  Physics Informed Neural Networks (PINNs) is a promising application of deep
learning. The smooth architecture of a fully connected neural network is
appropriate for finding the solutions of PDEs; the corresponding loss function
can also be intuitively designed and guarantees the convergence for various
kinds of PDEs. However, the rate of convergence has been considered as a
weakness of this approach. This paper proposes Sobolev-PINNs, a novel loss
function for the training of PINNs, making the training substantially
efficient. Inspired by the recent studies that incorporate derivative
information for the training of neural networks, we develop a loss function
that guides a neural network to reduce the error in the corresponding Sobolev
space. Surprisingly, a simple modification of the loss function can make the
training process similar to \textit{Sobolev Training} although PINNs is not a
fully supervised learning task. We provide several theoretical justifications
that the proposed loss functions upper bound the error in the corresponding
Sobolev spaces for the viscous Burgers equation and the kinetic Fokker--Planck
equation. We also present several simulation results, which show that compared
with the traditional $L^2$ loss function, the proposed loss function guides the
neural network to a significantly faster convergence. Moreover, we provide the
empirical evidence that shows that the proposed loss function, together with
the iterative sampling techniques, performs better in solving high dimensional
PDEs.

Numerical Analysis;Numerical Analysis