Jordan Murphy, Slingshot Aerospace; Conner Grey, Slingshot Aerospace; Joshua Horwood, Slingshot Aerospace; Jason Stauch, Slingshot Aerospace; Navraj Singh, Slingshot Aerospace; Timothy Olson, Slingshot Aerospace; Jeffrey Shaddix, Slingshot Aerospace; Belinda Marchand, Slingshot Aerospace
https://doi.org/10.64861/BTAE1611
Keywords: Space Weather, Drag Modeling, Scientific Machine Learning, Compressive Sensing, LEO
Abstract:
Accurate modeling of Earth’s atmosphere remains critical for precise satellite propagation and conjunction analysis in low Earth orbit (LEO). At low altitudes, atmospheric drag is often the dominant propagation error source, and models such as Jacchia-Roberts 1971 (JR71) continue to suffer from systemic biases due to their fixed parameterizations. While the High Accuracy Satellite Drag Model (HASDM) achieves state-of-the-art performance through real-time data assimilation, it remains inaccessible to the broader space community. At AMOS 2025, we introduced a hybrid atmospheric correction framework using Universal Differential Equations (UDEs) combined with compressive sensing, demonstrating significant bias reduction when trained against higher-fidelity density models. This work extends that foundation toward operational deployment by transitioning from idealized density-based supervision to trajectory-based learning using sparse observations from multiple calibration satellites.
The key advancement presented in this paper is the adoption of the Prediction Error Method (PEM) for training the UDE-based atmospheric correction. In our prior work, the correction was optimized by directly comparing densities between the corrected JR1971 and JB2008 models, an approach that assumes access to an oracle density truth. In operational settings, however, no such oracle exists. The PEM framework addresses this limitation by learning corrections directly from observed satellite trajectories. During training, the dynamics include observer feedback terms that inject measurement information into the propagation, keeping predicted states close to observations. The observer gains, which balance model trust against measurement trust, are themselves learned parameters. This formulation produces a well-conditioned optimization landscape that avoids the local minima problems common in trajectory-matching approaches and naturally handles the phase and frequency mismatches that plague pure simulation error minimization.
A second critical extension is the ability to learn a global correction from multiple calibration satellites with sparse, irregularly-spaced observations. The previous methodology required dense observation histories and was demonstrated on single-trajectory fits. The current framework handles batched training across many satellites, each with only a handful of state observations, by exploiting the differential equations formulation which places no restrictions on observation timing. Combining multiple exposed trajectories, the correction naturally identifies only the most physically relevant spatial correction terms while preventing overfitting to any single calibration object. The learned correction thus generalizes across orbital regimes rather than being tuned to specific ephemerides.
The correction architecture also preserves the hybrid design validated in our prior work. A compact spatial expansion captures persistent solar-relative density variations, while a small neural network modulates this spatial pattern based on space weather indices. The compact neural network provides implicit regularization, and the overall design ensures that the correction remains interpretable and computationally efficient. Through the UDE embedding, the correction integrates directly into the drag force computation, enabling the full power of modern numerical solvers and adjoint sensitivity methods for efficient gradient computation.
The experimental results in this paper will demonstrate successful training of the global atmospheric correction using only sparse trajectory observations, validating the operational viability of the approach. We will show propagation accuracy improvements comparable to our density-supervised results while using dramatically less information per calibration satellite. Furthermore, we will present analysis of the learned observer gains and their physical interpretation. Comparisons against both uncorrected JR71 and higher-fidelity baselines will quantify improvements in predicted state accuracy and covariance realism.
This work represents a significant step toward an operationally deployable atmospheric correction framework. By removing the requirement for oracle density access and enabling learning from the types of sparse tracking data available in practice, the methodology becomes applicable to real-world calibration using ground-based observations or publicly available ephemerides. The resulting corrections can improve conjunction assessment, reduce predicted state uncertainty, and enhance filtering and data association performance across space situational awareness applications.
Date of Conference: September 15-18, 2026
Track: Atmospherics/Space Weather