Enhancing the stability of physics-informed neural networks applied to convection problems

Physics-Informed Neural Networks (PINNs) represent an innovative method for solving a wide range of problems in mathematics, physics, and engineering. PINNs combine the neural networks concepts and physical equations aimed at modeling and analyzing various physical processes. In particular, PINNs can be applied to solve differential equations, including the one-dimensional convection equation. The research shows that the standard implementation of PINNs efficiently solves a one-dimensional convection equation at relatively small convection velocity values, but diverges for higher values of this parameter. This paper provides an overview of existing approaches for solving the one-dimensional convection equation using PINNs and demonstrates improvement for model performance through different methods. The results of comparison indicate the superiority of the approach based on dynamically adjusting collocation points according to the residual at the current training step.