发表机构
School of Mathematics and Statistics, Xidian University; National University of Singapore（西安电子科技大学数学与统计学学院; 新加坡国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对高维椭圆型偏微分方程，提出分层方差分析随机特征方法（HA-RFM），通过选择坐标块、识别低秩特征等步骤，建立误差界并推导相关保证，在随机岭测试等方面有显著效果，扩展了半线性计算维度并描绘了坐标族。
AI 中文摘要
随机特征方法将高维椭圆型偏微分方程的配置问题简化为线性系数问题，但全维试验空间忽略了低维结构。我们引入了分层方差分析随机特征方法（HA-RFM），它利用偏微分方程残差的封闭Sobol指标选择坐标块，从拟合预测器梯度中识别倾斜低秩特征，并在一个正则化最小二乘求解中耦合所有保留的特征。在结构和稳定性假设下，我们建立了一个\(L^2\)误差界，将解和残差截断与有限宽度逼近和正则化有限样本拟合联系起来，并推导了宽度和结构恢复的保证。在固定交互阶数下，所得宽度在维度上是多项式的，在均匀结构控制下具有与维度无关的高阶贡献。残差筛选实现了规定三对支撑的精确恢复，而拟合预测器梯度在维度50以内恢复倾斜方向。在随机岭测试中，额外宽度小于\(1\%\)时，误差比坐标块减少了\(14\) - \(39\)倍，比等宽全维RFM减少了\(34\) - \(100\)倍。半线性计算将HA-RFM扩展到维度100，而密集和分布式交互描绘了更广泛结构所需的坐标族。
英文摘要
Random-feature methods reduce high-dimensional elliptic PDE collocation to linear coefficient problems, but full-dimensional trial spaces overlook lower-dimensional structure. We introduce the Hierarchical Analysis-of-Variance Random Feature Method (HA-RFM), which selects coordinate blocks using closed Sobol indices of the PDE residual, identifies oblique low-rank features from fitted-predictor gradients, and couples all retained features in one regularized least-squares solve. Under structural and stability hypotheses, we establish an $L^2$ error bound that links solution and residual truncation to finite-width approximation and regularized finite-sample fitting, and we derive guarantees for width and structure recovery. The resulting width is polynomial in the dimension at fixed interaction order, with dimension-independent higher-order contributions under uniform structural control. Residual screening achieves exact recovery of the prescribed three-pair support, while fitted-predictor gradients recover oblique directions through dimension $50$. In random-ridge tests, less than $1\%$ additional width reduces errors by factors of $14$-$39$ over coordinate blocks and $34$-$100$ over equal-width full-dimensional RFM. Semilinear computations extend HA-RFM through dimension $100$, while dense and distributed interactions delineate the coordinate families required for broader structure.
