TY - JOUR
T1 - A Stabilized Physics Informed Neural Networks Method for Wave Equations
AU - Jiao , Yuling
AU - Liu , Yuhui
AU - Yang , Jerry Zhijian
AU - Yuan , Cheng
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 4
SP - 1100
EP - 1127
PY - 2024
DA - 2024/12
SN - 17
DO - http://doi.org/10.4208/nmtma.OA-2024-0044
UR - https://global-sci.org/intro/article_detail/nmtma/23653.html
KW - PINNs, $ReLU^3$ neural network, wave equations, error analysis.
AB - <p style="text-align: justify;">In this article, we propose a novel stabilized physics informed neural networks method (SPINNs) for solving wave equations. In general, this method not
only demonstrates theoretical convergence but also exhibits higher efficiency compared to the original PINNs. By replacing the $L^2$ norm with $H^1$ norm in the learning
of initial condition and boundary condition, we theoretically proved that the error of
solution can be upper bounded by the risk in SPINNs. Based on this, we decompose
the error of SPINNs into approximation error, statistical error and optimization error. Furthermore, by applying the approximating theory of $ReLU^3$ networks and the
learning theory on Rademacher complexity, covering number and pseudo-dimension
of neural networks, we present a systematical non-asymptotic convergence analysis
on our method, which shows that the error of SPINNs can be well controlled if the
number of training samples, depth and width of the deep neural networks have
been appropriately chosen. Two illustrative numerical examples on 1-dimensional
and 2-dimensional wave equations demonstrate that SPINNs can achieve a faster
and better convergence than classical PINNs method.</p>