Spectral connvergece of random feature method in one dimension

TL;DR

This paper analyzes the spectral convergence properties of the Random Feature Method (RFM) for solving 1D elliptic PDEs, showing spectral convergence under certain function space conditions and the impact of PUM on convergence and matrix properties.

Contribution

It provides the first spectral convergence analysis of RFM for 1D elliptic equations and demonstrates how PUM improves convergence and matrix stability.

Findings

RFM exhibits spectral convergence for solutions in Gevrey or Sobolev spaces.

Incorporating PUM enhances convergence rates and mitigates singular value decay.

Singular values of the RFM matrix decay exponentially, with condition numbers increasing exponentially.

Abstract

Among the various machine learning methods solving partial differential equations, the Random Feature Method (RFM) stands out due to its accuracy and efficiency. In this paper, we demonstrate that the approximation error of RFM exhibits spectral convergence when it is applied to the second-order elliptic equations in one dimension, provided that the solution belongs to Gevrey classes or Sobolev spaces. We highlight the significant impact of incorporating the Partition of Unity Method (PUM) to enhance the convergence of RFM by establishing the convergence rate in terms of the maximum patch size. Furthermore, we reveal that the singular values of the random feature matrix (RFMtx) decay exponentially, while its condition number increases exponentially as the number of the features grows. We also theoretically illustrate that PUM may mitigate the excessive decay of the singular values of RFMtx.