用于带误差界的轨道不确定性传播的物理信息学习
Physics-informed Learning for Orbital Uncertainty Propagation with Error Bounds
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中文总结 AI 辅助
针对轨道动力学中FP-PDE求解的挑战，提出PINN-GMM结合误差PINN的方法，实现带严格误差界的不确定性传播，其前向传播速度快于基准方法，可靠性更高。
中文摘要 AI 辅助
福克-普朗克偏微分方程（FP-PDE）控制随机动力系统中的不确定性演化。在轨道动力学中，求解FP-PDE极具挑战，原因在于非线性运动、高维状态及大时空域。我们提出一种物理信息神经网络（PINN）方法，将FP-PDE的解近似为单一时空概率密度，同时量化其最坏情况近似误差。该方法原则上与状态坐标和神经网络架构的选择无关。具体而言，为将概率密度函数（PDF）特性引入神经网络，我们设计了物理信息高斯混合模型（PINN-GMM）；随后，伴随误差PINN学习近似误差的动力学，并生成随时间变化的界，这些界定义了PDF的模糊集。该模糊集可通过易处理的线性规划严格计算事件概率的上下界。对说明性1D示例及若干4D至6D轨道测试案例的数值研究表明，该方法能实现准确的不确定性传播，得到正确且具信息量的误差界，且相较于常见的不确定性传播基准方法（高斯近似、无迹变换、高斯混合模型）可靠性更高。构建PINN-GMM需要离线训练，因此比基准近似方法成本更高；但训练完成后，单次前向传播即可在亚毫秒级时间内返回任意时刻的密度（我们的实现中为0.16毫秒）。
英文摘要
The Fokker-Planck partial differential equation (FP-PDE) governs uncertainty evolution in stochastic dynamical systems. In orbital dynamics, solving the FP-PDE is challenging because of nonlinear motion, high-dimensional states, and large space-time domains. We develop a physics-informed neural network (PINN) approach that approximates the FP-PDE solution as a single space-time probability density, while also quantifying its worst-case approximation error. This approach is, in principle, independent of the choice of state coordinates and neural network architecture. Specifically, to enforce probability density function (PDF) properties into the neural network, we design a Physics-informed Gaussian mixture model (PINN-GMM). Then a companion error PINN learns the dynamics of the approximation error and yields time-dependent bounds that define an ambiguity set of PDFs. This ambiguity set enables rigorous computation of upper and lower bounds on event probabilities through tractable linear programs. Numerical studies on illustrative 1D examples and several 4D--6D orbital test cases demonstrate accurate uncertainty propagation, correct and informative error bounds, and improved reliability over common uncertainty-propagation baseline methods (Gaussian approximation, unscented transform, and Gaussian mixture model). Constructing the PINN-GMM requires offline training, making it costlier than the baseline approximations; once trained, however, a single forward pass returns the density at any time in sub-millisecond time $(0.16~\mathrm{ms}$ in our implementation).
发表机构
- University of Colorado, Boulder（科罗拉多大学博尔德分校）
机构由 AI 辅助整理，请以论文原文为准。
