用于求解高维偏微分方程的PINN中的退火自适应重要性采样方法
Annealed adaptive importance sampling method in PINNs for solving high dimensional partial differential equations
AI总结：
本文提出退火自适应重要性采样（AAIS）方法，通过在残差较高区域策略性采样额外点来改进PINNs求解高维偏微分方程的精度与效率，尤其适用于奇异性问题。
摘要：
物理信息神经网络（PINNs）已成为求解各类偏微分方程（PDEs）的强大工具。然而，尽管PINNs具有用户友好的接口和广泛的适用性，它们在精确求解PDEs方面仍面临挑战，尤其是在处理可能导致不理想局部极小值的奇异情形时。为了解决这些挑战并提高求解精度，我们提出了一种创新方法，称为退火自适应重要性采样（AAIS），用于计算代价函数的离散化PDE残差，其灵感来源于有限混合模型中用于模拟目标密度的期望最大化算法。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，从而提高PINNs的有效性和准确性。我们的AAIS算法与PINNs中一种直接的再采样策略一起实现，在一系列测试的PDEs中展示了效率的显著提升，即使在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方面展现出有前景的能力。这里引入的自适应采样框架可以集成到各种PINN框架中。
英文摘要：
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.
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