A novel auxiliary equation neural networks method for exactly explicit solutions of nonlinear partial differential equations

TL;DR

This paper introduces an innovative neural network-based method, AENNM, that combines auxiliary equations and neural networks to efficiently find exact solutions of nonlinear partial differential equations, enhancing accuracy and broadening application scope.

Contribution

The paper presents a novel auxiliary equation neural networks method integrating Riccati-based activation functions for solving NLPDEs with high precision and efficiency, including new solution types.

Findings

Successfully solved several NLPDEs with exact solutions

Constructed new trial functions within neural network models

Provided detailed visualizations of solutions' dynamic characteristics

Abstract

In this study, we firstly propose an auxiliary equation neural networks method (AENNM), an innovative analytical method that integrates neural networks (NNs) models with the auxiliary equation method to obtain exact solutions of nonlinear partial differential equations (NLPDEs). A key novelty of this method is the introduction of a novel activation function derived from the solutions of the Riccati equation, establishing a new mathematical link between differential equations theory and deep learning. By combining the strong approximation capability of NNs with the high precision of symbolic computation, AENNM significantly enhances computational efficiency and accuracy. To demonstrate the effectiveness of the AENNM in solving NLPDEs, three numerical examples are investigated, including the nonlinear evolution equation, the Korteweg-de Vries-Burgers equation, and the (2+1)-dimensional Boussinesq equation. Furthermore, some new trial functions are constructed by setting specific activation functions within the "2-2-2-1" and "3-2-2-1" NNs models. By embedding the auxiliary equation method into the NNs framework, we derive previously unreported solutions. The exact analytical solutions are expressed in terms of hyperbolic functions, trigonometric functions, and rational functions. Finally, three-dimensional plots, contour plots, and density plots are presented to illustrate the dynamic characteristics of the obtained solutions. This research provides a novel methodological framework for addressing NLPDEs, with broad applicability across scientific and engineering fields.