NeuralODEs for VLEO simulations: Introducing thermoNET for Thermosphere Modeling

TL;DR

The paper addresses large uncertainties in thermospheric drag modeling for satellite propagation by introducing thermoNET, a compact, physics-informed neural representation of density with $\rho(h) = \sum_{i=0}^3 \alpha_i \exp(-\beta_i(h-\gamma_i))$, whose coefficients are neural-network outputs. It demonstrates two training paths: regression to ground-truth empirical models (NRLMSISE-00, JB08) and learning from observed dynamics via NeuralODEs, yielding a differentiable, asymptotically well-behaved density capable of fast propagation. By exploiting differentiability, it combines Taylor-based integration and NeuralODE sensitivities to achieve substantial speedups and accurate trajectory decay predictions, including a successful gradient-based update to recover orbital decay from synthetic observations. The work provides practical tools for improved VLEO propagation, re-entry analysis, and debris studies, with openly available code for reproducibility and extension.

Abstract

We introduce a novel neural architecture termed thermoNET, designed to represent thermospheric density in satellite orbital propagation using a reduced amount of differentiable computations. Due to the appearance of a neural network on the right-hand side of the equations of motion, the resulting satellite dynamics is governed by a NeuralODE, a neural Ordinary Differential Equation, characterized by its fully differentiable nature, allowing the derivation of variational equations (hence of the state transition matrix) and facilitating its use in connection to advanced numerical techniques such as Taylor-based numerical propagation and differential algebraic techniques. Efficient training of the network parameters occurs through two distinct approaches. In the first approach, the network undergoes training independently of spacecraft dynamics, engaging in a pure regression task against ground truth models, including JB-08 and NRLMSISE-00. In the second paradigm, network parameters are learned based on observed dynamics, adapting through ODE sensitivities. In both cases, the outcome is a flexible, compact model of the thermosphere density greatly enhancing numerical propagation efficiency while maintaining accuracy in the orbital predictions.

Figures (6)

• Figure 1: Thermonets at a glance. A lightweight artificial neural network transforms geodetic coordinates and space weather data into the predefined air density model. The network parameters are learned via backpropagating the error via two distinct learning pipelines: a) to match some ground truth model (regression learning) or b) to match orbital observations (neural ODE learning).
• Figure 2: Left: Ground truth (NRLMSISE-00), and model predictions fitted assuming a fixed longitude, latitude, and solar weather. Each case corresponds to an independent fit of the coefficients in Eq. \ref{['eq:rho_functional_form']} and the continuous line represents the ground truth, while the overlaid markers represent the fits. Right: relative error introduced by the fits. All interesting cases (above 180km) are mostly well below a 1% error as highlighted by the dotted line.
• Figure 3: Prediction (diagonal) and distribution of the absolute percentage difference between models (off-diagonal) at an altitude of 400 km and on the 22nd of April 2018 at 5:13:35 (GMT). The mean is also indicated. The ability of the neural models to approximate well atmospheric trend is evident as the error associated is one order of magnitude smaller than the difference between models.
• Figure 4: Altitude trends along various numerical propagations of VLEO satellites where the thermospheric density is represented by a) the NRLMSISE-00 empirical model, b) the thermoNET $\mathcal{N}_{NRL00}$, or c) a fit accounting only for altitude dependence. For all practical purposes the thermoNET yields equivalent numerical results.
• Figure 5: The updated thermospheric model learns to match orbital observations using the NeuralODE approach and captures the orbital decay rate accurately.