用QR-DEIM动态选择关键点，提升物理信息神经网络求解精度。
Adaptive Collocation Point Strategies For Physics Informed Neural Networks via the QR Discrete Empirical Interpolation Method
- 基于QR-DEIM构建自适应采样策略，聚焦高梯度区域。
- 在典型PDE测试中，相比传统方法精度显著提升。
- 适合需高精度求解复杂偏微分方程的研究者。
物理信息神经网络（PINNs）在求解偏微分方程（PDE）的正向与反向问题中备受关注。尽管损失函数和网络结构的改进提升了精度，但采样点的选择对性能的影响仍缺乏深入研究。固定采样方式（如均匀随机或等距网格）难以捕捉解梯度大的关键区域，限制了其在复杂PDE中的表现。受传统数值方法自适应网格加密启发，现有自适应方法虽能动态更新采样点，但在更新间可能忽略残差动态，导致信息丢失。为此，本文提出两种基于QR离散经验插值法（QR-DEIM）的自适应采样策略，该方法是一种高效近似非线性函数的降阶建模技术。在基准PDE上的实验表明，所提方法相较现有方法显著提升了PINN的准确性，为自适应采样策略提供了新方向。
原文摘要 · Abstract (English)
Physics-informed neural networks (PINNs) have gained significant attention for solving forward and inverse problems related to partial differential equations (PDEs). While advancements in loss functions and network architectures have improved PINN accuracy, the impact of collocation point sampling on their performance remains underexplored. Fixed sampling methods, such as uniform random sampling and equispaced grids, can fail to capture critical regions with high solution gradients, limiting their effectiveness for complex PDEs. Adaptive methods, inspired by adaptive mesh refinement from traditional numerical methods, address this by dynamically updating collocation points during training but may overlook residual dynamics between updates, potentially losing valuable information. To overcome this limitation, we propose two adaptive collocation point selection strategies utilizing the QR Discrete Empirical Interpolation Method (QR-DEIM), a reduced-order modeling technique for efficiently approximating nonlinear functions. Our results on benchmark PDEs demonstrate that our QR-DEIM-based approaches improve PINN accuracy compared to existing methods, offering a promising direction for adaptive collocation point strategies.
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