Soliton profiles: Classical Numerical Schemes vs. Neural Network - Based Solvers
Chandler Haight, Svetlana Roudenko, Zhongming Wang
TL;DR
The work systematically benchmarks classical numerical solvers against neural-network-based solvers for one-dimensional solitary-wave profiles governed by $Q''=bQ-\gamma Q^p$ across NLS, gKdV, and NLKG reductions. It finds that traditional finite-difference and Petviashvili methods deliver high accuracy and efficiency for single instances, while PINNs can approximate qualitative features but incur substantial training cost and slower convergence. Operator-learning approaches like DeepONet and FNO offer offline training to enable rapid, mesh-free inference across parameter regimes, with DeepONet generally outperforming FNO in this setting. The study highlights practical trade-offs and suggests avenues such as MoE architectures and physics-based activations to enhance neural solvers, while also detailing Dirichlet-boundary-induced order reduction in finite-difference schemes.
Abstract
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.