Learning Traveling Solitary Waves Using Separable Gaussian Neural Networks
This addresses the propagation failure issue in PINNs for large computational domains in computational physics, though it is incremental as it builds on existing PINN frameworks.
The paper tackled the problem of learning traveling solitary waves in various PDE families by integrating a novel Separable Gaussian Neural Network (SGNN) into Physics-Informed Neural Networks (PINNs), achieving comparable accuracy with fewer than a tenth of the neurons compared to MLPs.
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.