Weak Random Feature Method for Solving Partial Differential Equations
Mikhail Kuvakin, Zijian Mei, Jingrun Chen
TL;DR
This paper introduces Weak Random Feature Method (WRFM), a mesh-free PDE solver designed to find weak solutions by reformulating the governing equations in weak form and approximating test functions through a finite sine-basis expansion. WRFM retains the linear least-squares structure of standard RFM but replaces strong-form residuals with weak-form integrals, yielding a linear system that directly enforces the weak formulation. Numerical experiments in 2D and 3D, including Helmholtz, static heat, Poisson, and heat equations, show that WRFM can achieve comparable or superior accuracy to PINN and WAN while using significantly fewer parameters and reducing computational resources, particularly on 2D problems; WRFM runs on CPUs, contrasting with GPU-accelerated baselines. A key limitation is the scalability of the test-function set in higher dimensions, which grows the linear system size; the authors propose exploring adversarially generated test functions as future work to mitigate the dimensionality challenge. Overall, WRFM offers a robust, efficient approach for solving PDEs with weak solutions, expanding the applicability of random-feature-based PDE solvers to problems with low regularity.
Abstract
The random feature method (RFM) has demonstrated great potential in bridging traditional numerical methods and machine learning techniques for solving partial differential equations (PDEs). It retains the advantages of mesh-free approaches while achieving spectral accuracy for smooth solutions, without the need for iterative procedures. However, the implementation of RFM in the identification of weak solutions remains a subject of limited comprehension, despite crucial role of weak solutions in addressing numerous applied problems. While the direct application of RFM to problems without strong solutions is fraught with potential challenges, we propose an enhancement to the original random feature method that is specifically suited for finding weak solutions and is termed as Weak RFM. Essentially, Weak RFM reformulates the original RFM by adopting the weak form of the governing equations and constructing a new linear system through the use of carefully designed test functions, ensuring that the resulting solution satisfies the weak form by default. To rigorously evaluate the performance of the proposed method, we conduct extensive experiments on a variety of benchmark problems, including challenging three-dimensional cases, and compare its performance with state of the art machine learning-based approaches. The results demonstrate that Weak RFM achieves comparable or superior accuracy while significantly reducing computational time and memory consumption, highlighting its potential as a highly efficient and robust tool for finding weak solutions to various PDE problems.