Challenges in Training PINNs: A Loss Landscape Perspective
Pratik Rathore, Weimu Lei, Zachary Frangella, Lu Lu, Madeleine Udell
Abstract
This paper explores challenges in training Physics-Informed Neural Networks (PINNs), emphasizing the role of the loss landscape in the training process. We examine difficulties in minimizing the PINN loss function, particularly due to ill-conditioning caused by differential operators in the residual term. We compare gradient-based optimizers Adam, L-BFGS, and their combination Adam+L-BFGS, showing the superiority of Adam+L-BFGS, and introduce a novel second-order optimizer, NysNewton-CG (NNCG), which significantly improves PINN performance. Theoretically, our work elucidates the connection between ill-conditioned differential operators and ill-conditioning in the PINN loss and shows the benefits of combining first- and second-order optimization methods. Our work presents valuable insights and more powerful optimization strategies for training PINNs, which could improve the utility of PINNs for solving difficult partial differential equations.
BibTeX
@inproceedings{
rathore2024challenges,
title={Challenges in Training {PINN}s: A Loss Landscape Perspective},
author={Pratik Rathore and Weimu Lei and Zachary Frangella and Lu Lu and Madeleine Udell},
booktitle={Forty-first International Conference on Machine Learning},
year={2024},
url={https://openreview.net/forum?id=mJGiFr8jLa}
}