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Although physics-informed neural networks(PINNs) have progressed a lot inmany real applications recently, there remains problems to be further studied,such as achieving more accurate results, taking less training time, andquantifying the uncertainty of the predicted results. Recent advances in PINNshave indeed significantly improved the performance of PINNs in many aspects,but few have considered the effect of variance in the training process. In thiswork, we take into consideration the effect of variance and propose ourVI-PINNs to give better predictions. We output two values in the final layer ofthe network to represent the predicted mean and variance respectively, and thelatter is used to represent the uncertainty of the output. A modified negativelog-likelihood loss and an auxiliary task are introduced for fast and accuratetraining. We perform several experiments on a wide range of different problemsto highlight the advantages of our approach. The results convey that our methodnot only gives more accurate predictions but also converges faster.

VI-PINNs: Variance-involved Physics-informed Neural Networks for Fast and Accurate Prediction of Partial Differential Equations