用不平衡学习优化神经网络采样，提升物理信息模型精度与效率
An Imbalanced Learning-based Sampling Method for Physics-informed Neural Networks
- 基于残差识别高误差区域，结合过采样策略自适应调整采样点分布
- 在多维度微分方程测试中，精度不低于现有最优方法，内存消耗显著降低
- 适合计算资源受限的高维偏微分方程求解场景
本文提出RSmote，一种面向物理信息神经网络（PINNs）的新型局部自适应采样方法，通过不平衡学习策略提升模型性能。传统残差自适应采样虽能提高精度，但在高维问题中常面临效率低、内存占用高的挑战。RSmote聚焦高残差区域，引入不平衡学习中的过采样技术优化采样过程。理论分析证明其在资源管理上的有效性。大量实验表明，RSmote在多种维度和微分方程上均达到或超越当前最优的残差自适应分布（RAD）方法，且内存使用显著减少，尤其适用于高维复杂偏微分方程求解，是兼顾精度与资源效率的可靠方案。
原文摘要 · Abstract (English)
This paper introduces Residual-based Smote (RSmote), an innovative local adaptive sampling technique tailored to improve the performance of Physics-Informed Neural Networks (PINNs) through imbalanced learning strategies. Traditional residual-based adaptive sampling methods, while effective in enhancing PINN accuracy, often struggle with efficiency and high memory consumption, particularly in high-dimensional problems. RSmote addresses these challenges by targeting regions with high residuals and employing oversampling techniques from imbalanced learning to refine the sampling process. Our approach is underpinned by a rigorous theoretical analysis that supports the effectiveness of RSmote in managing computational resources more efficiently. Through extensive evaluations, we benchmark RSmote against the state-of-the-art Residual-based Adaptive Distribution (RAD) method across a variety of dimensions and differential equations. The results demonstrate that RSmote not only achieves or exceeds the accuracy of RAD but also significantly reduces memory usage, making it particularly advantageous in high-dimensional scenarios. These contributions position RSmote as a robust and resource-efficient solution for solving complex partial differential equations, especially when computational constraints are a critical consideration.
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