映射多块谱极限学习机用于偏微分方程
Mapped Multi-Patch Spectral Extreme Learning Machine for Partial Differential Equations
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中文总结 AI 辅助
本文提出映射多块谱极限学习机（Spectral-ELM）求解偏微分方程，通过CGL微分矩阵和映射处理弯曲域，实验表明在光滑问题中与直接配点法精度相当，适用于一至三维。
中文摘要 AI 辅助
我们研究了一种固定特征求解器，称为谱极限学习机（Spectral-ELM），其中将切比雪夫-高斯-洛巴托（CGL）微分矩阵应用于极限学习机（ELM）试验函数的节点值。弯曲域通过映射多块公式处理。每个弯曲四边形块表示为参考正方形的像，物理导数由相关的度量项计算，相邻块通过强制解及其法向通量的连续性进行耦合。使用局部QR正交化来减少离散特征之间的近似线性相关性。数值研究包括与直接CGL配点法、解析微分ELM、全局随机特征方法和TransNet采样策略的比较。还包括一个匹配的KdV测试、一个小型三维示例以及一个包含孔的弯曲域上的泊松问题。对于正方形上的光滑椭圆问题，直接CGL配点法给出的误差最小。当应用于相同的固定特征试验空间时，谱微分和解析微分产生的误差相当。对于弯曲域问题，映射直接CGL配点法和映射Spectral-ELM在未知数数量相当的情况下实现了相似的精度。本公式特别适用于一至三维中的光滑偏微分方程，其中张量积网格能够实现精确的高阶离散化，而稀疏网格和维度自适应策略为扩展到更高维问题提供了有前景的方向。
英文摘要
We study a fixed-feature solver, referred to as Spectral-ELM, in which Chebyshev--Gauss--Lobatto (CGL) differentiation matrices are applied to the nodal values of an extreme learning machine (ELM) trial function. Curved domains are treated using a mapped multi-patch formulation. Each curved quadrilateral patch is represented as the image of a reference square, physical derivatives are computed from the associated metric terms, and adjacent patches are coupled by enforcing continuity of the solution and its normal flux. Local QR orthogonalization is used to reduce near-linear dependence among the discrete features. The numerical study includes comparisons with direct CGL collocation, an analytically differentiated ELM, a global Random Feature Method, and a TransNet sampling strategy. It also includes a matched KdV test, a small three-dimensional example, and a Poisson problem on a curved domain containing a hole. For a smooth elliptic problem on a square, direct CGL collocation gives the smallest error. When applied to the same fixed-feature trial space, spectral and analytic differentiation produce comparable errors. For the curved-domain problem, mapped direct CGL collocation and mapped Spectral-ELM achieve similar accuracy with comparable numbers of unknowns. The present formulation is particularly well suited to smooth PDEs in one to three dimensions, where tensor-product grids enable accurate high-order discretizations, while sparse-grid and dimension-adaptive strategies offer promising directions for extension to higher-dimensional problems.
