Local Feature Filtering for Scalable and Well-Conditioned Domain-Decomposed Random Feature Methods

TL;DR

This paper introduces a block RRQR filtering and preconditioning strategy for Random Feature Methods, significantly improving the conditioning and efficiency of PDE solvers using localized neural network basis functions.

Contribution

The work presents a novel RRQR-based filtering and preconditioning approach that reduces redundancy, improves conditioning, and accelerates convergence in domain-decomposed RFM PDE solvers.

Findings

Condition numbers reduced by up to eleven orders of magnitude

LSQR convergence improved by factors of 10-1000

Achieved higher accuracy with lower computational cost

Abstract

Random Feature Methods (RFMs) and their variants such as extreme learning machine finite-basis physics-informed neural networks (ELM-FBPINNs) offer a scalable approach for solving partial differential equations (PDEs) by using localized, overlapping and randomly initialized neural network basis functions to approximate the PDE solution and training them to minimize PDE residuals through solving structured least-squares problems. This combination leverages the approximation power of randomized neural networks and the parallelism of domain decomposition. However, the resulting least-squares systems are often severely ill-conditioned, due to local redundancy among random basis functions, which significantly affects the convergence of standard solvers. In this work, we introduce a block rank-revealing QR (RRQR) filtering and preconditioning strategy that operates directly on the structured least-squares problem. First, local RRQR factorizations identify and remove redundant basis functions while preserving numerically informative ones, reducing problem size, and improving conditioning. Second, we use these factorizations to construct a right preconditioner for the global problem which preserves block-sparsity and numerical stability. Third, we derive deterministic bounds of the condition number of the preconditioned system, with probabilistic refinements for small overlaps. We validate our approach on challenging, multi-scale PDE problems in 1D, 2D, and (2+1)D, demonstrating reductions in condition numbers by up to eleven orders of magnitude, LSQR convergence speedups by factors of 10-1000, and higher accuracy than both unpreconditioned and additive Schwarz-preconditioned baselines, all at significantly lower memory and computational cost. These results establish RRQR-based preconditioning as a scalable, accurate, and efficient enhancement for RFM-based PDE solvers.