Partial Differential Equations Numerical Modeling Using Dynamic Neural Networks

In this paper a strategy based on differential neural networks (DNN) for the identification of the parameters in a mathematical model described by partial differential equations is proposed. The identification problem is reduced to finding an exact expression for the weights dynamics using the DNNs properties. The adaptive laws for weights ensure the convergence of the DNN trajectories to the PDE states. To investigate the qualitative behavior of the suggested methodology, here the non parametric modeling problem for a distributed parameter plant is analyzed: the anaerobic digestion system

[1]  Nam Mai-Duy,et al.  Numerical solution of differential equations using multiquadric radial basis function networks , 2001, Neural Networks.

[2]  Neil E. Cotter,et al.  The Stone-Weierstrass theorem and its application to neural networks , 1990, IEEE Trans. Neural Networks.

[3]  George Cybenko,et al.  Approximation by superpositions of a sigmoidal function , 1992, Math. Control. Signals Syst..

[4]  Alexander S. Poznyak,et al.  Differential Neural Networks for Robust Nonlinear Control: Identification, State Estimation and Trajectory Tracking , 2001 .

[5]  Mohamad H. Hassoun.  Guest Editorial Neural Networks Council Awards , 1996, IEEE Trans. Neural Networks.

[6]  G. Smith,et al.  Numerical Solution of Partial Differential Equations: Finite Difference Methods , 1978 .

[7]  Alexander S. Poznyak.  Deterministic output noise effects in sliding mode observation , 2004 .

[8]  Simon Haykin,et al.  Neural Networks: A Comprehensive Foundation , 1998 .

[9]  Frank L. Lewis,et al.  Multilayer neural-net robot controller with guaranteed tracking performance , 1996, IEEE Trans. Neural Networks.

[10]  J. Z. Zhu,et al.  The finite element method , 1977 .

[11]  Konrad Reif,et al.  Multilayer neural networks for solving a class of partial differential equations , 2000, Neural Networks.

[12]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[13]  Bernard Delyon,et al.  Accuracy analysis for wavelet approximations , 1995, IEEE Trans. Neural Networks.

[14]  Dimitrios I. Fotiadis,et al.  Artificial neural networks for solving ordinary and partial differential equations , 1997, IEEE Trans. Neural Networks.

[15]  N. Phan-Thien,et al.  Neural-network-based approximations for solving partial differential equations , 1994 .