微分欧拉：设计一个神经网络逼近器解决混沌三体问题
Differential Euler: Designing a Neural Network approximator to solve the Chaotic Three Body Problem
摘要：
三体问题是n体问题的一个特例，其中一个需要三个点质量的初始位置和速度，并试图根据牛顿运动定律和万有引力来预测它们随时间的运动。虽然已经找到了特殊情况下的解析解，但一般问题仍未解决;存在的解决方案是不切实际的。幸运的是，对于许多应用程序，我们可能不需要完全解决问题，即，对于某些时间步长以合理的精度进行预测可能就足够了。最近，Breen等人试图用一个简单的神经网络来近似求解三体问题。虽然他们的方法似乎在减少计算开销方面取得了一些成功，但他们的模型非常有限，适用于专门的2D情况。作者没有解释他们在实验设计中所做的关键决定，也没有详细介绍他们的模型或架构，也没有公布他们的代码。此外，该模型不能很好地推广到看不见的情况。在本文中，我们提出了一个详细的实验装置，以确定使用神经网络来解决三体问题的可行性，达到一定数量的时间步长。我们在数据集大小上建立了一个基准，并设置了一个准确度阈值来衡量我们的结果在实际应用中的可行性。然后，我们使用标准数值积分器生成的数据集，根据列出的NN类构建我们的模型。我们逐渐增加数据集的复杂性，以确定NN是否可以很好地学习混沌三体问题的表示，以取代现实生活中的数值积分器。
英文摘要：
The three body problem is a special case of the n body problem where one takes the initial positions and velocities of three point masses and attempts to predict their motion over time according to Newtonian laws of motion and universal gravitation. Though analytical solutions have been found for special cases, the general problem remains unsolved; the solutions that do exist are impractical. Fortunately, for many applications, we may not need to solve the problem completely, i.e., predicting with reasonable accuracy for some time steps, may be sufficient. Recently, Breen et al attempted to approximately solve the three body problem using a simple neural network. Although their methods appear to achieve some success in reducing the computational overhead, their model is extremely restricted, applying to a specialized 2D case. The authors do not provide explanations for critical decisions taken in their experimental design, no details on their model or architecture, and nor do they publish their code. Moreover, the model does not generalize well to unseen cases. In this paper, we propose a detailed experimental setup to determine the feasibility of using neural networks to solve the three body problem up to a certain number of time steps. We establish a benchmark on the dataset size and set an accuracy threshold to measure the viability of our results for practical applications. Then, we build our models according to the listed class of NNs using a dataset generated from standard numerical integrators. We gradually increase the complexity of our data set to determine whether NNs can learn a representation of the chaotic three body problem well enough to replace numerical integrators in real life scenarios.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
