2006). https://doi.org/10.1016/j.neucom.2005.12.126
Kaula, W.M.: Theory of Satellite Geodesy: Applications of Satellites to Geodesy. Blaisdell Publishing Co, New York (1966)
Kingma, DP., Ba, J.: Adam: a method for stochastic optimization. In: 3rd International Conference on Learning Representations, ICLR 2015 - Conference Track Proceedings pp 1–15, http://arxiv.org/abs/1412.6980 (2014)
Koch, K.R., Morrison, F.: A simple layer model of the geopotential from a combination of satellite and gravity data. J. Geophys. Res. 75, 1483–1492 (1970). https://doi.org/10.1029/JB075i008p01483
LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521, 436–444 (2015). https://doi.org/10.1038/nature14539
Lemoine, F.G., Goossens, S., Sabaka, T.J., Nicholas, J.B., Mazarico, E., Rowlands, D.D., et al.: GRGM900C: a degree 900 lunar gravity model from GRAIL primary and extended mission data. Geophys. Res. Lett. 41, 3382–3389 (2014). https://doi.org/10.1002/2014GL060027
Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: 7th International Conference on Learning Representations, ICLR 2019 http://arxiv.org/abs/1711.05101 (2017)
Manzi, M., Vasile, M.: Discovering Unmodeled Components in Astrodynamics with Symbolic Regression. IEEE, pp. 1–7, https://doi.org/10.1109/CEC48606.2020.9185534 (2020)
Martin, J.R., Schaub, H.: GPGPU Implementation of Pines’ Spherical Harmonic Gravity Model. Univelt Inc., Escondido (2020)
Mertikopoulos, P., Papadimitriou, C., Piliouras, G.: Cycles in adversarial regularized learning. Soc. Ind. Appl. Math. (2018). https://doi.org/10.1137/1.9781611975031.172
Pavlis, N.K., Holmes, S.A., Kenyon, S.C., Factor, J.K.: The development and evaluation of the Earth gravitational model 2008 (EGM2008). J. Geophys. Res. Solid Earth (2012). https://doi.org/10.1029/2011JB008916
Pines, S.: Uniform representation of the gravitational potential and its derivatives. AIAA J. 11, 1508–1511 (1973). https://doi.org/10.2514/3.50619
Raissi, M., Perdikaris, P., Karniadakis, G.: Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707 (2019). https://doi.org/10.1016/j.jcp.2018.10.045
Romain, G., Jean-Pierre, B.: Ellipsoidal harmonic expansions of the gravitational potential: theory and application. Celest. Mech. Dyn. Astron. 79, 235–275 (2001). https://doi.org/10.1023/A:1017555515763
Ruder, S.: An Overview of Gradient Descent Optimization Algorithms, pp. 1–14, arXiv arXiv:1609.04747 (2016)
Russell, R.P., Arora, N.: Global point mascon models for simple, accurate, and parallel geopotential computation. J. Guid. Control Dyn. 35, 1568–1581 (2012). https://doi.org/10.2514/1.54533
Swinbank, R., Purser, R.J.: Fibonacci grids: a novel approach to global modelling. Q. J. R. Meteorol. Soc. 132, 1769–1793 (2006). https://doi.org/10.1256/qj.05.227
Takahashi, Y., Scheeres, D.: Morphology driven density distribution estimation for small bodies. Icarus 233, 179–193 (2014), doi: 10.1016/j.icarus.2014.02.004
Tapley, B.D.: Gravity model determination from the GRACE mission. J. Astronaut. Sci. 56, 273–285 (2008). https://doi.org/10.1007/BF03256553
Tardivel, S.: The Limits of the Mascons Approximation of the Homogeneous Polyhedron. Am. Inst. Aeronaut. Astronaut. (2016) https://doi.org/10.2514/6.2016-5261
Wang, S., Teng, Y., Perdikaris, P.: Understanding and mitigating gradient pathologies in physics-informed neural networks, pp. 1–28, arXiv arXiv:2001.04536 (2020)
Werner, R., Scheeres, D.: Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 Castalia. Celest. Mech. Dyn. Astron. 65, 313–344 (1997). https://doi.org/10.1007/BF00053511
Wittick, P.T., Russell, R.P.: Mixed-model gravity representations for small celestial bodies using mascons and spherical harmonics. Celestial Mechanics and Dynamical Astronomy 131, 31 (2019). https://doi.org/10.1007/s10569-019-9904-6
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This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. 2040434.
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Martin, J., Schaub, H. Physics-informed neural networks for gravity field modeling of the Earth and Moon. Celest Mech Dyn Astr 134, 13 (2022). https://doi.org/10.1007/s10569-022-10069-5
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DOI: https://doi.org/10.1007/s10569-022-10069-5