@Article{EAJAM-13-3,
author = {Jing-Run, Chen and Weinan, E and Yi-Xin, Luo},
title = {The Random Feature Method for Time-Dependent Problems},
journal = {East Asian Journal on Applied Mathematics},
year = {2023},
volume = {13},
number = {3},
pages = {435--463},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {2079-7370},
doi = {https://doi.org/10.4208/eajam.2023-065.050423},
url = {https://global-sci.com/article/82435/the-random-feature-method-for-time-dependent-problems}
}