精确保守的物理信息的神经网络和深度操作员网络用于动态系统
Elsa Cardoso-Bihlo1, Alex Bihlo1
1Department of Mathematics and Statistics Memorial University of Newfoundland St. John's, NL, A1C 5S7, Canada.
概括
我们开发了一种基于投影的新方法,用于训练动态系统的保守物理信息的神经网络. 这种方法显著提高了解决者的准确性和在现实世界问题的表现.
科学领域:
- 计算数学是指计算数学.
- 应用数学 应用数学 应用数学
- 科学机器学习科学机器学习
背景情况:
- 动态系统和普通微分方程 (ODEs) 是科学中的基础.
- 基于物理学的神经网络 (PINNs) 为解决ODE提供了一个强大的工具.
- 在神经网络解决器中确保保存规律对于准确性至关重要.
研究的目的:
- 引入一种用于训练完全保守的PINN和物理知情深度操作员网络 (IPDON) 的新方法.
- 为了提高神经网络解决器的准确性和稳定性,用于动态系统.
- 为了证明保守的解决方法比非保守的解决方法的优越性.
主要方法:
- 采用基于投影的技术,将候选解决方案映射到不变的多元体上.
- 该方法确保神经网络解决方案严格遵守系统的第一个整数.
- 该方法适用于具有至少一个第一个积分的动态系统.
主要成果:
- 与非保守方法相比,准确保守的PINNs和IPDON表现出极其优异的性能.
- 投影技术在训练过程中有效地执行了保护法则.
- 在几个现实世界的数学问题中观察到精度和稳定性的显著改善.
结论:
- 提出的基于投影的方法为开发高精度的保守神经网络解决方案提供了一个强大的框架.
- 这一进步对于各种科学领域的动态系统的可靠模拟至关重要.
- 这些发现强调了将物理保存定律纳入用于科学计算的神经网络架构的重要性.
相关概念视频
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