提出一种高效融合全局与局部采样的新方法,提升物理信息神经网络求解精度与速度。
Efficient Global-Local Fusion Sampling for Physics-Informed Neural Networks
- 根据残差自适应生成采样点,重点覆盖难解区域
- 在10个基准PDE上测试,精度和效率均优于传统采样策略
- 适合求解复杂高维偏微分方程的科研与工程场景
物理信息神经网络(PINNs)的精度高度依赖于配点点的位置选择,因偏微分方程损失通过解域采样近似。全局采样虽能保证稳定性,但需大量样本且计算成本高;局部采样聚焦高残差区域以提升效率,但可能忽略已充分学习区域,降低鲁棒性。本文提出全局-局部融合(GLF)采样策略,通过将训练点加高斯噪声并按残差反比缩放,使采样集中在困难区域的同时保持探索性。为进一步降低计算开销,引入轻量级线性代理模型近似全局残差分布,在极低代价下实现相近效果。两者结合——残差自适应采样与残差驱动近似——在保持全局方法稳定性的同时,继承局部优化的高效性。大量实验在标准偏微分方程(benchmark PDEs)上验证,相比全局与局部采样,GLF持续提升准确率与效率。本研究为复杂高维偏微分方程求解提供了可扩展、实用的增强框架。
原文摘要 · Abstract (English)
The accuracy of Physics-Informed Neural Networks (PINNs) critically depends on the placement of collocation points, as the PDE loss is approximated through sampling over the solution domain. Global sampling ensures stability by covering the entire domain but requires many samples and is computationally expensive, whereas local sampling improves efficiency by focusing on high-residual regions but may neglect well-learned areas, reducing robustness. We propose a Global-Local Fusion (GLF) Sampling Strategy that combines the strengths of both approaches. Specifically, new collocation points are generated by perturbing training points with Gaussian noise scaled inversely to the residual, thereby concentrating samples in difficult regions while preserving exploration. To further reduce computational overhead, a lightweight linear surrogate is introduced to approximate the global residual-based distribution, achieving similar effectiveness at a fraction of the cost. Together, these components, residual-adaptive sampling and residual-based approximation, preserve the stability of global methods while retaining the efficiency of local refinement. Extensive experiments on benchmark PDEs demonstrate that GLF consistently improves both accuracy and efficiency compared with global and local sampling strategies. This study provides a practical and scalable framework for enhancing the reliability and efficiency of PINNs in solving complex and high-dimensional PDEs.
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