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arXiv · 2402.01868

Challenges in Training PINNs: A Loss Landscape Perspective

Pratik Rathore·Weimu Lei·Zachary Frangella·Lu Lu·Madeleine Udell
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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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Pratik Rathore, Weimu Lei, Zachary Frangella, Lu Lu, Madeleine Udell. 2024-02-02. Challenges in Training PINNs: A Loss Landscape Perspective. https://arxiv.org/abs/2402.01868

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