High-precision physics-informed extreme learning machines for evolving interface problems

Abstract

Neural network (NN) methods have been developed to solve interface problems recently. In comparison with conventional techniques (e.g., finite element method), the NN method enjoys the merits of meshless features, powerful ability to approximate complex interface geometries, and high accuracy. The current NN studies are mostly focused on elliptic interface problems. The methodology will cause difficulties for evolving (time-dependent) and moving interface problems. (a) The accuracy is also characterized by time discretization strategies; if the time step $\tau$ is small, the training time is unbearable. (b) At each time step, it is difficult to design the high-precision NNs because of involvement of complex interfaces, especially for the moving interfaces. This study is devoted to proposing the high-precision NN methods for the evolving and moving interface problems. The first method is based on the time stepping scheme. In every time step we develop piecewise extreme learning machine (ELM) to improve the accuracy of space discretization and reduce the training time. As a consequence, the optimal overall error with respect to the time, $O\left(\tau \right)$, is achieved (for the backward Euler method). Evidently, the accuracy is still limited by $\tau$. To improve the accuracy further, the second method is to treat the time dimension as an additional space dimension to formulate the equation in an extended time–space domain $\tilde{\Omega }$. A time–space piecewise ELM in $\tilde{\Omega }$ is designed. The new method avoids the time stepping so that the training time is saved essentially, and the approximation errors are also reduced significantly. We note that the increase of dimension in $\tilde{\Omega }$ does not yield additional computational complexities because of the dimensionless feature of NN techniques. A great many numerical experiments are executed to verify the accuracy and efficiency of the proposed methods, including two- and three-dimensional evolving and moving interface problems with the complex interface geometries. The comparisons with other NN methods, such as PINN, fully-connected NN, are also made.