# VS-PINN: A fast and efficient training of physics-informed neural networks using variable-scaling methods for solving PDEs with stiff behavior

arXiv: https://arxiv.org/abs/2406.06287 · HTML: https://mlsanity.com/paper/2406.06287/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 34 references that are in the map.
- First posted on arXiv 10 Jun 2024.

## Builds on (most cited first, 20 of 20)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Towards Understanding the Spectral Bias of Deep Learning](https://mlsanity.com/paper/1912.01198/) (2019)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Deep Learning of Vortex Induced Vibrations](https://mlsanity.com/paper/1808.08952/) (2018)
- [Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks](https://mlsanity.com/paper/1905.04817/) (2019)
- [Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks](https://mlsanity.com/paper/2005.03596/) (2020)
- [Physics-informed deep learning for incompressible laminar flows](https://mlsanity.com/paper/2002.10558/) (2020)
- [Semi-analytic PINN methods for singularly perturbed boundary value problems](https://mlsanity.com/paper/2208.09145/) (2022)
- [EPINN-NSE: Enhanced Physics-Informed Neural Networks for Solving Navier-Stokes Equations](https://mlsanity.com/paper/2304.03689/) (2023)

## Built on by (most cited first, 13 of 13)

- [Kernel-Adaptive PI-ELMs for Forward and Inverse Problems in PDEs with Sharp Gradients](https://mlsanity.com/paper/2507.10241/) (2025)
- [Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometries](https://mlsanity.com/paper/2512.14941/) (2025)
- [Physics-Informed Neural Networks for Optimal Vaccination Plan in SIR Epidemic Models](https://mlsanity.com/paper/2502.19890/) (2025)
- [Physics-Informed Deep B-Spline Networks](https://mlsanity.com/paper/2503.16777/) (2025)
- [Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEs](https://mlsanity.com/paper/2510.01039/) (2025)
- [Engineering application of physics-informed neural networks for Saint-Venant torsion](https://mlsanity.com/paper/2505.12389/) (2025)
- [A decomposition-based robust training of physics-informed neural networks for nearly incompressible linear elasticity](https://mlsanity.com/paper/2505.21994/) (2025)
- [The compressible Neural Particle Method for Simulating Compressible Viscous Fluid Flows](https://mlsanity.com/paper/2508.16916/) (2025)
- [Soft Partition-based KAPI-ELM for Multi-Scale PDEs](https://mlsanity.com/paper/2601.08719/) (2026)
- [Solving stiff dark matter equations via Jacobian Normalization with Physics-Informed Neural Networks](https://mlsanity.com/paper/2602.21988/) (2026)
- [A scaled TW-PINN: A physics-informed neural network for traveling wave solutions of reaction-diffusion equations with general coefficients](https://mlsanity.com/paper/2603.15331/) (2026)
- [Physics-Informed Stochastic Configuration Machine: A Backpropagation-Free Neural Network with Fast Training for Nonlinear Differential Equations](https://mlsanity.com/paper/2608.26549/) (2026)
- [A Systematic Analysis of Automatic Differentiation versus Discretization-based Constraints for Physics-Informed PDE Solvers](https://mlsanity.com/paper/2609.07437/) (2026)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) have recently emerged as a promising way to compute the solutions of partial differential equations (PDEs) using deep neural networks. However, despite their significant success in various fields, it remains unclear in many aspects how to effectively train PINNs if the solutions of PDEs exhibit stiff behaviors or high frequencies. In this paper, we propose a new method for training PINNs using variable-scaling techniques. This method is simple and it can be applied to a wide range of problems including PDEs with rapidly-varying solutions. Throughout various numerical experiments, we will demonstrate the effectiveness of the proposed method for these problems and confirm that it can significantly improve the training efficiency and performance of PINNs. Furthermore, based on the analysis of the neural tangent kernel (NTK), we will provide theoretical evidence for this phenomenon and show that our methods can indeed improve the performance of PINNs.