Soliton profiles: Classical Numerical Schemes vs. Neural Network - Based Solvers
This work addresses the trade-offs between classical and neural solvers for PDEs, providing insights for researchers in computational physics and machine learning, though it is incremental as it confirms known limitations.
The study compared classical numerical solvers and neural network-based methods for computing solitary-wave solutions in one-dimensional dispersive PDEs, finding that classical methods retain high-order accuracy and efficiency for single-instance problems, while neural methods like PINNs and operator-learning are less accurate and efficient in low dimensions but offer advantages for repeated simulations.
We present a comparative study of classical numerical solvers, such as Petviashvili's method or finite difference with Newton iterations, and neural network-based methods for computing ground states or profiles of solitary-wave solutions to the one-dimensional dispersive PDEs that include the nonlinear Schrödinger, the nonlinear Klein-Gordon and the generalized KdV equations. We confirm that classical approaches retain high-order accuracy and strong computational efficiency for single-instance problems in the one-dimensional setting. Physics-informed neural networks (PINNs) are also able to reproduce qualitative solutions but are generally less accurate and less efficient in low dimensions than classical solvers due to expensive training and slow convergence. We also investigate the operator-learning methods, which, although computationally intensive during training, can be reused across many parameter instances, providing rapid inference after pretraining, making them attractive for applications involving repeated simulations or real-time predictions. For single-instance computations, however, the accuracy of operator-learning methods remains lower than that of classical methods or PINNs, in general.