Physics-informed neural networks are a powerful deep-learning framework that integrates physical laws to solve partial differential equations, yet achieving fast convergence and high prediction accuracy remains challenging due to the ongoing issue of obtaining high-quality training data. In this study, we introduce a sampling-enhanced framework to unify residual-based sampling methods of PINNs. To address the limitations of existing adaptive sampling methods, we propose MRD (Maximum Residual-based continuous Distribution) to navigate PINNs and move training points towards the high-residual region. It not only adopts a continuous form for more precise residual indication, but also effectively generates a high-quality training dataset. Our method is generic, straightforward, and easily extended to high-dimensional, dynamic and nonlinear PDEs. Experimental results across all five equations demonstrate a significant improvement in relative error, validating the generality and efficacy of our MRD method.

中文翻译：

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通过基于最大残差的连续分布导航 PINN


物理信息神经网络是一个强大的深度学习框架，它集成了物理定律来求解偏微分方程，但由于获得高质量训练数据的持续问题，实现快速收敛和高预测精度仍然具有挑战性。在这项研究中，我们引入了一个采样增强框架来统一基于残差的 PINN 采样方法。为了解决现有自适应采样方法的局限性，我们提出了 MRD （Maximum Residual-based continuous Distribution） 来导航 PINN 并将训练点移动到高残差区域。它不仅采用连续形式进行更精确的残差指示，而且有效地生成了高质量的训练数据集。我们的方法是通用的、直接的，并且很容易扩展到高维、动态和非线性偏微分方程。所有五个方程的实验结果表明，相对误差有显著改善，验证了我们的 MRD 方法的通用性和有效性。