Numerical implementation of continuous Hopfield networks for optimization

A novel approach is presented to implement continuous Hop eld neural networks, which are modelled by a system of ordinary di erential equations (ODEs). The simulation of a continuous network in a digital computer implies the discretization of the ODE, which is usually carried out by simply substituting the derivative by the di erence, without any further theoretical justi cation. Instead, the numerical solution of the ODE is proposed. Among the existing numerical methods for ODEs, we have selected the modi ed trapezoidal rule. The Hamiltonian Cycle Problem is used as an illustrative example to compare the novel method to the standard implementation. Simulation results show that this "numerical neural technique" obtains valid solutions of the problem and it is more eÆcient than other simulation algorithms. This technique opens a promising way to optimization neural networks that could be competitive with nonlinear programming methods.