AI 中文总结
该研究针对等几何分析提出基于残差最小化的r自适应神经算法，利用物理信息神经网络的强形式残差加权得到能量误差估计器，通过反向模式自动微分获网格梯度，实验表明此方法能集中自由度提升精度。
AI 中文摘要
我们提出了一种基于残差最小化的等几何分析（IGA）的r自适应神经算法。使用标准协调伽辽金公式求解边值问题，同时神经网络重新定位内部节点。物理信息神经网络（PINNs）意义下的强形式残差控制比能量（H^1）误差更强的范数。我们根据经典后验理论对其加权：由局部网格尺寸缩放的单元残差、界面通量跳跃和诺伊曼边界残差产生能量误差的可计算估计器，我们对节点进行最小化。对于允许网格族上的强制问题，该估计器在存在振荡项的情况下是可靠且局部有效的；在此范围之外，相同的损失仍然定义良好，并将可微r自适应扩展到不定和对流主导问题。在参数设置中，网络在单次评估中将每个参数映射到节点密度函数；由于它输出的是密度而不是节点位置的固定维向量，一个训练好的网络可以在任何细化级别生成可允许的网格。通过离散解方程式的反向模式自动微分获得网格梯度。一维和二维的数值实验表明，该方法将自由度集中在奇点、材料界面和边界层附近，在固定自由度数量的情况下提高了精度。
英文摘要
We propose an r-adaptive neural algorithm for Isogeometric Analysis (IGA) based on residual minimization. The boundary-value problem is solved using a standard conforming Galerkin formulation, while a neural network relocates the interior knots. A strong-form residual in the sense of physics-informed neural networks (PINNs) controls a norm stronger than the energy (H^1) error. We therefore weight it by classical a posteriori theory: element residuals scaled by the local mesh size, interface flux jumps, and Neumann boundary residuals yield a computable estimator of the energy error, which we minimize with respect to the knots. For coercive problems on admissible mesh families, this estimator is reliable and locally efficient up to oscillation terms; beyond that regime, the same loss remains well-defined and extends differentiable r-adaptivity to indefinite and advection-dominated problems. In the parametric setting, the network maps each parameter to a knot-density function in a single evaluation; since it outputs a density rather than a fixed-dimensional vector of knot locations, one trained network produces an admissible mesh at any refinement level. Mesh gradients are obtained by reverse-mode automatic differentiation through the discrete solution equation. Numerical experiments in one and two dimensions illustrate that the method concentrates degrees of freedom near singularities, material interfaces, and boundary layers, improving accuracy for a fixed number of degrees of freedom.
Comments38 pages, 21 figures, 5 tables
