(324e) Parameter Estimation of Partial Differential Equations Using Artificial Neural Network

A wide range of real physical systems in many areas of applied science and engineering belong to distributed parameter systems (DPS) and their pertinent mathematical models often take the form of partial differential equations (PDEs) describing the spatial-temporal dynamics of the system. In order to develop a high-fidelity spatial-temporal mathematical model, an efficient and reliable approach for parameter estimation of PDE systems is required to obtain accurate parameter values with fast convergence rates that takes into account the domains with irregular boundaries so as to be capable of dealing with any arbitrarily complex geometrical shape. While previous contributions in the inverse problems of estimating unknown parameters for PDE systems have investigated extensively the parameter estimation properties, such as accuracy and computing time (Coca and Billings, 2000; Muller and Timmer, 2004); these contributions discussed cases where methods mainly considered functions over a uniform grid discretisation, thus PDE models with irregular boundaries, which widely exist in real-world phenomena, were largely ignored.

This work aims at developing a novel meshless parameter estimation framework for a system of partial differential equations using Artificial Neural Network (ANN) approximations. Since the approximation capabilities of the feedforward neural networks have been widely acknowledged (Hornik et al., 1989; Lagaris et al., 1998, 2000; Leshno et al., 1993), and the ANN-based methodology for parameter estimation was successfully examined for ordinary differential equation (ODE) systems (Dua, 2011; Dua and Dua, 2012), it is therefore of interest to consider this meshless scheme as a candidate for the estimation of parameters in partial differential equations. One of the main advantages of this method is that the ANN-based formulation offers a meshless framework to consider arbitrarily complex boundaries. The developed methodology is able to deal with linear and nonlinear PDEs, with Dirichlet and Neumann boundary conditions, considering both regular and irregular boundaries. This work focuses on testing the applicability of neural networks for estimating the process model parameters while simultaneously computing the model predictions of the state variables in the system of PDEs representing the process. The capability of the proposed methodology is demonstrated with different numerical problems, showing that the ANN-based approach is very efficient by providing accurate solutions in reasonable computing times.

References:

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Dua, V. (2011) An Artificial Neural Network approximation based decomposition approach for parameter estimation of system of ordinary differential equations. Computers & Chemical Engineering, 35(3), 545-553.

Dua, V. & Dua, P. (2012) A simultaneous approach for parameter estimation of a system of ordinary differential equations, using artificial neural network approximation. Industrial & Engineering Chemistry Research, 51(4), 1809-1814.

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Lagaris, I. E., Likas, A. C. & Papageorgiou, D. G. (2000) Neural-network methods for boundary value problems with irregular boundaries. IEEE Transactions on Neural Networks, 11(5), 1041-1049.

Leshno, M., Lin, V. Y., Pinkus, A. & Schocken, S. (1993) Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Neural Networks, 6(6), 861-867.

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