An artificial neural network‐based method for the optimal control problem governed by the fractional parabolic equation

Abstract

In this paper, we propose an artificial neural network model (ANN) to solve a partial differential equation (PDE) constrained optimization problem. Here, the discretize then optimize approach is used. At first, the Legendre polynomials are used to discretize the optimization problem and transform it into a quadratic optimization problem with linear constraint. Then an ANN model is proposed to solve the obtained quadratic optimization problem. Finally, several examples are presented to illustrate the abilities and efficiency of the proposed approach.

First-Optimize-Then-Discretize Strategy for the Parabolic PDE Constrained Optimization Problem With Application to the Reheating Furnace

“…On the one hand, the FDTO approach is that we discretize the PDE and the cost functional first and then obtain the first order optimality conditions or Karush-Kuhn-Tucker (KKT) conditions. For instance, the FDTO approach combined with an artificial neural network model is proposed to solve a PDE constrained optimization problem in the paper [6]. On the other hand, the FOTD approach needs to derive the infinite-dimensional first order conditions and then select an appropriate discretization method.…”