Wavelets based physics informed neural networks to solve non-linear differential equations

Area of Science:

• Computational fluid dynamics
• Applied mathematics
• Numerical analysis

Background:

• Solving complex differential equations is crucial in science and engineering.
• Traditional numerical methods can be computationally intensive and may face challenges with non-linear problems.
• Physics-informed neural networks (PINNs) offer a novel approach by integrating physical laws into neural network training.

Purpose of the Study:

• To investigate the efficacy of PINNs utilizing wavelets as activation functions for solving differential equations.
• To demonstrate the versatility of this approach across different types of equations, including those in fluid dynamics.
• To analyze the impact of key design factors on the accuracy of the wavelet-based PINN model.

Main Methods:

• Implementation of physics-informed neural networks (PINNs).
• Utilization of wavelets as activation functions within the neural network architecture.
• Application to solve the Blasius viscous flow problem, linear and non-linear coupled differential equations, and partial differential equations.
• Investigation of critical parameters influencing model performance.

Main Results:

• The wavelet-based PINN approach successfully solved various differential equations, including the Blasius viscous flow problem.
• The method demonstrated high accuracy and efficiency when compared to existing techniques.
• Key factors influencing the neural network's accuracy were identified and analyzed.

Conclusions:

• Physics-informed neural networks with wavelets are a viable and effective tool for solving non-linear differential equations.
• The proposed method offers a computationally efficient and accurate alternative for problems in fluid dynamics and beyond.
• Further investigation into model design can optimize performance for specific applications.