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.

通过引入名为 thermoNET 的新型神经网络架构，我们能够以较少的可微分计算来表示卫星轨道传播中的热层密度。该网络结构通过 NeuralODE（神经常微分方程）控制卫星动力学，实现完全可微分的变分方程的推导，并能够与先进的数值技术（如基于泰勒级数的数值传播和微分代数技术）进行连接，从而实现网络参数的高效训练。通过纯回归任务和基于观察到的动态的 ODE 敏感性的自适应学习，我们得到了一个灵活而紧凑的热层密度模型，极大地提高了数值传播的效率，并在轨道预测中保持了准确性。

用于 VLEO 模拟的神经 ODE：介绍 thermoNET 对热层建模

NeuralODEs for VLEO simulations: Introducing thermoNET for Thermosphere  Modeling

The high computational cost associated with solving for detailed chemistry
poses a significant challenge for predictive computational fluid dynamics (CFD)
simulations of turbulent reacting flows. These models often require solving a
system of coupled stiff ordinary differential equations (ODEs). While deep
learning techniques have been experimented with to develop faster surrogate
models, they often fail to integrate reliably with CFD solvers. This
instability arises because deep learning methods optimize for training error
without ensuring compatibility with ODE solvers, leading to accumulation of
errors over time. Recently, NeuralODE-based techniques have offered a promising
solution by effectively modeling chemical kinetics. In this study, we extend
the NeuralODE framework for stiff chemical kinetics by incorporating mass
conservation constraints directly into the loss function during training. This
ensures that the total mass and the elemental mass are conserved, a critical
requirement for reliable downstream integration with CFD solvers. Our results
demonstrate that this enhancement not only improves the physical consistency
with respect to mass conservation criteria but also ensures better robustness
and makes the training process more computationally efficient.

本研究扩展了神经 ODE 框架，通过在训练期间直接将质量守恒约束纳入损失函数，实现了对刚性化学动力学的建模，在与 CFD 求解器的下游集成中确保了总质量和元素质量的守恒，提高了物理一致性、稳定性和计算效率。