Enforcing Continuous Physical Symmetries in Deep Learning Network for Solving Partial Differential Equations

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 of PDEs are embedded into the loss function of PINN, for improving the accuracy of PINN. We test the effectiveness of SPINN via two groups of ten independent numerical experiments for the heat equation, Korteweg-de Vries (KdV) equation and potential Burgers {equations} respectively, which shows 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 some cases.