A Structure-Adaptive Random Feature Method for High-Dimensional Elliptic PDEs
For researchers solving high-dimensional PDEs, HA-RFM offers a principled way to exploit hidden low-dimensional structure, significantly improving accuracy over standard random feature methods.
The paper introduces HA-RFM, a random feature method for high-dimensional elliptic PDEs that adapts to lower-dimensional structure via Sobol index screening and oblique low-rank feature recovery. It achieves error reductions of 14-100x over baselines with less than 1% additional width in random-ridge tests.
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.