Li-Sheng Zhang

Zhi-Yong Zhang, Hui Zhang, Li-Sheng Zhang et al.

As a typical application of deep learning, physics-informed neural network (PINN) {has been} successfully used to find numerical solutions of partial differential equations (PDEs), but how to improve the limited accuracy is still a great challenge for PINN. In this work, we introduce a new method, symmetry-enhanced physics informed neural network (SPINN) where the invariant surface conditions induced by the Lie symmetries or non-classical symmetries of PDEs are embedded into the loss function in PINN, to improve the accuracy of PINN for solving the forward and inverse problems of PDEs. We test the effectiveness of SPINN for the forward problem via two groups of ten independent numerical experiments using different numbers of collocation points and neurons for the heat equation, Korteweg-de Vries (KdV) equation and potential Burgers {equations} respectively, and for the inverse problem by considering different layers and neurons as well as different training points for the Burgers equation in potential form. The numerical results show that SPINN performs better than PINN with fewer training points and simpler architecture of neural network. Furthermore, we discuss the computational overhead of SPINN in terms of the relative computational cost to PINN and show that the training time of SPINN has no obvious increases, even less than PINN for certain cases.

Ye Liu, Jie-Ying Li, Li-Sheng Zhang et al.

Domain decomposition provides an effective way to tackle the dilemma of physics-informed neural networks (PINN) which struggle to accurately and efficiently solve partial differential equations (PDEs) in the whole domain, but the lack of efficient tools for dealing with the interfaces between two adjacent sub-domains heavily hinders the training effects, even leads to the discontinuity of the learned solutions. In this paper, we propose a symmetry group based domain decomposition strategy to enhance the PINN for solving the forward and inverse problems of the PDEs possessing a Lie symmetry group. Specifically, for the forward problem, we first deploy the symmetry group to generate the dividing-lines having known solution information which can be adjusted flexibly and are used to divide the whole training domain into a finite number of non-overlapping sub-domains, then utilize the PINN and the symmetry-enhanced PINN methods to learn the solutions in each sub-domain and finally stitch them to the overall solution of PDEs. For the inverse problem, we first utilize the symmetry group acting on the data of the initial and boundary conditions to generate labeled data in the interior domain of PDEs and then find the undetermined parameters as well as the solution by only training the neural networks in a sub-domain. Consequently, the proposed method can predict high-accuracy solutions of PDEs which are failed by the vanilla PINN in the whole domain and the extended physics-informed neural network in the same sub-domains. Numerical results of the Korteweg-de Vries equation with a translation symmetry and the nonlinear viscous fluid equation with a scaling symmetry show that the accuracies of the learned solutions are improved largely.