Rethinking Input Domains in Physics-Informed Neural Networks via Geometric Compactification Mappings

TL;DR

This paper introduces a geometric compactification mapping framework for physics-informed neural networks (PINNs) to better handle multi-scale PDEs with complex structures, improving training stability and accuracy.

Contribution

The paper proposes a novel geometric compactification paradigm and three mapping strategies for PINNs, enhancing their ability to model multi-scale PDEs with complex geometries.

Findings

More uniform residual distributions achieved

Higher solution accuracy demonstrated

Improved training stability and convergence speed

Abstract

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.