通过自适应采样与加权结合，提升PINNs求解PDE的精度与效率。
Self-adaptive weighting and sampling for physics-informed neural networks
- 动态选择解变化剧烈区域的训练点，提升关键区域覆盖。
- 自适应调整各点权重，平衡不同位置的收敛速度。
- 适合数据稀缺或解变化剧烈的复杂PDE求解任务。
物理信息深度学习已成为求解偏微分方程（PDE）的有前景框架。然而，在复杂问题上训练这些模型仍具挑战性，常导致精度和效率受限。本文提出一种混合自适应采样与加权方法，以提升物理信息神经网络（PINNs）的性能。自适应采样组件识别解快速变化的区域，自适应权重组件平衡各训练点的收敛速率。数值实验表明，仅使用自适应采样或仅使用自适应权重均无法在训练点稀疏时持续获得准确预测，因两者侧重解的不同方面，效果依赖具体问题。结合二者后，新框架在多种场景下均显著提升预测精度与训练效率，为用PINNs求解PDE提供更鲁棒的方案。
原文摘要 · Abstract (English)
Physics-informed deep learning has emerged as a promising framework for solving partial differential equations (PDEs). Nevertheless, training these models on complex problems remains challenging, often leading to limited accuracy and efficiency. In this work, we introduce a hybrid adaptive sampling and weighting method to enhance the performance of physics-informed neural networks (PINNs). The adaptive sampling component identifies training points in regions where the solution exhibits rapid variation, while the adaptive weighting component balances the convergence rate across training points. Numerical experiments show that applying only adaptive sampling or only adaptive weighting is insufficient to consistently achieve accurate predictions, particularly when training points are scarce. Since each method emphasizes different aspects of the solution, their effectiveness is problem dependent. By combining both strategies, the proposed framework consistently improves prediction accuracy and training efficiency, offering a more robust approach for solving PDEs with PINNs.
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