  Various phenomena in biology, physics, and engineering are modeled by
differential equations. These differential equations including partial
differential equations and ordinary differential equations can be converted and
represented as integral equations. In particular, Volterra Fredholm Hammerstein
integral equations are the main type of these integral equations and
researchers are interested in investigating and solving these equations. In
this paper, we propose Legendre Deep Neural Network (LDNN) for solving
nonlinear Volterra Fredholm Hammerstein integral equations (VFHIEs). LDNN
utilizes Legendre orthogonal polynomials as activation functions of the Deep
structure. We present how LDNN can be used to solve nonlinear VFHIEs. We show
using the Gaussian quadrature collocation method in combination with LDNN
results in a novel numerical solution for nonlinear VFHIEs. Several examples
are given to verify the performance and accuracy of LDNN.
