对于 PDE 定义的 PINNs,严格的 a posteriori 错误极限.
IEEE transactions on neural networks and learning systems
|December 1, 2023
概括
我们为物理信息神经网络 (PINN) 预测错误制定了严格的上限. 这个边界只使用关于动态系统的先验信息,而不是真正的解决方案,帮助PDE-governed模型分析.
科学领域:
- 计算数学 计算数学 计算数学
- 机器学习 机器学习
- 科学计算科学计算
背景情况:
- 预测错误量化在神经网络 (NN) 研究中经常被忽视.
- 现有的NN方法,无论是数据驱动的还是基于物理的,都缺乏严格的误差界限.
- 基于物理学的神经网络 (PINNs) 为解决部分微分方程 (PDEs) 提供了一个有前途的方法.
研究的目的:
- 为PINNs的预测错误引入一个严格的a posteriori上限.
- 提供一种不需要了解真实解决方案的错误量化方法.
- 为了证明拟议错误的适用性,绑定到各种PDE-governed系统.
主要方法:
- 为PINN预测错误推导一个理论上限.
- 边界仅依赖于关于由PDE控制的动态系统的先验信息.
- 在基准PDE问题上应用和验证绑定错误.
主要成果:
- 在PINN预测错误上提出了一个新的,可计算的上限.
- 错误边界被证明在各种 PDE 中是有效的,包括运输,热量,纳维埃-斯托克斯方程和克莱因-戈登方程.
- 该方法量化不确定性而不需要基本真相解决方案.
结论:
- 拟议的后续错误限制在PINN的可靠应用方面取得了重大进展.
- 这项工作通过提供可靠的错误量化来解决PINNs的方法研究中的一个关键差距.
- 开发的技术提高了PINN模型在科学应用中的可靠性和可解释性.
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