A dynamic mesh refinement in Physic-Informed Neural Networks for handling Stiff Partial Differential Equations

Stiff Partial Differential Equations (PDEs) pose considerable challenges due to the presence of steep gradients, which often lead to large residuals when employing coarse training mesh grids, while excessively refined grids substantially increase computational costs. To address this trade off, we propose an Augmented Mesh Refinement Physics Informed Neural Network (Aug-MR PINN), a novel framework that dynamically allocates mesh points in regions with sharp gradients. Unlike existing approaches, Aug-MR PINN adaptively adjusts the mesh distribution during training by evaluating residual based criteria, thereby maintaining a balance between accuracy and efficiency. The method was validated on three representative stiff PDEs, namely Burgers’, Telegraph, and Flow Mixing equations, under two scenarios which are percentile and threshold based refinement. Our key findings show that Aug-MR PINN consistently achieves lower loss values and improved convergence compared to state of the art methods such as Adaptive Mesh Refinement with Finite Element Method (AMR-FEM), basic PINN, Restarting PINN (R-PINN), and Extended PINN (X-PINN). Moreover, both refinement strategies yielded comparable performance, confirming the robustness of the proposed approach. Beyond stiff PDEs, this framework provides a promising foundation for broader applications in computational physics, such as fluid dynamics, wave propagation, and multiphase systems. • Stiff PDEs pose challenges due to steep gradients requiring adaptive solutions. • AugMR-PINN optimizes mesh refinement dynamically for better accuracy and efficiency. • Mesh points are adjusted iteratively based on loss function evaluations. • Two refinement methods: percentile-based and threshold-based approaches. • AugMR-PINN surpasses PINN, R-PINN, X-PINN and AMR-FEM in minimizing loss.