  This paper proposes a plane wave activation based neural network (PWNN) for
solving Helmholtz equation, the basic partial differential equation to
represent wave propagation, e.g. acoustic wave, electromagnetic wave, and
seismic wave. Unlike using traditional activation based neural network (TANN)
or $sin$ activation based neural network (SIREN) for solving general partial
differential equations, we instead introduce a complex activation function
$e^{\mathbf{i}{x}}$, the plane wave which is the basic component of the
solution of Helmholtz equation. By a simple derivation, we further find that
PWNN is actually a generalization of the plane wave partition of unity method
(PWPUM) by additionally imposing a learned basis with both amplitude and
direction to better characterize the potential solution. We firstly investigate
our performance on a problem with the solution is an integral of the plane
waves with all known directions. The experiments demonstrate that: PWNN works
much better than TANN and SIREN on varying architectures or the number of
training samples, that means the plane wave activation indeed helps to enhance
the representation ability of neural network toward the solution of Helmholtz
equation; PWNN has competitive performance than PWPUM, e.g. the same
convergence order but less relative error. Furthermore, we focus a more
practical problem, the solution of which only integrate the plane waves with
some unknown directions. We find that PWNN works much better than PWPUM at this
case. Unlike using the plane wave basis with fixed directions in PWPUM, PWNN
can learn a group of optimized plane wave basis which can better predict the
unknown directions of the solution. The proposed approach may provide some new
insights in the aspect of applying deep learning in Helmholtz equation.
