Learning Traveling Solitary Waves Using Separable Gaussian Neural Networks
<|reference_start|>Learning Traveling Solitary Waves Using Separable Gaussian Neural Networks: In this paper, we apply a machine-learning approach to learn traveling solitary waves across various families of partial differential equations (PDEs). Our approach integrates a novel interpretable neural network (NN) architecture, called Separable Gaussian Neural Networks (SGNN) into the framework of Physics-Informed Neural Networks (PINNs). Unlike the traditional PINNs that treat spatial and temporal data as independent inputs, the present method leverages wave characteristics to transform data into the so-called co-traveling wave frame. This adaptation effectively addresses the issue of propagation failure in PINNs when applied to large computational domains. Here, the SGNN architecture demonstrates robust approximation capabilities for single-peakon, multi-peakon, and stationary solutions within the (1+1)-dimensional, $b$-family of PDEs. In addition, we expand our investigations, and explore not only peakon solutions in the $ab$-family but also compacton solutions in (2+1)-dimensional, Rosenau-Hyman family of PDEs. A comparative analysis with MLP reveals that SGNN achieves comparable accuracy with fewer than a tenth of the neurons, underscoring its efficiency and potential for broader application in solving complex nonlinear PDEs.<|reference_end|>
@article{xing2024learning,
    title={Learning Traveling Solitary Waves Using Separable Gaussian Neural
  Networks},
    author={Siyuan Xing and Efstathios G. Charalampidis},
    journal={arXiv preprint arXiv:2403.04883},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.04883},
    primaryClass={nlin.PS cs.LG}
}