Abstract
This paper presents the state identification study of 3D partial differential equations (PDEs) using the differential neural networks (DNNs) approximation. There are so many physical situations in applied mathematics and engineering that can be described by PDEs; these models possess the disadvantage of having many sources of uncertainties around their mathematical representation. Moreover, to find the exact solutions of those uncertain PDEs is not a trivial task especially if the PDE is described in two or more dimensions. Given the continuous nature and the temporal evolution of these systems, differential neural networks are an attractive option as nonparametric identifiers capable of estimating a 3D distributed model. The adaptive laws for weights ensure the “practical stability” of the DNN trajectories to the parabolic three-dimensional (3D) PDE states. To verify the qualitative behavior of the suggested methodology, here a nonparametric modeling problem for a distributed parameter plant is analyzed.
Highlights
This paper suggests a different numerical solution for uncertain systems based on the Neural Network approach [4]
Given by the vector Partial differential equations (PDEs) (20), to study the quality of the differential neural networks (DNNs) identifier supplied with the adjustment laws (22), estimate the upper bound of the identification error δ given by
This implicates that the reduction of the identification error δ means that the differential neural network has converged to the solution of the 3D PDE; this can be observed in the matching of the DNN to the PDE state
Summary
The main idea behind the application of DNN [11] to approximate the 3D PDEs solution is to use a class of finite-difference methods but for uncertain nonlinear functions. F (x, t) ∈ Rn represents the modelling error term, A and V k (k = 1, 6) any constant matrices and the set of sigmoidal functions have the corresponding size (σ(x, y, z) ∈. In the sum we have that s = 1, 3, φ represents functions σ, φ, γ, ψ, and U can be taken as the corresponding ui,j,k, ui−1,j,k, ui−2, j,k, ui, j−1,k, ui, j−2,k, ui, j,k−1, ui, j,k−2, ui−1, j−1,k, ui, j−1,k−1, ui−1,j,k−1, ui−1,j−1,k−1 In this equation the term f i,j,k(t), which is usually recognized as the modeling error, satisfies the following identify, and here, it has been omitted the dependence to xi, y j, zk of each sigmoidal function:.
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