  We present a method for solving linear and nonlinear PDEs based on the
variable projection (VarPro) framework and artificial neural networks (ANN).
For linear PDEs, enforcing the boundary/initial value problem on the
collocation points leads to a separable nonlinear least squares problem about
the network coefficients. We reformulate this problem by the VarPro approach to
eliminate the linear output-layer coefficients, leading to a reduced problem
about the hidden-layer coefficients only. The reduced problem is solved first
by the nonlinear least squares method to determine the hidden-layer
coefficients, and then the output-layer coefficients are computed by the linear
least squares method. For nonlinear PDEs, enforcing the boundary/initial value
problem on the collocation points leads to a nonlinear least squares problem
that is not separable, which precludes the VarPro strategy for such problems.
To enable the VarPro approach for nonlinear PDEs, we first linearize the
problem with a Newton iteration, using a particular form of linearization. The
linearized system is solved by the VarPro framework together with ANNs. Upon
convergence of the Newton iteration, the network coefficients provide the
representation of the solution field to the original nonlinear problem. We
present ample numerical examples with linear and nonlinear PDEs to demonstrate
the performance of the method herein. For smooth field solutions, the errors of
the current method decrease exponentially as the number of collocation points
or the number of output-layer coefficients increases. We compare the current
method with the ELM method from a previous work. Under identical conditions and
network configurations, the current method exhibits an accuracy significantly
superior to the ELM method.
