TY - JOUR
T1 - On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs
AU - Shin , Yeonjong
AU - Darbon , Jérôme
AU - Em Karniadakis , George
JO - Communications in Computational Physics
VL - 5
SP - 2042
EP - 2074
PY - 2020
DA - 2020/11
SN - 28
DO - http://doi.org/10.4208/cicp.OA-2020-0193
UR - https://global-sci.org/intro/article_detail/cicp/18404.html
KW - Physics informed neural networks, convergence, Hölder regularization, elliptic and
parabolic PDEs, Schauder approach.
AB - <p style="text-align: justify;">Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encountered in computational
science and engineering. Guided by data and physical laws, PINNs find a neural network that approximates the solution to a system of PDEs. Such a neural network is
obtained by minimizing a loss function in which any prior knowledge of PDEs and
data are encoded. Despite its remarkable empirical success in one, two or three dimensional problems, there is little theoretical justification for PINNs.<br/>As the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We want to answer the question: Does the sequence of minimizers converge to the solution to the PDE? We consider two classes of
PDEs: linear second-order elliptic and parabolic. By adapting the Schauder approach
and the maximum principle, we show that the sequence of minimizers strongly converges to the PDE solution in $C^0$. Furthermore, we show that if each minimizer satisfies
the initial/boundary conditions, the convergence mode becomes $H^1$. Computational
examples are provided to illustrate our theoretical findings. To the best of our knowledge, this is the first theoretical work that shows the consistency of PINNs.</p>