VS-PINN: A fast and efficient training of physics-informed neural networks using variable-scaling methods for solving PDEs with stiff behavior
This addresses a bottleneck in PINN training for PDEs with stiff solutions, offering a simple and broadly applicable method, though it appears incremental as an enhancement to existing PINN frameworks.
The paper tackles the challenge of training physics-informed neural networks (PINNs) for solving partial differential equations (PDEs) with stiff or high-frequency behaviors, proposing a variable-scaling method that significantly improves training efficiency and performance, as demonstrated through numerical experiments and theoretical analysis.
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.