Lie Symmetry-Guided Adaptive Sampling for Physics-Informed Neural Networks
Abstract
Physics-informed neural networks (PINNs) have emerged as a promising mesh-free framework for solving partial differential equations (PDEs), with broad applications in physics, engineering, and medicine. However, their performance is highly sensitive to the sampling strategy for training points. Conventional uniform sampling often fails to adequately resolve regions with complex solution features, leading to low efficiency and limited accuracy. To address these challenges, we propose a Lie symmetry-driven adaptive sampling framework for PINNs, termed Lie-AS-ISC. For PDEs admitting Lie symmetry groups, symmetry transformations are exploited to derive intrinsic functional relationships between the solution and its variables, enabling the construction of basis points with exact solutions that span the solution domain. Building on this, an adaptive sampling strategy is developed by jointly considering absolute error and invariant surface conditions, while a dual-index loss is introduced to enhance training constraints. The proposed method significantly improves prediction accuracy. The experimental results show that this method consistently outperforms existing methods in accuracy.
Related
- AKAPINN: Adaptive Kolmogorov-Arnold Physics-Informed Neural Networks for approximating solutions to quasilinear partial differential equations
- GAC-PINN: Geometry-Adaptive and Constraint-Enhanced Physics-Informed Neural Networks
- Lazy training of quantum physics informed neural networks
- SS-ESOAP: Self-Scaled Adaptive Preconditioning for Physics-Informed Learning
- Solving Fredholm Integral Equations with Physics-Informed Neural Networks: Single-and double-network approaches
BibTeX
@misc{zhao2026symmetryguided,
title = {{Lie Symmetry-Guided Adaptive Sampling for Physics-Informed Neural Networks}},
author = {Runze Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34345},
eprint = {2610.34345},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34345},
}
RIS
TY - UNPB
TI - Lie Symmetry-Guided Adaptive Sampling for Physics-Informed Neural Networks
AU - Runze Zhao
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34345
UR - https://arxiv.org/abs/2610.34345
AB - Physics-informed neural networks (PINNs) have emerged as a promising mesh-free framework for solving partial differential equations (PDEs), with broad applications in physics, engineering, and medicine. However, their performance is highly sensitive to the sampling strategy for training points. Conventional uniform sampling often fails to adequately resolve regions with complex solution features, leading to low efficiency and limited accuracy. To address these challenges, we propose a Lie symmetry-driven adaptive sampling framework for PINNs, termed Lie-AS-ISC. For PDEs admitting Lie symmetry groups, symmetry transformations are exploited to derive intrinsic functional relationships between the solution and its variables, enabling the construction of basis points with exact solutions that span the solution domain. Building on this, an adaptive sampling strategy is developed by jointly considering absolute error and invariant surface conditions, while a dual-index loss is introduced to enhance training constraints. The proposed method significantly improves prediction accuracy. The experimental results show that this method consistently outperforms existing methods in accuracy.
ER -