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

Enforcing continuous symmetries in physics-informed neural network for solving forward and inverse problems of partial differential equations