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

2026-02-18

AI Summary

This work addresses the challenges of gradient stiffness and ill-conditioning in physics-informed neural networks (PINNs) when solving multiscale partial differential equations, which often arise from geometric mismatches between input coordinates and local high-frequency solution structures, leading to poor convergence. To overcome this, the authors propose the GC-PINN framework, which introduces a differentiable geometric compactification mapping that adaptively aligns complex solution features by coupling the PDE’s geometric structure with the spectral properties of the residual operator—without altering the original PINN architecture. Three input-domain mapping strategies are specifically designed for representative scenarios involving periodic boundaries, far-field extensions, and local singularities. Experiments on one- and two-dimensional benchmark problems demonstrate that GC-PINN significantly enhances training stability, convergence speed, and solution accuracy, while yielding a more uniform residual distribution.

Technology Category

• Machine Learning: Learning with Manifolds
• Natural Language Processing: Language Grounding & Multi-modal NLP
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• physics-informed neural networks
• multi-scale PDEs
• gradient stiffness
• geometric misalignment
• ill-conditioning

Innovation

• Geometric Compactification
• Physics-Informed Neural Networks
• Input Domain Mapping
• Multi-scale PDEs
• Residual Operator Spectral Alignment