论文标题
动态系统神经网络集成商的确切保护法
Exact conservation laws for neural network integrators of dynamical systems
论文作者
论文摘要
最近，与神经网络的时间相关微分方程的解决方案最近引起了很多关注。核心思想是学习控制解决方案从数据演变的法律，这些法律可能会被随机噪声污染。但是，与其他机器学习应用相比，通常对手头的系统知之甚少。例如，对于许多动态系统，诸如能量或（角度）动量之类的物理量是完全保守的。因此，神经网络必须从数据中学习这些保护法，并且仅由于有限的训练时间和随机噪声而被满足。在本文中，我们提出了一种替代方法，该方法使用Noether的定理将保护定律本质地纳入神经网络的架构。我们证明，这可以更好地预测三个模型系统：在三维牛顿引力电位中非忠实粒子的运动，这是Schwarzschild指标中庞大的相对论粒子的运动，以及在四个维度中的两个相互作用粒子的系统。
The solution of time dependent differential equations with neural networks has attracted a lot of attention recently. The central idea is to learn the laws that govern the evolution of the solution from data, which might be polluted with random noise. However, in contrast to other machine learning applications, usually a lot is known about the system at hand. For example, for many dynamical systems physical quantities such as energy or (angular) momentum are exactly conserved. Hence, the neural network has to learn these conservation laws from data and they will only be satisfied approximately due to finite training time and random noise. In this paper we present an alternative approach which uses Noether's Theorem to inherently incorporate conservation laws into the architecture of the neural network. We demonstrate that this leads to better predictions for three model systems: the motion of a non-relativistic particle in a three-dimensional Newtonian gravitational potential, the motion of a massive relativistic particle in the Schwarzschild metric and a system of two interacting particles in four dimensions.
