发表机构
School of Science and Technology, Hellenic Open University; School of Pedagogical and Technological Education (ASPETE)（希腊开放大学科学与技术学院; 雅典教育与技术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究引力三体问题中周期轨道的发现，利用物理信息神经网络（PINNs）在无初始条件的稀疏噪声观测上训练，通过二阶ODE公式等方法恢复周期轨道，实验表明其能发现训练数据外的轨道族，且训练数据影响恢复轨道的分布。
AI 中文摘要
定位混沌动力系统的周期解通常需要初始猜测足够接近目标轨道，以便数值延拓或基于梯度的搜索收敛。我们表明，在无初始条件的稀疏、噪声观测上训练的物理信息神经网络（PINNs）能恢复引力三体问题的周期轨道，包括训练数据中不存在的轨道族。该方法基于二阶常微分方程公式、固定频率傅里叶特征、基于百分位数的自适应细化和可训练缩放参数，每个都在前向问题上得到验证。在两个100种子集上，23%-25%的运行收敛到训练数据中不存在的族。然后我们探究是什么决定了出现哪个族。两个卡方检验给出了一致答案：改变训练数据源会显著改变恢复族的分布（p<0.001，克莱默V=0.339），而在测试的两种初始化分布之间切换则不会（p=0.620，V=0.094）。随机种子决定给定运行恢复哪个族；权重所取自的分布不会改变总体频率，但训练数据会。实验证明恢复的轨道是可验证的而非仅仅是似是而非的：识别出的轨道细化为真正的周期解，在拉格朗日数据上训练的网络恢复了八字形编排（李 - 廖I.A.1类，在T*上与七位有效数字匹配），在八字形数据上训练的一个网络恢复了接近δT<10^-9的布劳克 - 哈吉德梅特里乌 - 亨农轨道。
英文摘要
Locating periodic solutions of chaotic dynamical systems normally requires an initial guess close enough to the target orbit for numerical continuation or gradient-based search to converge. We show that Physics-Informed Neural Networks (PINNs) trained on sparse, noisy observations \emph{without} initial conditions recover periodic orbits of the gravitational three-body problem, including orbit families absent from the training data. The method rests on a second-order ODE formulation, fixed-frequency Fourier features, percentile-based adaptive refinement, and a trainable scaling parameter, each validated on forward problems. Across two 100-seed ensembles, $23$--$25\%$ of runs converge to families not present in the training data. We then ask what determines which family emerges. Two $χ^2$ tests give a consistent answer: changing the training data source significantly shifts the distribution of recovered families ($p < 0.001$, Cramér's $V = 0.339$), whereas switching between the two initialization distributions tested does not ($p = 0.620$, $V = 0.094$). The random seed selects which family a given run recovers; the \emph{distribution} the weights are drawn from does not shift the aggregate frequencies, but the training data does. The evidence is empirical: we do not characterize the loss landscape analytically, and PINNs remain slower than conventional integrators on well-posed initial-value problems. What the experiments establish is that the recovered orbits are verifiable rather than merely plausible: the identified ones refine to genuine periodic solutions, a network trained on Lagrange data recovers the figure-eight choreography (Li--Liao class I.A.1, matched to seven significant digits in $T^*$), and one trained on figure-eight data recovers a Broucke--Hadjidemetriou--Hénon orbit closing to $δ_T < 10^{-9}$.
Comments39 pages, 16 figures, 16 tables
