Weak Random Feature Method for Solving Partial Differential Equations

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.