Robust identification of uncertain Schrödinger type complex partial differential equations

Schrödinger equation is a well known example of the so-called complex partial differential equations (C-PDE). This paper presents a technique based in the Differential Neural Networks (DNN) methodology to solve the nonparametric identification problem of systems described by C.PDE. In this case, the identification scheme is proposed as the composition of two coupled DNN: the first one is used to approximate the real part of the complex valued equation and the second reproduces the complementary imaginary part. The convergence of the identification is obtained by a modified Lyapunov function in infinite dimensional spaces. The adaptive laws for complex weights ensure the convergence of the DNN trajectories to the sates of the PDE complex-valued. In order to investigate the qualitative behavior of the suggested technique, it is analyzed, as an example, the approximation of Schrödinger equation. The suggested no parametric identifier converge to the trajectories of the uncertain complex systems. This novel methodology that explores the application of the DNN method for the identification of complex PDE has shown its ability to produce a numerical model of an uncertain complex valued system.