用多头网络和PCA构建非线性偏微分方程解的低维嵌入空间
Learning embeddings of non-linear PDEs: the Burgers' equation
- 通过多头结构与正交约束，学习解空间的共享低维嵌入
- 在黏性伯格斯方程上，少数潜变量即可捕获主导动态特征
- 结果具物理可解释性，适合研究复杂系统动力学的学者
嵌入能提供复杂函数空间的低维表示，支持高效检索、比较与泛化。本文将该思想推广至物理信息神经网络，提出一种基于多头结构构造非线性偏微分方程解嵌入空间的方法，并利用主成分分析（PCA）提取非退化的信息。我们以黏性伯格斯方程为例，同时求解一组初值条件与不同黏度下的解。共享网络体学习解空间的潜在嵌入，线性头将该嵌入映射到具体解。通过施加头间的正交约束，获得对训练退化鲁棒的主成分分解，具有直接物理意义。结果显示，伯格斯方程的潜成分快速饱和，表明少量潜模态即可捕捉主要动态特征。
原文摘要 · Abstract (English)
Embeddings provide low-dimensional representations that organize complex function spaces and support generalization. They provide a geometric representation that supports efficient retrieval, comparison, and generalization. In this work we generalize the concept to Physics Informed Neural Networks. We present a method to construct solution embedding spaces of nonlinear partial differential equations using a multi-head setup, and extract non-degenerate information from them using principal component analysis (PCA). We test this method by applying it to viscous Burgers' equation, which is solved simultaneously for a family of initial conditions and values of the viscosity. A shared network body learns a latent embedding of the solution space, while linear heads map this embedding to individual realizations. By enforcing orthogonality constraints on the heads, we obtain a principal-component decomposition of the latent space that is robust to training degeneracies and admits a direct physical interpretation. The obtained components for Burgers' equation exhibit rapid saturation, indicating that a small number of latent modes captures the dominant features of the dynamics.
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