Physics-informed Neural Network: The Effect of Reparameterization in Solving Differential Equations

TL;DR

This paper compares physics-informed neural networks with and without reparameterization, demonstrating that reparameterization reduces approximation error in solving complex differential equations in engineering problems.

Contribution

It provides a quantitative comparison of reparameterization versus penalty functions in physics-informed neural networks for differential equations.

Findings

Reparameterization lowers approximation error in complex differential equations.

Reparameterization outperforms penalty functions in benchmark mechanical problems.

Reparameterization is effective despite its implementation complexity.

Abstract

Differential equations are used to model and predict the behaviour of complex systems in a wide range of fields, and the ability to solve them is an important asset for understanding and predicting the behaviour of these systems. Complicated physics mostly involves difficult differential equations, which are hard to solve analytically. In recent years, physics-informed neural networks have been shown to perform very well in solving systems with various differential equations. The main ways to approximate differential equations are through penalty function and reparameterization. Most researchers use penalty functions rather than reparameterization due to the complexity of implementing reparameterization. In this study, we quantitatively compare physics-informed neural network models with and without reparameterization using the approximation error. The performance of reparameterization is demonstrated based on two benchmark mechanical engineering problems, a one-dimensional bar problem and a two-dimensional bending beam problem. Our results show that when dealing with complex differential equations, applying reparameterization results in a lower approximation error.