Challenges in Training PINNs: A Loss Landscape Perspective

• URL: http://arxiv.org/abs/2402.01868v2
• Date: Mon, 3 Jun 2024 23:35:42 GMT
• Title: Challenges in Training PINNs: A Loss Landscape Perspective
• Authors: Pratik Rathore, Weimu Lei, Zachary Frangella, Lu Lu, Madeleine Udell,
• Abstract summary: This paper explores challenges in training Physics-Informed Neural Networks (PINNs) We examine difficulties in minimizing the PINN loss function, particularly due to ill-conditioning caused by differential operators in the residual term. We introduce a novel second-order gradient, NysNewton-CG (NNCG), which significantly improves PINN performance.
• Score: 16.89714536706181
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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