A Structure-Adaptive Random Feature Method for High-Dimensional Elliptic PDEs

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 L2 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.