提出高效自适应采样法，提升求解偏微分方程的精度与效率。
RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations
- 基于残差梯度方向动态迁移样本点，聚焦高误差区域。
- 在高维偏微分方程与算子学习任务中显著提升精度。
- 首个适用于算子学习的高效自适应采样方法，适合科学计算研究者。
物理信息神经网络（PINNs）和神经算子是求解偏微分方程（PDEs）的两类主流科学机器学习（SciML）范式。尽管增加训练样本量通常能提升模型性能，但也会带来更高的计算成本。为缓解这一权衡，已有采样策略通过在高残差区域集中采样来优化。然而，现有方法在高维问题（如高维PDE或算子学习任务）中计算开销大。本文提出残差驱动的对抗梯度移动样本方法（RAMS），通过梯度优化沿对抗梯度方向移动样本点，以最大化PDE残差。RAMS可无缝集成至现有采样框架。大量实验覆盖从高维PDE的PINN到物理信息与数据驱动的算子学习任务，验证了其有效性。值得注意的是，RAMS是首个适用于算子学习的高效自适应采样方法，在科学机器学习领域具有重要意义。
原文摘要 · Abstract (English)
Physics-informed neural networks (PINNs) and neural operators, two leading scientific machine learning (SciML) paradigms, have emerged as powerful tools for solving partial differential equations (PDEs). Although increasing the training sample size generally enhances network performance, it also increases computational costs for physics-informed or data-driven training. To address this trade-off, different sampling strategies have been developed to sample more points in regions with high PDE residuals. However, existing sampling methods are computationally demanding for high-dimensional problems, such as high-dimensional PDEs or operator learning tasks. Here, we propose a residual-based adversarial-gradient moving sample (RAMS) method, which moves samples according to the adversarial gradient direction to maximize the PDE residual via gradient-based optimization. RAMS can be easily integrated into existing sampling methods. Extensive experiments, ranging from PINN applied to high-dimensional PDEs to physics-informed and data-driven operator learning problems, have been conducted to demonstrate the effectiveness of RAMS. Notably, RAMS represents the first efficient adaptive sampling approach for operator learning, marking a significant advancement in the SciML field.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。