通过几何压缩映射提升物理神经网络的求解精度与稳定性
Rethinking Input Domains in Physics-Informed Neural Networks via Geometric Compactification Mappings
- 用可微分的几何压缩映射重构输入坐标，匹配PDE的多尺度结构
- 在1D/2D典型方程上实现更均匀残差分布和更高求解精度
- 适用于有周期边界、远场扩展或局部奇点的复杂物理系统
多个复杂物理系统由具有多尺度特性的偏微分方程（PDE）支配，其包含平滑的低频成分和局域的高频结构。现有物理信息神经网络（PINN）通常采用固定坐标系输入，当几何结构与这些特征不匹配时，会导致梯度僵化和病态条件，阻碍收敛。为此，我们提出一种映射范式，通过可微分的几何压缩映射重塑输入坐标，并将PDE的几何结构与残差算子的谱特性相耦合。基于此，我们构建了几何压缩（GC）-PINN框架，引入三种无需修改原有架构的映射策略：周期边界、远场尺度扩展和局部奇异结构处理。大量实验表明，该方法在代表性1D和2D PDE上实现了更均匀的残差分布、更高的解精度，同时提升了训练稳定性和收敛速度。
原文摘要 · Abstract (English)
Several complex physical systems are governed by multi-scale partial differential equations (PDEs) that exhibit both smooth low-frequency components and localized high-frequency structures. Existing physics-informed neural network (PINN) methods typically train with fixed coordinate system inputs, where geometric misalignment with these structures induces gradient stiffness and ill-conditioning that hinder convergence. To address this issue, we introduce a mapping paradigm that reshapes the input coordinates through differentiable geometric compactification mappings and couples the geometric structure of PDEs with the spectral properties of residual operators. Based on this paradigm, we propose Geometric Compactification (GC)-PINN, a framework that introduces three mapping strategies for periodic boundaries, far-field scale expansion, and localized singular structures in the input domain without modifying the underlying PINN architecture. Extensive empirical evaluation demonstrates that this approach yields more uniform residual distributions and higher solution accuracy on representative 1D and 2D PDEs, while improving training stability and convergence speed.
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