Equidistribution of Solution Gradients in Physics-Informed Neural Networks via Coordinate Transformations

Abstract

Physics-Informed Neural Networks (PINNs) often struggle to approximate solutions with sharp gradients due to optimization difficulties and spectral bias. Motivated by the principle of gradient equidistribution from the stretching approach in grid generation, which redistributes solution gradients to reduce functional complexity, we propose Coordinate Transformation PINNs (CTPINNs), which incorporate learnable coordinate transformations to map the physical domain into a computational domain where the physical solutions become approximately affine, making them easier for neural networks to learn. We propose two constructions of the transformation: an explicit construction based on Majorants and an implicit construction based on invertable neural networks. Experimentally, the explicit construction can improve accuracy for PDEs with available explicit coordinate transformations, and the implicit construction can be combined with other advanced PINN models as an efficient plug-in module to improve performance across a range of challenging PDE benchmarks.