  Physics informed neural networks (PINNs) are deep learning based techniques
for solving partial differential equations (PDEs) encounted 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.
  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.
