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.

翻译：生物学、物理学和工程学中的各种现象都以不同方程式为模型。 这些差异方程式, 包括部分差异方程式和普通差异方程式, 可以转换为整体方程式, 并被作为整体方程式表示。 特别是, Volterra Fredholm Hammerstein 集成方程式是这些整体方程式的主要类型, 研究人员感兴趣的是调查和解决这些方程式。 在本文中, 我们提议Tlulesre Deep神经网络( LDNNN) 用于解决非线性方程式Volterra Fredholm Hammerstein 集成方程式( VFHIEs ) 。 LDNN 使用传说或直观多边多边方程式作为深层结构的激活功能。 我们展示了如何将LDNN 用于解决非线性方程式中的 VFHIE 。 我们用高斯四面方程式和LDNN 组合组合法来演示非线性方程式的新型数字解决方案。 以几个实例来验证 LDN的性能和准确性。