Generalized multi-symplectic PINN method and its application in quantum tunneling simulations

Focusing on the conservative/nonconservative physical information of infinite-dimensional dissipative system, the generalized multi-symplectic physics informed neural network (GMPINN) method is proposed to improve the reproducing ability of dissipative effects for the physics informed neural network (PINN) in this paper. The core idea of GMPINN is to embed the geometric structures of dissipative systems into the loss function, thereby preserving the geometric structures as well as intrinsic conservation laws of the physical system to reproduce the dissipative effects reliably. The above merits of GMPINN are illustrated in the simulations on the dissipative quantum tunneling problem formulated by the dissipative Schrödinger equation. In the simulations by using GMPINN, the influences of the dissipation coefficient and the square potential barrier thickness on the probability distribution are presented. From the numerical results, the inverse proportion laws between the dissipation coefficient/square potential barrier thickness and the probability distribution density on right side of the square potential barrier are found. Comparing with the effect of the square potential barrier thickness, it is found that the effect of the dissipation coefficient on the quantum coherence on the left side of the square potential barrier is more remarkable. The above findings illustrate the excellent ability of GMPINN in reproducing the dissipative effects of the nonconservative systems and give guidance on the parameter design of the devices based on the principle of the quantum tunneling effect.