标题： 近奇解偏微分方程的自适应特征捕捉方法 显示英文标题

标题： Adaptive feature capture method for solving partial differential equations with near singular solutions

摘要： 偏微分方程（PDEs）具有接近奇异解的情况对传统数值方法构成了重大挑战，特别是在复杂几何中，网格生成和自适应细化变得计算成本高昂。 尽管基于深度学习的方法，如物理信息神经网络（PINNs）和随机特征方法（RFM），提供了无网格的替代方案，但它们在关键区域往往缺乏自适应分辨率，限制了其在具有陡峭梯度或奇异性解中的准确性。 在本工作中，我们提出了自适应特征捕捉方法（AFCM），这是一种新的机器学习框架，能够自适应地重新分配高梯度区域中的神经元和配点，以增强局部表达能力。 受自适应移动网格技术的启发，AFCM采用近似解的梯度模作为监控函数，以指导特征函数参数的重新初始化。 这确保了分割超平面和配点在最需要的地方聚集，实现了更高的分辨率而不会增加计算开销。 AFCM扩展了RFM的能力，使其能够处理接近奇异解的PDEs，同时保留其无网格效率。 数值实验表明该方法在准确解决接近奇异问题上的有效性，即使在复杂几何中也是如此。 通过弥合自适应网格细化与随机神经网络之间的差距，AFCM为科学和工程应用中解决具有挑战性的PDEs提供了一种强大且可扩展的方法。 显示英文摘要

摘要： Partial differential equations (PDEs) with near singular solutions pose significant challenges for traditional numerical methods, particularly in complex geometries where mesh generation and adaptive refinement become computationally expensive. While deep-learning-based approaches, such as Physics-Informed Neural Networks (PINNs) and the Random Feature Method (RFM), offer mesh-free alternatives, they often lack adaptive resolution in critical regions, limiting their accuracy for solutions with steep gradients or singularities. In this work, we propose the Adaptive Feature Capture Method (AFCM), a novel machine learning framework that adaptively redistributes neurons and collocation points in high-gradient regions to enhance local expressive power. Inspired by adaptive moving mesh techniques, AFCM employs the gradient norm of an approximate solution as a monitor function to guide the reinitialization of feature function parameters. This ensures that partition hyperplanes and collocation points cluster where they are most needed, achieving higher resolution without increasing computational overhead. The AFCM extends the capabilities of RFM to handle PDEs with near-singular solutions while preserving its mesh-free efficiency. Numerical experiments demonstrate the method's effectiveness in accurately resolving near-singular problems, even in complex geometries. By bridging the gap between adaptive mesh refinement and randomized neural networks, AFCM offers a robust and scalable approach for solving challenging PDEs in scientific and engineering applications.