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

arXiv: https://arxiv.org/abs/2607.19786 · HTML: https://mlsanity.com/paper/2607.19786/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 36 references that are in the map.
- First posted on arXiv 22 Jul 2026.

## Builds on (most cited first, 9 of 9)

- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions](https://mlsanity.com/paper/1502.06800/) (2015)
- [A Priori Estimates of the Population Risk for Two-layer Neural Networks](https://mlsanity.com/paper/1810.06397/) (2018)
- [Adaptive Deep Learning for High-Dimensional Hamilton-Jacobi-Bellman Equations](https://mlsanity.com/paper/1907.05317/) (2019)
- [Algorithms for Overcoming the Curse of Dimensionality for Certain Hamilton-Jacobi Equations Arising in Control Theory and Elsewhere](https://mlsanity.com/paper/1605.01799/) (2016)
- [An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensions](https://mlsanity.com/paper/2309.07049/) (2023)

## Built on by (most cited first, 1 of 1)

- [Spectral Convergence of Random Feature Method in Multiple Dimensions](https://mlsanity.com/paper/2609.03401/) (2026)

## Abstract (arXiv)

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.