标题： RAMS：基于残差的对抗梯度移动样本方法用于求解偏微分方程的科学机器学习 显示英文标题

标题： RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations

摘要： 物理信息神经网络（PINNs）和神经算子，两种领先的科学机器学习（SciML）范式，已成为求解偏微分方程（PDEs）的强大工具。 尽管增加训练样本数量通常可以提高网络性能，但也增加了物理信息或数据驱动训练的计算成本。 为解决这一权衡，已开发了不同的采样策略，以在PDE残差较高的区域采样更多点。 然而，现有的采样方法对于高维问题（如高维PDE或算子学习任务）计算成本较高。 在此，我们提出了一种基于残差的对抗梯度移动样本（RAMS）方法，该方法根据对抗梯度方向移动样本，通过基于梯度的优化最大化PDE残差。 RAMS可以轻松集成到现有的采样方法中。 进行了广泛的实验，从PINN应用于高维PDE到物理信息和数据驱动的算子学习问题，以证明RAMS的有效性。 值得注意的是，RAMS代表了算子学习的第一个高效自适应采样方法，在SciML领域标志着重要的进展。 显示英文摘要

摘要： 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.