Learning Traveling Solitary Waves Using Separable Gaussian Neural Networks

TL;DR

This approach integrates a novel interpretable neural network architecture, called Separable Gaussian Neural Networks (SGNN) into the framework of Physics-Informed Neural Networks (PINNs), and leverages wave characteristics to transform data into the so-called co-traveling wave frame.

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.

Figures (15)

• Figure S1: Computational results of the spatio-temporal evolution of a peakon in the Camassa-Holm (CH) equation using PINNs. The propagation failure of PINNs occurs when a large spatio-temporal domain, e.g., $[-30,30]\times[0,10]$ is utilized. In smaller domains, e.g., $[-5,5]\times[0,1]$ (left panel), PINNs are able to roughly capture the correct solution. However, in larger domains (right panel), it tends to converge to trivial solutions. a) and b): ground truth; c) and d): NN approximation. The configuration of the loss function is referred from WangandYanPhysicaD2021.
• Figure S2: The architecture of SGNN with input transformed into the co-traveling frame whose coordinates are $\boldsymbol{\zeta}=\boldsymbol{x}-\boldsymbol{c} t$, where $\boldsymbol{c}$ represents the velocity of the wave. Different from MLPs, the transformed input is then split and fed sequentially to hidden layers of an SGNN that consist of univariate functions. The multiplication and addition of such univariate functions in feedforward propagation can eventually lead to the summation of a set of multivariate functions used to approximate the solution of a PDE.
• Figure S3: A one-peakon solution in the CH equation ($b=2$) with $c=1$. a): $\bar{u}(x,t)$ inferred by SGNN; b): error $e(x,t)=u(x,t)-\bar{u}(x,t)$; c): $\bar{u}(x,t)$ at two time instants. In c), "x" markers represent the exact solution while lines represent the prediction by SGNN. The training loss is $8.43e-3$, with $\lambda_{ic}=1,000$, $\lambda_{bc}=1$. Validation error: $\Vert e \Vert=3.90e-2$, $\Vert e \Vert_2=7.21e-6$.
• Figure S4: Same as Fig. \ref{['fig:fig2']} but for an one-antipeakon solution in the CH equation ($b=2$) with $c=-1$. a): $\bar{u}(x,t)$ inferred by SGNN; b): error $e(x,t)=u(x,t)-\bar{u}(x,t)$; c): $\bar{u}(x,t)$ at two time instants. In c), "x" represents the exact solution while lines represent the prediction by SGNN. The training loss is $1.94e-11$, with $\lambda_{ic}=1,000, \lambda_{bc}=1$. Validation error: $\Vert e \Vert=1.02e-5$, $\Vert e \Vert_2=9.59e-13$.
• Figure S5: Same as Fig. \ref{['fig:fig3']} but for a one-peakon solution of the $b$-family with $b=0.8$ and $c=1.5$. a): $\bar{u}(x,t)$ inferred by SGNN; b): error $e(x,t)=u(x,t)-\bar{u}(x,t)$; c): $\bar{u}(x,t)$ at two time instants. In c), "x" represents the analytical solution while curves represent the prediction by SGNN. The training loss is $9.18e-2$, with $\lambda_{ic}=1,000, \lambda_{bc}=1$. Validation error: $\Vert e \Vert=2.74e-2$, $\Vert e \Vert_2=4.09e-6$.