Sobolev Training for Physics Informed Neural Network

Abstract

Physics-Informed Neural Networks (PINNs) are promising applications 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 convergence for various kinds of PDEs. However, the high computational cost required to train neural networks 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 Sobolev Training although PINNs are not fully supervised learning tasks. 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 L2 loss function, the proposed loss function guides the neural network to a significantly faster convergence. Moreover, we provide empirical evidence that shows that the proposed loss function, together with the iterative sampling techniques, performs better in solving high-dimensional PDEs. © 2023 International Press