Physics-informed neural networks (PINNs) have emerged as powerful tools for solving a wide range of partial differential equations (PDEs). However, despite their user-friendly interface and broad applicability, PINNs encounter challenges in accurately resolving PDEs, especially when dealing with singular cases that may lead to unsatisfactory local minima. To address these challenges and improve solution accuracy, we propose an innovative approach called Annealed Adaptive Importance Sampling (AAIS) for computing the discretized PDE residuals of the cost functions, inspired by the Expectation Maximization algorithm used in finite mixtures to mimic target density. Our objective is to approximate discretized PDE residuals by strategically sampling additional points in regions with elevated residuals, thus enhancing the effectiveness and accuracy of PINNs. Implemented together with a straightforward resampling strategy within PINNs, our AAIS algorithm demonstrates significant improvements in efficiency across a range of tested PDEs, even with limited training datasets. Moreover, our proposed AAIS-PINN method shows promising capabilities in solving high-dimensional singular PDEs. The adaptive sampling framework introduced here can be integrated into various PINN frameworks.

翻译：物理信息神经网络（PINNs）已成为求解各类偏微分方程（PDEs）的有力工具。然而，尽管其用户界面友好且适用范围广泛，PINNs在精确求解PDEs时仍面临挑战，特别是在处理可能导致不理想局部极小值的奇异情形时。为应对这些挑战并提高求解精度，我们提出了一种创新方法——退火自适应重要性采样（AAIS），用于计算代价函数中离散化PDE残差，该方法受有限混合模型中用于模拟目标密度的期望最大化算法启发。我们的目标是通过在残差较高的区域策略性地采样额外点来近似离散化PDE残差，从而提升PINNs的有效性和准确性。将AAIS算法与PINNs内一种简单的重采样策略结合实施，我们的AAIS算法在测试的多种PDEs上展示了显著的效率提升，即使训练数据集有限的情况下也是如此。此外，我们提出的AAIS-PINN方法在求解高维奇异PDEs方面展现了良好的能力。本文介绍的自适应采样框架可整合到各类PINN框架中。