VI-PINNs: Variance-involved Physics-informed Neural Networks for Fast and Accurate Prediction of Partial Differential Equations
This work addresses challenges in PINNs for solving partial differential equations, offering incremental improvements in speed and accuracy for researchers and practitioners in computational science.
The authors tackled the problem of improving accuracy, reducing training time, and quantifying uncertainty in physics-informed neural networks (PINNs) by proposing VI-PINNs, which incorporate variance into the training process, resulting in more accurate predictions and faster convergence as demonstrated in experiments.
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.