FDM data driven U-Net as a 2D Laplace PINN solver.

Area of Science:

• Computational Mathematics
• Machine Learning
• Image Analysis

Background:

• Traditional numerical methods for solving partial differential equations (PDEs), like Finite Difference (FDM) and Finite Element (FEM), are computationally intensive and difficult to adapt for new applications.
• Physically Informed Neural Networks (PINNs) have emerged as a promising alternative, offering simpler application and potentially better performance for solving PDEs.

Purpose of the Study:

• To develop and evaluate a novel data-driven approach for solving the 2D Laplace partial differential equation (PDE) with arbitrary boundary conditions.
• To demonstrate the efficacy of deep learning models, specifically PINNs, trained on Finite Difference Method (FDM) solutions for both forward and inverse PDE problems.

Main Methods:

• A deep learning framework utilizing Physically Informed Neural Networks (PINNs) was developed.
• The PINN models were trained on a comprehensive dataset of reference solutions generated by the Finite Difference Method (FDM).
• The approach was tested on various boundary value problems for the 2D Laplace equation.

Main Results:

• The proposed PINN approach achieved nearly real-time performance for solving the 2D Laplace PDE.
• An average accuracy of 94% was obtained when comparing PINN solutions to FDM results across different boundary conditions.
• Both forward and inverse 2D Laplace problems were efficiently solved using the data-driven deep learning method.

Conclusions:

• The developed deep learning-based PINN solver offers an efficient and accurate tool for addressing the 2D Laplace PDE.
• This approach shows significant potential for applications in image analysis and the computational simulation of physical problems with image-based boundary conditions.