用李对称群提升物理信息神经网络求解微分方程的精度与效率。
Enhancing PINN Performance Through Lie Symmetry Group
- 将李对称群的无穷小生成元引入PINN，改进求解机制。
- 三种案例均显示性能逐步提升，优于当前主流数值方法。
- 适合研究科学计算与深度学习交叉问题的研究者参考。
本文将物理信息神经网络（PINNs）与李对称群相结合，以提升求解偏微分方程（PDEs）的准确性和效率。李对称群是一种高效方法，可为具有李对称性的PDEs提供精确解。本文创新性地在PINN中引入李对称群的无穷小生成元，显著提升了PDE求解效果。研究共分析三种不同情形，每种均通过李对称修正与自适应技术实现性能递进提升。采用当前最先进的数值方法进行对比验证。数值实验表明，李对称在增强PINN性能中起关键作用，凸显将抽象数学概念融入深度学习以应对复杂科学问题的重要性。
原文摘要 · Abstract (English)
This paper presents intersection of Physics informed neural networks (PINNs) and Lie symmetry group to enhance the accuracy and efficiency of solving partial differential equation (PDEs). Various methods have been developed to solve these equations. A Lie group is an efficient method that can lead to exact solutions for the PDEs that possessing Lie Symmetry. Leveraging the concept of infinitesimal generators from Lie symmetry group in a novel manner within PINN leads to significant improvements in solution of PDEs. In this study three distinct cases are discussed, each showing progressive improvements achieved through Lie symmetry modifications and adaptive techniques. State-of-the-art numerical methods are adopted for comparing the progressive PINN models. Numerical experiments demonstrate the key role of Lie symmetry in enhancing PINNs performance, emphasizing the importance of integrating abstract mathematical concepts into deep learning for addressing complex scientific problems adequately.
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