当物理学失败时的食谱:恢复物理信息的神经网络的强大学习
Chandrajit Bajaj1, Luke McLennan1, Timothy Andeen2
1Department of Computer Science & Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, 78712, United States of America.
概括
物理信息神经网络 (PINNs) 可以在训练数据中超出错误. 高斯过程 (GP) 平滑增强了PINN对杂数据的稳定性,并量化了不确定性,改进了部分微分方程的解决方案.
科学领域:
- 计算数学 计算数学 计算数学
- 机器学习 机器学习
背景情况:
- 基于物理学的神经网络 (PINNs) 将物理定律集成到神经网络训练中,用于解决部分微分方程 (PDEs).
- 标准PINN可能对杂的训练数据敏感,导致错误传播和低于最佳的解决方案.
研究的目的:
- 调查PINNs对数据错误的敏感性以及传统物理规范化的局限性.
- 开发一个强大的PINN框架,减轻噪音数据的影响,量化解决方案的不确定性.
主要方法:
- 在PINN培训之前引入基于高斯过程 (GP) 的平滑来提高数据质量.
- 使用稀疏诱导的GP来有效量化边界数据上的不确定性.
- 与依赖时间的施罗丁格方程和市民方程的基准模型进行比较.
主要成果:
- 基于GP的平滑显著提高了PINN的性能和对测量错误的稳定性.
- 物理规范化可能会无意中导致局部最小值的趋同.
- 拟议的方法有效量化不确定性演变,并实现稳健的性能.
结论:
- 高斯过程平滑提供了一个强大的解决方案,用于在有噪音数据的情况下增强物理信息的神经网络.
- 开发的技术为解决不确定或错误测量的PDEs提供了可靠的方法.
- 这种方法促进了PINNs在科学计算中的实际应用.
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