Challenges in Training PINNs: A Loss Landscape Perspective
It addresses training difficulties for PINN users, offering incremental improvements in optimization strategies for solving partial differential equations.
This paper tackles challenges in training Physics-Informed Neural Networks (PINNs) by analyzing the loss landscape, showing that a novel optimizer, NysNewton-CG, significantly improves PINN performance compared to methods like Adam and L-BFGS.
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.