The Random Feature Method for Time-dependent Problems

TL;DR

This paper introduces a novel space-time random feature method for solving time-dependent PDEs, achieving spectral accuracy and mesh-free computation, with theoretical error bounds and practical demonstrations on complex domains.

Contribution

The paper develops the space-time random feature method (ST-RFM) for PDEs, analyzing two construction strategies and providing theoretical error bounds along with numerical validation.

Findings

ST-RFM achieves spectral accuracy in space and time.

The method is mesh-free and suitable for complex geometries.

Error bounds show sublinear growth in the number of subdomains.

Abstract

We present a framework for solving time-dependent partial differential equations (PDEs) in the spirit of the random feature method. The numerical solution is constructed using a space-time partition of unity and random feature functions. Two different ways of constructing the random feature functions are investigated: feature functions that treat the spatial and temporal variables (STC) on the same footing, or functions that are the product of two random feature functions depending on spatial and temporal variables separately (SoV). Boundary and initial conditions are enforced by penalty terms. We also study two ways of solving the resulting least-squares problem: the problem is solved as a whole or solved using the block time-marching strategy. The former is termed ``the space-time random feature method'' (ST-RFM). Numerical results for a series of problems show that the proposed method, i.e. ST-RFM with STC and ST-RFM with SoV, have spectral accuracy in both space and time. In addition, ST-RFM only requires collocation points, not a mesh. This is important for solving problems with complex geometry. We demonstrate this by using ST-RFM to solve a two-dimensional wave equation over a complex domain. The two strategies differ significantly in terms of the behavior in time. In the case when block time-marching is used, we prove a lower error bound that shows an exponentially growing factor with respect to the number of blocks in time. For ST-RFM, we prove an upper bound with a sublinearly growing factor with respect to the number of subdomains in time. These estimates are also confirmed by numerical results.