Search papers, labs, and topics across Lattice.
The paper introduces the Hierarchical Analysis-of-Variance Random Feature Method (HA-RFM), which enhances random-feature methods for solving high-dimensional elliptic PDEs by leveraging lower-dimensional structures identified through closed Sobol indices and fitted-predictor gradients. This method couples selected features in a single regularized least-squares solve, achieving significant error reductions in random-ridge tests, with improvements of factors between 14 to 100 compared to traditional approaches. The authors establish theoretical guarantees for the method's performance, demonstrating its effectiveness in both structural recovery and computational efficiency across high dimensions.
HA-RFM achieves up to 100 times error reduction in high-dimensional elliptic PDEs by intelligently selecting features based on their structural relevance.
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.
