  We present two effective methods for solving high-dimensional partial
differential equations (PDE) based on randomized neural networks. Motivated by
the universal approximation property of this type of networks, both methods
extend the extreme learning machine (ELM) approach from low to high dimensions.
With the first method the unknown solution field in $d$ dimensions is
represented by a randomized feed-forward neural network, in which the
hidden-layer parameters are randomly assigned and fixed while the output-layer
parameters are trained. The PDE and the boundary/initial conditions, as well as
the continuity conditions (for the local variant of the method), are enforced
on a set of random interior/boundary collocation points. The resultant linear
or nonlinear algebraic system, through its least squares solution, provides the
trained values for the network parameters. With the second method the
high-dimensional PDE problem is reformulated through a constrained expression
based on an Approximate variant of the Theory of Functional Connections
(A-TFC), which avoids the exponential growth in the number of terms of TFC as
the dimension increases. The free field function in the A-TFC constrained
expression is represented by a randomized neural network and is trained by a
procedure analogous to the first method. We present ample numerical simulations
for a number of high-dimensional linear/nonlinear stationary/dynamic PDEs to
demonstrate their performance. These methods can produce accurate solutions to
high-dimensional PDEs, in particular with their errors reaching levels not far
from the machine accuracy for relatively lower dimensions. Compared with the
physics-informed neural network (PINN) method, the current method is both
cost-effective and more accurate for high-dimensional PDEs.
