用广义对称性提升物理信息神经算子的训练效率
Generalized Lie Symmetries in Physics-Informed Neural Operators
- 引入演化代表性对称性增强损失函数
- 显著提高训练数据效率与模型精度
- 适合研究偏微分方程求解的算法开发者
物理信息神经算子（PINOs）已成为学习偏微分方程（PDEs）解算子的强大工具。近期研究显示，融入点对称性信息可显著提升PINOs的训练效率，主要通过数据、结构和损失增强实现。本文聚焦于损失增强策略，指出点对称性常导致无训练信号，限制其在多数问题中的有效性。为此，我们提出一种新方法，利用底层PDE的演化代表性对称性——一类广义对称性——作为生成器，提供比标准对称性更丰富的生成集合，从而产生更具信息量的训练信号。实验表明，该策略能有效提升神经算子性能，实现更高的数据效率与训练精度。
原文摘要 · Abstract (English)
Physics-informed neural operators (PINOs) have emerged as powerful tools for learning solution operators of partial differential equations (PDEs). Recent research has demonstrated that incorporating Lie point symmetry information can significantly enhance the training efficiency of PINOs, primarily through techniques like data, architecture, and loss augmentation. In this work, we focus on the latter, highlighting that point symmetries oftentimes result in no training signal, limiting their effectiveness in many problems. To address this, we propose a novel loss augmentation strategy that leverages evolutionary representatives of point symmetries, a specific class of generalized symmetries of the underlying PDE. These generalized symmetries provide a richer set of generators compared to standard symmetries, leading to a more informative training signal. We demonstrate that leveraging evolutionary representatives enhances the performance of neural operators, resulting in improved data efficiency and accuracy during training.
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