Rethinking Input Domains in Physics-Informed Neural Networks via Geometric Compactification Mappings
This addresses convergence issues in PINNs for multi-scale physical systems, though it appears incremental as it builds on existing PINN architectures with new mapping strategies.
The paper tackles the problem of gradient stiffness and ill-conditioning in physics-informed neural networks (PINNs) when solving multi-scale PDEs with both smooth and localized structures, by introducing geometric compactification mappings that reshape input coordinates to align with PDE structures. The approach yields more uniform residual distributions, higher solution accuracy on 1D and 2D PDEs, and improves training stability and convergence speed.
Several complex physical systems are governed by multi-scale partial differential equations (PDEs) that exhibit both smooth low-frequency components and localized high-frequency structures. Existing physics-informed neural network (PINN) methods typically train with fixed coordinate system inputs, where geometric misalignment with these structures induces gradient stiffness and ill-conditioning that hinder convergence. To address this issue, we introduce a mapping paradigm that reshapes the input coordinates through differentiable geometric compactification mappings and couples the geometric structure of PDEs with the spectral properties of residual operators. Based on this paradigm, we propose Geometric Compactification (GC)-PINN, a framework that introduces three mapping strategies for periodic boundaries, far-field scale expansion, and localized singular structures in the input domain without modifying the underlying PINN architecture. Extensive empirical evaluation demonstrates that this approach yields more uniform residual distributions and higher solution accuracy on representative 1D and 2D PDEs, while improving training stability and convergence speed.