Organizations: School of Information and Software Engineering, University of Electronic Science and Technology of China, Chengdu, China · Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, China · Institute of Software Chinese Academy of Sciences, Beijing, China · School of Robotics and Advanced Manufacture, Harbin Institute of Technology, Shenzhen, China
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 & tables
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.
PDE
Metric
GC-Torus
GC-Radial
GC-Local
gPINN
RAR
FF
SA
PINN
1D Burgers
MSE
3.25E-10
8.69E-11
1.69E-10
1.73E-10
1.96E-10
2.65E-09
5.13E-10
3.14E-09
RelL2
2.44E-05
1.26E-05
1.76E-05
1.78E-05
1.90E-05
6.97E-05
3.07E-05
7.59E-05
RelH1
3.97E-05
1.18E-04
9.15E-05
4.22E-05
6.16E-05
8.07E-05
8.51E-05
3.91E-04
1D Conv-Diff
MSE
3.99E-06
6.00E-06
9.32E-08
1.96E-04
2.27E-06
1.39E-06
7.91E-07
2.43E-04
RelL2
2.00E-03
2.45E-03
3.06E-04
1.40E-02
1.51E-03
1.18E-03
8.90E-04
1.56E-02
RelH1
2.16E-03
3.55E-03
6.66E-04
1.57E-02
1.06E-03
5.15E-04
1.08E-03
1.57E-02
Table 1: Performance comparison across benchmark PDEs. (GC-* denote GC-PINN with Torus Mapping, Radial Mapping, and Local Stretching; RAR, FF, SA denote PINN+RAR, FF-PINN, and SA-PINN. PDE names are also abbreviated.)
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 RelL2 ..
Parameter β
Parameter α
β
MSE
Rel L2
α
MSE
Rel L2
5
1.09E-09
4.47E-05
10
1.46E-09
5.19E-05
10
3.58E-09
8.11E-05
20
8.17E-10
3.87E-05
15
2.69E-10
2.22E-05
30
4.60E-10
2.91E-05
20
1.69E-10
1.76E-05
40
3.44E-10
2.51E-05
25
2.08E-10
1.95E-05
50
8.69E-11
1.26E-05
Table 2: Parameter sensitivity result for the 1D Burgers equation.
Figure 6: Visualization of the error distribution of predicted solutions for the 1D Burgers equation.
Appendix figures & tables11 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 7: Spectral analysis of residual neural tangent kernels under geometric compactification. (a) Eigenvalue distribution of the kernel matrix after training convergence. (b) Temporal evolution of the effective rank during training.
Figure 8: Comparison of heatmaps of residual NTK kernel matrices.
1D Convection-Diffusion Equation
Parameter α
Parameter β
Param
MSE
Rel L2
Rel H1
Param
MSE
Rel L2
Rel H1
5
8.22E-06
2.87E-03
2.99E-03
10
6.66E-06
2.58E-03
1.21E-03
10
6.22E-06
2.50E-03
6.05E-03
20
5.66E-06
2.38E-03
2.02E-03
15
6.07E-06
2.47E-03
1.34E-02
30
2.22E-06
1.49E-03
5.35E-04
20
9.32E-08
3.06E-04
6.66E-04
40
2.01E-07
4.49E-04
2.07E-04
Appendix
Table 3: Parameter sensitivity analysis for the 1D convection-diffusion and 2D Navier-Stokes equations
Problem
Mapping Method
MSE
RelL2
RelH1
1D Burgers
Torus Mapping
3.25E-10
2.44E-05
3.97E-05
Radial Mapping
8.69E-11
1.26E-05
1.18E-04
Local Stretching
1.69E-10
1.76E-05
9.15E-05
Saturating Mapping
2.75E-01
7.11E-01
8.33E-01
PWL Mapping
2.56E-06
2.17E-03
1.57E-03
1D Convection- Diffusion
Torus Mapping
3.99E-06
2.00E-03
2.16E-03
Appendix
Table 4: Performance comparison of different mapping methods across multiple benchmark problems.
Model
Parameters
GC-PINN(Torus Mapping)
19,841
GC-PINN(Radial Mapping)
19,682
GC-PINN(Local Stretching)
19,684
PINN
19,681
FF-PINN
20,641
PINN+RAR
19,681
Appendix
Table 5: Parameter counts of compared PINNs.
Figure 9: Predicted solutions and error distributions across models for the 1D convection-diffusion equation (1).
Figure 10: Predicted solutions and error distributions across models for the 1D convection-diffusion equation (2).
Figure 11: Predicted solutions and error distributions across models for the 2D convection-diffusion equation (1).
Figure 12: Predicted solutions and error distributions across models for the 2D convection-diffusion equation (2).
Figure 13: Predicted solutions and error distributions across models for the 2D convection-diffusion equation (3).
Figure 14: Ground-truth solution for the 1/2D convection-diffusion equation.
Neural PDE solvers use trainable models to approximate solutions or solution operators for partial differential equations, often with physical constraints.