Exploring the exact solutions to the nonlinear systems with neural networks method
Jan Muhammad1, Ali H Tedjani2, Ejaz Hussain3
1Department of Mathematics, Shanghai University, No. 99 Shangda Road, 200444, Shanghai, China.
Scientific Reports
|October 21, 2025
Summary
This study introduces Riccati subequation neural networks for solving complex biological nonlinear partial differential equations. The method integrates Riccati equation solutions into neural networks, yielding exact analytical solutions for modeling biological systems.
Area of Science:
- Computational Biology
- Applied Mathematics
- Biophysics
Background:
- Nonlinear partial differential equations (PDEs) model complex biological processes.
- Challenges exist in finding exact analytical solutions for these PDEs.
- Spatially varying biochemical signaling and tissue regeneration require advanced modeling techniques.
Purpose of the Study:
- To explore Riccati subequation neural networks for solving nonlinear PDEs in biology.
- To integrate Riccati problem solutions into neural network architectures.
- To derive and verify exact analytical solutions for biological models.
Main Methods:
- Implementing neural networks with the first hidden layer solving the Riccati equation.
- Deriving hyperbolic, trigonometric, and rational function solutions.
- Utilizing graphical illustrations to analyze wave-like solution dynamics.
Main Results:
- Demonstrated the derivation of exact analytical solutions for nonlinear PDEs.
- Successfully applied Riccati subequation neural networks to biological models.
- Visualized and analyzed the dynamic properties of wave solutions.
Conclusions:
- The proposed method effectively solves nonlinear PDEs in biological systems.
- Riccati subequation neural networks offer a powerful tool for understanding complex biological dynamics.
- Confirms the efficacy of the implemented neural network technique for scientific modeling.
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