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

TL;DR

This work proposes 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.

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.

Figures (14)

• Figure 1: Conceptual comparison of standard PINNs and the proposed GC-PINN. Existing PINNs operate on the original Euclidean geometry, leading to unstable spectral conditioning and training. GC-PINN introduces a geometric compactification mapping paradigm that transforms the input domain, resulting in stable spectral properties and accelerated convergence while preserving predictive accuracy.
• Figure 2: Coordinate distributions under the three mappings.
• Figure 3: Overview of the GC-PINN pipeline. Physical domain coordinates are transformed via a learnable geometric compactification mapping and subsequently fed into an MLP backbone for solution prediction. The PDE residuals are computed through automatic differentiation, enabling end-to-end training of both the mapping parameters and the network weights.
• Figure 4: Comparison between predicted and ground-truth solutions for the 2D Navier-Stokes equation.
• Figure 5: Training convergence on five PDE benchmark tasks, measured by the evolution of $\mathrm{Rel}_{L^2}$..