Qiumei Huang

Jianqing Zhu, Juncai He, Qiumei Huang

This study used a multigrid-based convolutional neural network architecture known as MgNet in operator learning to solve numerical partial differential equations (PDEs). Given the property of smoothing iterations in multigrid methods where low-frequency errors decay slowly, we introduced a low-frequency correction structure for residuals to enhance the standard V-cycle MgNet. The enhanced MgNet model can capture the low-frequency features of solutions considerably better than the standard V-cycle MgNet. The numerical results obtained using some standard operator learning tasks are better than those obtained using many state-of-the-art methods, demonstrating the efficiency of our model.Moreover, numerically, our new model is more robust in case of low- and high-resolution data during training and testing, respectively.

Qiumei Huang, Shanghui Jia, Fei Xu et al.

In this paper, we consider the wave propagation with Debye polarization in nonlinear dielectric materials. For this model, the Rother's method is employed to derive the well-posedness of the electric fields and the existence of the polarized fields by monotonicity theorem as well as the boundedness of the two fields are established. Then, the time errors are derived for the semi-discrete solutions by the order $O(Δt)$. Subsequently, decoupled the full-discrete scheme of the Euler in time and Raviart-Thomas-N$\acute{e}$d$\acute{e}$lec element $k\geq 2$ in spatial is established. Based on the truncated error, we present the convergent analysis with the order $O(Δt+h^s) $ under the technique of a-prior $L^\infty$ assumption. For the $k=1$, we employ the superconvergence technique to ensure the a-prior $L^\infty$ assumption. In the end, we give some numerical examples to demonstrate our theories.

Qiumei Huang, Jiaxuan Ma, Zhen Xu

As one kind of important phase field equations, Cahn-Hilliard equations involve high-order spatial derivatives, strong nonlinearities, and even solution singularities when certain bulk potentials are used. When using the physics informed neural network (PINN) to simulate the long time evolution of the solution, it is necessary to decompose the time domain to capture the transition of solutions in different time. Moreover, the standard PINN cannot maintain the mass conservation property for the equations exactly. We propose a novel mass-preserving spatiotemporal adaptive PINN, which adaptively divides the time domain according to the rate of energy decrease, and solves the Cahn-Hilliard equation within each subinterval. To improve the prediction accuracy, spatial adaptive sampling is employed in the subdomain to select points with large residual value which are added to the training samples. Notably, a mass constraint is added to the loss function to compensate the mass degradation problem of the PINN method when solving Cahn-Hilliard equations. Numerical experiments are presented to illustrate the effectiveness of the proposed method in solving complex phase field models, including the Cahn-Hilliard equations with different bulk potentials, the three-dimensional Cahn-Hilliard equation with singularities, and the system of Cahn-Hilliard equations.