Summary of Challenges in Training Pinns: a Loss Landscape Perspective, by Pratik Rathore et al.
Challenges in Training PINNs: A Loss Landscape Perspective
by Pratik Rathore, Weimu Lei, Zachary Frangella, Lu Lu, Madeleine Udell
First submitted to arxiv on: 2 Feb 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: Optimization and Control (math.OC); Machine Learning (stat.ML)
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| Summary difficulty | Written by | Summary |
|---|---|---|
| High | Paper authors | High Difficulty Summary Read the original abstract here |
| Medium | GrooveSquid.com (original content) | Medium Difficulty Summary This paper delves into the challenges of training Physics-Informed Neural Networks (PINNs) by investigating the role of the loss landscape in the training process. The authors highlight difficulties in minimizing the PINN loss function, particularly due to ill-conditioning caused by differential operators in the residual term. They compare various gradient-based optimizers, including Adam, L-BFGS, and their combination Adam+L-BFGS, demonstrating the superiority of Adam+L-BFGS, and introduce a novel second-order optimizer, NysNewton-CG (NNCG), which significantly improves PINN performance. Theoretical insights reveal the connection between ill-conditioned differential operators and ill-conditioning in the PINN loss, highlighting the benefits of combining first- and second-order optimization methods. This work presents valuable contributions to the development of more powerful optimization strategies for training PINNs, potentially enhancing their utility for solving complex partial differential equations. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary This paper looks at how to make Physics-Informed Neural Networks (PINNs) work better by understanding what happens during the training process. The authors found it’s hard to get PINNs to learn because of a problem called ill-conditioning that comes from using math operators in the network. They tested different ways to optimize the training process, like Adam and L-BFGS, and found that combining them worked best. They also created a new way to optimize called NysNewton-CG (NNCG) that made PINNs work even better. The paper explains why this problem happens and how it can be fixed, which could make PINNs more useful for solving tricky math problems. |
Keywords
* Artificial intelligence * Loss function * Optimization
