  Although physics-informed neural networks(PINNs) have progressed a lot in
many real applications recently, there remains problems to be further studied,
such as achieving more accurate results, taking less training time, and
quantifying the uncertainty of the predicted results. Recent advances in PINNs
have indeed significantly improved the performance of PINNs in many aspects,
but few have considered the effect of variance in the training process. In this
work, we take into consideration the effect of variance and propose our
VI-PINNs to give better predictions. We output two values in the final layer of
the network to represent the predicted mean and variance respectively, and the
latter is used to represent the uncertainty of the output. A modified negative
log-likelihood loss and an auxiliary task are introduced for fast and accurate
training. We perform several experiments on a wide range of different problems
to highlight the advantages of our approach. The results convey that our method
not only gives more accurate predictions but also converges faster.
