Challenges in Training PINNs: A Loss Landscape Perspective
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910470203506688 |
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| author | Rathore, Pratik Lei, Weimu Frangella, Zachary Lu, Lu Udell, Madeleine |
| author_facet | Rathore, Pratik Lei, Weimu Frangella, Zachary Lu, Lu Udell, Madeleine |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_01868 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Challenges in Training PINNs: A Loss Landscape Perspective Rathore, Pratik Lei, Weimu Frangella, Zachary Lu, Lu Udell, Madeleine Machine Learning Optimization and Control 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. |
| title | Challenges in Training PINNs: A Loss Landscape Perspective |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2402.01868 |