Learning Traveling Solitary Waves Using Separable Gaussian Neural
  Networks
Siyuan Xing Efstathios G. Charalampidis
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.