发表机构
Northwest University; Institute of Photonics & Photon Technology（西北大学; 光子与光子技术研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对PINNs在非线性问题中收敛慢和捕捉局部结构不足的问题，提出3GAS-PINNs，结合均匀与高斯平滑分布进行自适应采样，在三个基准方程上最高提升精度14倍。
AI 中文摘要
物理信息神经网络（PINNs）提供了一种无网格框架用于求解偏微分方程，然而其在非线性问题中的性能常常受到收敛速度慢、梯度不平衡以及分辨率不足的限制，难以捕捉诸如激波等局部间歇结构[1]。这些问题主要源于使用固定的损失权重和均匀的配点分布，无法在训练过程中适应解场不断变化的复杂性。为解决这些挑战，本文提出了梯度引导高斯自适应采样物理信息神经网络（3GAS-PINNs），该方法结合均匀概率分布和由解的梯度导出的高斯平滑概率分布，以保持全局约束满足，同时将配点集中在高梯度区域。因此，可以准确捕捉激波和孤子等间歇性结构。该方法在三个基准非线性问题上进行了评估，包括一维受迫Burgers方程、Korteweg-de Vries（KdV）方程和非线性薛定谔方程，这些问题均表现出陡峭梯度或强非线性。与基线PINNs相比，3GAS-PINNs能有效促进间歇区域的物理一致性。数值模拟的精度最高可提升14倍。
英文摘要
Physics-informed neural networks (PINNs) provide a mesh-free framework for solving partial differential equations, yet their performance in nonlinear problems is often limited by slow convergence, gradient imbalance, and insufficient resolution to capture localized intermittent structures such as shock waves[1]. These issues arise primarily from the use of fixed weights of loss and uniform collocation point distributions, which cannot adapt to the evolving complexity of the solution field during training. To address these challenges, Gradient-Guided Gaussian Adaptive Sampling Physics-Informed Neural Networks (3GAS-PINNs) is proposed in this paper, which combines uniform probability distribution and Gaussian-smoothed probability distribution derived from the spatial gradients of solution, to maintain global constraint satisfaction as well as concentrating collocation points in regions of high gradient. Thus, intermittency structures like shock wave and solitons can be accurately captured. The method is evaluated on three benchmark nonlinear problems, including one-dimensional forced Burgers equation, Korteweg-de Vries (KdV) equation and nonlinear Schrodinger equation, all of which exhibit steep gradients or strong nonlinearity. In comparison with baseline PINNs, 3GAS-PINNs can effectively promote the physical consistency in intermittent regions. The accuracy of the numerical simulation can be improved by a factor of up to 14.
