The famous three-body problem can be traced back to Newton in 1687, but quite few families of periodic orbits were found in 300 years thereafter. As proved by Poincar\`{e}, the first integral does not exist for three-body systems, which implies that numerical approach had to be used in general. In this paper, we propose an effective approach and roadmap to numerically gain planar periodic orbits of three-body systems with arbitrary masses by means of machine learning based on an artificial neural network (ANN) model. Given any a known periodic orbit as a starting point, this approach can provide more and more periodic orbits (of the same family name) with variable masses, while the mass domain having periodic orbits becomes larger and larger, and the ANN model becomes wiser and wiser. Finally we have an ANN model trained by means of all obtained periodic orbits of the same family, which provides a convenient way to give accurate enough predictions of periodic orbits with arbitrary masses for physicists and astronomers. It suggests that the high-performance computer and artificial intelligence (including machine learning) should be the key to gain periodic orbits of the famous three-body problem.

中文翻译：

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三体问题——从牛顿到超级计算机加机器学习

著名的三体问题可以追溯到 1687 年的牛顿，但在此后的 300 年里发现的周期轨道家族很少。正如 Poincar\`{e} 所证明的，三体系统不存在第一积分，这意味着必须普遍使用数值方法。在本文中，我们提出了一种有效的方法和路线图，通过基于人工神经网络 (ANN) 模型的机器学习，以数值方式获得具有任意质量的三体系统的平面周期轨道。给定任何已知的周期轨道作为起点，这种方法可以提供越来越多的具有可变质量的周期轨道（同名），而具有周期轨道的质量域变得越来越大，ANN模型变得更加明智和更聪明。最后，我们有一个 ANN 模型，通过所有获得的同一族的周期轨道训练，这提供了一种方便的方法，可以为物理学家和天文学家提供具有任意质量的周期轨道的足够准确的预测。这表明高性能计算机和人工智能（包括机器学习）应该是获得著名三体问题周期轨道的关键。