基于梯度统计的多目标优化在基于物理的神经网络中
Sai Karthikeya Vemuri1,2, Joachim Denzler1
1Computer Vision Group, Friedrich Schiller University Jena, 07743 Jena, Germany.
Sensors (Basel, Switzerland)
|November 14, 2023
概括
物理信息神经网络 (PINNs) 由于多个损失条款,可以在训练中扎. 引入了基于渐变统计的高级权重方案,以平衡培训并提高解决微分方程的准确性.
科学领域:
- 计算物理 计算物理
- 机器学习 机器学习
- 数字分析 数字分析
背景情况:
- 复杂非线性系统的建模在科学和工程方面至关重要.
- 神经网络擅长从数据中学习,但通常需要大型数据集.
- 物理信息神经网络 (PINNs) 将域知识 (数学模型) 与神经网络集成,以克服数据限制.
研究的目的:
- 为了应对在PINNs中优化多目标损失函数的挑战.
- 提出和评估基于高级梯度统计的权重方案,以改善PINN培训.
- 提高PINNs在解决微分方程中的准确性和可靠性.
主要方法:
- 开发了基于梯度统计的PINNs的新型权重方案 (kurtosis标准偏差,平均标准偏差).
- 利用反向传播的梯度统计数据来动态缩放和加重个体损失条款.
- 应用了提出的方法来解决二维波桑方程和克莱恩-戈登方程.
主要成果:
- 拟议的基于梯度统计的权重方案有效平衡PINN中的损失条款.
- 这些先进的方案导致了比标准方法更稳定的培训和更高的准确性.
- 在测试的微分方程的近似解决方案方面取得显著的改进.
结论:
- 基于高级梯度统计的权重方案在克服PINN的优化挑战方面是有效的.
- 引入的库尔托斯标准偏差和综合平均标准偏差方案为PDE近似提供了强大的解决方案.
- 这些发现增强了PINNs对复杂科学和工程问题的适用性.
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