Learning Traveling Solitary Waves Using Separable Gaussian Neural Networks

• URL: http://arxiv.org/abs/2403.04883v1
• Date: Thu, 7 Mar 2024 20:16:18 GMT
• Title: Learning Traveling Solitary Waves Using Separable Gaussian Neural Networks
• Authors: Siyuan Xing and Efstathios G. Charalampidis
• Abstract summary: 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 into the framework of Physics-Informed Neural Networks (PINNs)
• Score: 0.9065034043031668
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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