李·波松神经网络 (LPNets):以数据为基础的对称的哈密尔顿系统的计算
Christopher Eldred1, François Gay-Balmaz2, Sofiia Huraka3
1Computer Science Research Institute, Sandia National Laboratory, 1450 Innovation Pkwy SE, Albuquerque, NM, 87123, USA.
概括
我们开发了新的神经网络,以保持对哈密尔顿系统准确的长期预测的基本结构. 这些方法通过尊重系统的对称性,确保对复杂的物理现象进行精确的模拟.
科学领域:
- * 计算物理 计算物理
- * 应用数学 * 应用数学
- * 机器学习 * 机器学习
背景情况:
- * 哈密尔顿系统是描述物理现象的基础,包括卫星运动和流体动力学.
- * 准确的长期预测需要计算方法来保持系统内在的数学结构.
- *现有的方法往往因为难以保存这些结构而难以长期准确.
研究的目的:
- *为哈密尔顿系统开发基于数据的预测模型,这些模型精确地保留了它们的基础数学结构.
- * 创建神经网络架构,尊重Lie-Poisson系统的对称性和不变量 (Casimirs).
- * 为了实现复杂的物理动态的高精度,长期模拟.
主要方法:
- * 发展神经网络 (LPNets),学习转换,准确地保持李·波松括号和卡西米尔.
- * 引入G-LPNets,利用转换组合作为构建块,以提高结构保存.
- * 调整方法以处理更广泛的Poisson括号.
- *适用于各种物理系统,如刚体运动和磁场中的粒子动力学.
主要成果:
- * 在学习转换中实现了Poisson支架和卡西米尔的机器精确保存.
- * 证明了LPNets和G-LPNets在准确模拟长期动态方面的有效性.
- *成功地将这些方法应用于各种基准物理系统,验证了它们的稳定性.
结论:
- * 拟议的基于网络的方法为准确的,长期的哈密尔顿系统模拟提供了一个强大的工具.
- * 保持基本的数学结构,特别是对称性,对于可靠的预测建模至关重要.
- *这些方法促进了对复杂物理应用的数据驱动模拟技术的开发.
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