Physics Informed Neural Network using Finite Difference Method

TL;DR

This paper introduces a finite difference method-based physics-informed neural network (PINN) that simplifies implementation and improves computational efficiency while maintaining accuracy, demonstrated through experiments on Laplace's and Burger's equations.

Contribution

It proposes using finite difference methods for PDE loss estimation in PINNs, offering a low-cost, efficient alternative to automatic differentiation.

Findings

FDM-based PINN outperforms non-PINN deep learning models.

FDM PINN is faster than AD-based PINN.

FDM PINN achieves comparable error reduction to AD-based PINN.

Abstract

In recent engineering applications using deep learning, physics-informed neural network (PINN) is a new development as it can exploit the underlying physics of engineering systems. The novelty of PINN lies in the use of partial differential equations (PDE) for the loss function. Most PINNs are implemented using automatic differentiation (AD) for training the PDE loss functions. A lesser well-known study is the use of finite difference method (FDM) as an alternative. Unlike an AD based PINN, an immediate benefit of using a FDM based PINN is low implementation cost. In this paper, we propose the use of finite difference method for estimating the PDE loss functions in PINN. Our work is inspired by computational analysis in electromagnetic systems that traditionally solve Laplace's equation using successive over-relaxation. In the case of Laplace's equation, our PINN approach can be seen as taking the Laplacian filter response of the neural network output as the loss function. Thus, the implementation of PINN can be very simple. In our experiments, we tested PINN on Laplace's equation and Burger's equation. We showed that using FDM, PINN consistently outperforms non-PINN based deep learning. When comparing to AD based PINNs, we showed that our method is faster to compute as well as on par in terms of error reduction.