Multilayer Perceptron-Based Solutions of the Riccati Equation Demonstrating the Impact of Activation Functions and Network Design

Abstract

This paper presents a multilayer perceptron (MLP)-based technique for solving the nonlinear Riccati equation, which frequently arises in science and engineering applications. The study emphasizes how activation functions, the number of neurons, and the depth of hidden layers affect solution accuracy and convergence. The network is trained by minimizing the residual error of the governing equation, and its performance is validated using absolute error analysis and statistical methods. To demonstrate the proposed technique, four representative Riccati equations are solved and analyzed in detail. Results show that the choice of activation function significantly influences stability and accuracy. Likewise, the depth of hidden layers and the number of neurons directly impact the quality of approximation. A comparative analysis is also conducted between Gradient Descent and Stochastic Gradient Descent to highlight their effectiveness in solving the problem. The novelty of this work lies in the development of a systematic and comprehensive framework that links neural network design choices to te accurate solution of nonlinear ordinary differential equations. The findings demonstrate the efficiency, flexibility, and scalability of MLPs as powerful alternatives to traditional solvers for nonlinear differential equations, offering practical insights for future applications in science and engineering.