用稀疏的内部数据和循环神经网络来识别PDE的参数
Jie Long1, Abdul Khaliq2,3, Khaled M Furati4
1Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN, 37132, USA. Jie.Long@mtsu.edu.
Scientific reports
|September 30, 2025
概括
基于物理学的神经网络 (PINNs) 难以处理稀疏的数据. 这项研究引入了一种新的方法,将Gated Recurrent Units和隐性数值方法结合起来,以提高PINN的性能,有效地识别参数和解决方案,即使内部数据有限.
科学领域:
- 计算科学 计算科学
- 应用数学 应用数学 应用数学
- 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中,用于解决微分方程.
- 由于内部数据稀少,PINNs面临性能恶化,限制了它们在现实世界的场景中的应用.
- 现有的方法往往需要密集的数据,这给有限测量的反向问题带来了挑战.
研究的目的:
- 开发一种新的方法,在内部数据稀疏的场景中克服PINNs的局限性.
- 为了提高参数识别和部分微分方程解决方案检索的准确性和稳定性.
- 证明拟议方法在各种基准问题的有效性.
主要方法:
- 建议采用混合方法,将Gated Recurrent Units (GRUs) 与隐式数值方法相结合.
- 该GRU提供了一个初始的解决方案近似,然后通过一个隐性时间渐进方案来改进.
- 物理约束嵌入时间代,损失函数包含代方案和稀疏数据点的错误.
主要成果:
- 提出的方法成功地识别了未知的参数,并获得了完整的解决方案,即使内部数据稀少.
- 在汉堡方程,艾伦-卡恩方程和非线性施罗丁格方程上的数值实验验证实了算法的有效性.
- 这种方法在内部数据稀缺时,与标准PINN相比,表现更好.
结论:
- 集成GRU和隐式数值方法为科学计算中的数据稀缺反向问题提供了强大的解决方案.
- 这种混合方法显著提高了物理信息学习在复杂的物理系统中的适用性.
- 该方法为推进计算科学中的数值模拟和参数估计提供了一个有希望的方向.
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