  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.
