Symbolic Riccati-Based Exact Solutions and Physics-Informed Neural Network Validation of the (2+1)-Dimensional Chaffee–Infante Equation
TL;DR
A symbolic Riccati sub-equation neural network framework for obtaining exact solutions of the (2 + 1)-dimensional Chaffee–Infante equation is developed, which successfully reproduces the representative kink solution with small numerical errors in the considered parameter regime.
Abstract
Nonlinear partial differential equations require effective analytical and computational techniques for constructing, interpreting, and validating nonlinear wave solutions. In this work, we develop a symbolic Riccati sub-equation neural network framework for obtaining exact solutions of the (2 + 1)-dimensional Chaffee–Infante equation. The framework combines the classical Riccati sub-equation method with a network-inspired trial-function structure, in which network-like weights and biases serve as symbolic algebraic parameters rather than trainable neural-network parameters. The Riccati sub-equation provides nonlinear building blocks in the form of hyperbolic, trigonometric, and rational functions, which are systematically incorporated into the symbolic network structure. The unknown parameters are determined through direct substitution of the trial function into the governing equation followed by symbolic coefficient balancing, yielding fourteen exact solution families, including hyperbolic, trigonometric, rational, kink, dark, and singular structures. In contrast to the symbolic Riccati sub-equation neural network framework, a separate genuinely trained physics-informed neural network is employed for independent numerical validation. The trainable parameters of this neural network are optimized by minimizing a physics-informed loss function constructed from the governing-equation residual and boundary conditions, with derivatives evaluated using automatic differentiation and gradient-based optimization. The analytical solution obtained from the symbolic Riccati framework is used as a reference for evaluating the trained neural network, which successfully reproduces the representative kink solution with small numerical errors in the considered parameter regime. Furthermore, graphical analysis is performed to illustrate the diversity of the obtained nonlinear wave structures, while their modulation characteristics are investigated through modulational instability analysis.