Transformed Physics-Informed Neural Networks for The Convection-Diffusion Equation
This work addresses numerical instability in solving convection-diffusion equations for computational physics and engineering, but it is incremental as it builds on existing PINN methods with specific modifications.
The paper tackled the challenge of solving singularly perturbed convection-diffusion equations with steep boundary layers, which are difficult for traditional methods like Finite Difference Methods (FDMs), by using Physics-Informed Neural Networks (PINNs) to correct oscillatory FDM solutions and modify reduced solutions, with input transformations improving accuracy as analyzed via neural tangent kernels.
Singularly perturbed problems are known to have solutions with steep boundary layers that are hard to resolve numerically. Traditional numerical methods, such as Finite Difference Methods (FDMs), require a refined mesh to obtain stable and accurate solutions. As Physics-Informed Neural Networks (PINNs) have been shown to successfully approximate solutions to differential equations from various fields, it is natural to examine their performance on singularly perturbed problems. The convection-diffusion equation is a representative example of such a class of problems, and we consider the use of PINNs to produce numerical solutions of this equation. We study two ways to use PINNS: as a method for correcting oscillatory discrete solutions obtained using FDMs, and as a method for modifying reduced solutions of unperturbed problems. For both methods, we also examine the use of input transformation to enhance accuracy, and we explain the behavior of input transformations analytically, with the help of neural tangent kernels.