Development and optimization of physics-informed neural networks for solving partial differential equations

TL;DR

This paper compares Finite Difference Method and Physics-Informed Neural Networks for solving PDEs, analyzing their accuracy, advantages, and limitations across different problem scenarios, including forward and inverse problems.

Contribution

It provides a comparative analysis of FDM and PINNs, highlighting their respective strengths, limitations, and suitable application contexts for solving PDEs.

Findings

FDM achieves higher accuracy with lower relative error.

PINNs can handle inverse problems and source term prediction.

Visualization reveals different performance characteristics of FDM and PINNs.

Abstract

This work compares the advantages and limitations of the Finite Difference Method with Physics-Informed Neural Networks, showing where each can best be applied for different problem scenarios. Analysis on the L2 relative error based on one-dimensional and two-dimensional Poisson equations suggests that FDM gives far more accurate results with a relative error of 7.26 x 10-8 and 2.21 x 10-4, respectively, in comparison with PINNs, with an error of 5.63 x 10-6 and 6.01 x 10-3 accordingly. Besides forward problems, PINN is realized also for forward-inverse problems which reflect its ability to predict source term after its sufficient training. Visualization of the solution underlines different methodologies adopted by FDM and PINNs, yielding useful insights into their performance and applicability.