Physics-informed Neural Network: The Effect of Reparameterization in Solving Differential Equations
This work addresses the challenge of solving difficult differential equations for mechanical engineers, but it is incremental as it focuses on comparing existing methods rather than introducing new ones.
The study tackled the problem of solving complex differential equations in mechanical engineering by comparing physics-informed neural networks with and without reparameterization, finding that reparameterization leads to lower approximation error in benchmark problems like a one-dimensional bar and two-dimensional bending beam.
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.