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

2026-07-22

AI Summary

This work addresses the limitations of random feature methods for high-dimensional elliptic partial differential equations, which often fail to exploit underlying low-dimensional structures. The authors propose HA-RFM, a novel approach that uniquely integrates residual-driven Sobol sensitivity analysis with gradient-guided oblique low-rank subspace learning to adaptively identify influential coordinate blocks and their interaction patterns. These components are jointly optimized via regularized least squares. The method establishes a unified theoretical error bound encompassing truncation, width, and sampling errors. In 50-dimensional test cases, HA-RFM accurately recovers prescribed three-pair support structures, achieving error reductions of 14–39 times over coordinate-block baselines and 34–100 times over full-dimensional random feature methods of comparable width. The framework further demonstrates scalability by successfully solving 100-dimensional semilinear PDEs.

Abstract

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.

Problem

• high-dimensional elliptic PDEs
• random feature methods
• low-dimensional structure
• dimensionality reduction
• collocation

Innovation

• random feature method
• high-dimensional PDEs
• Sobol indices
• low-rank structure
• oblique features