This paper explores challenges in training Physics-Informed Neural Networks (PINNs), emphasizing the loss landscape’s role 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 perform a thorough comparison of 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 optimization strategies for training PINNs, which could broaden the application of PINNs in solving difficult partial differential equation-based problems.