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.

Separable Gaussian Neural Networks: Structure, Analysis, and Function Approximations

The Gaussian-radial-basis function neural network (GRBFNN) has been a popular choice for interpolation and classification. However, it is computationally intensive when the dimension of the input vector is high. To address this issue, we propose a new feedforward network - Separable Gaussian Neural Network (SGNN) by taking advantage of the separable property of Gaussian functions, which splits input data into multiple columns and sequentially feeds them into parallel layers formed by uni-variate Gaussian functions. This structure reduces the number of neurons from O(N^d) of GRBFNN to O(dN), which exponentially improves the computational speed of SGNN and makes it scale linearly as the input dimension increases. In addition, SGNN can preserve the dominant subspace of the Hessian matrix of GRBFNN in gradient descent training, leading to a similar level of accuracy to GRBFNN. It is experimentally demonstrated that SGNN can achieve 100 times speedup with a similar level of accuracy over GRBFNN on tri-variate function approximations. The SGNN also has better trainability and is more tuning-friendly than DNNs with RuLU and Sigmoid functions. For approximating functions with complex geometry, SGNN can lead to three orders of magnitude more accurate results than a RuLU-DNN with twice the number of layers and the number of neurons per layer.