复杂和合方程的多基准适应性采样物理信息的神经网络
Yabin Zhang1, Liang-Jian Deng1, Minyu Feng2
1School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China.
Chaos (Woodbury, N.Y.)
|October 24, 2025
概括
本研究介绍了一种多基准自适应采样物理信息神经网络 (MBAS-PINN),用于解决复杂方程. 该方法提高了部分微分方程 (PDEs) 的精度和趋同.
科学领域:
- 计算数学 计算数学 计算数学
- 应用物理 应用物理
- 机器学习 机器学习
背景情况:
- 解决复杂值和合部分微分方程 (PDEs) 提出了重要的计算挑战.
- 物理信息神经网络 (PINNs) 通过将物理定律整合到神经网络训练中,提供了一个有前途的方法.
- 现有的PINN方法可以在复杂的方程系统中与收和准确性作斗争.
研究的目的:
- 开发一种先进的PINN方法,能够有效地解决复杂值和合的PDEs.
- 为了提高基于神经网络的PDE解决器的准确性和融合速度.
- 引入智能自适应采样策略,以提高解决方案的可靠性.
主要方法:
- 引入多基准适应性采样物理信息神经网络 (MBAS-PINN) 框架.
- 实施适应性抽样策略,根据多个基准指标动态调整剩余点分布.
- 开发用于复杂PDEs的PINNs的神经触点内核.
- 使用两种不同的培训策略,以优化关注关键解决方案区域 (真实和虚构部分).
主要成果:
- MBAS-PINN方法在复杂的PDEs的准确性和趋同性方面取得了显著的改进.
- 对非线性施罗丁格方程,希罗塔方程和亚吉玛-奥卡瓦系统的实验验证证证了该方法的有效性.
- 适应性采样策略成功引导神经网络优先考虑解决方案准确性至关重要的区域.
结论:
- MBAS-PINN方法提供了一种新且有效的方法来解决复杂值和合PDEs.
- 这种技术增强了PINNs在解决具有挑战性的数学物理问题的能力.
- 适应性采样策略是改善物理信息神经网络性能的一个关键进步.
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