Hybrid explicit–implicit learning for multiscale problems with time dependent source

The splitting method is a powerful for solving partial differential equations. Various methods have been designed to separate different physics, nonlinearities, and so on. Recently, new approach has proposed where some degrees of freedom are handled implicitly while other explicitly. As result, the scheme contains two equations, one implicit explicit. stability this studied. It was shown that time step scales as coarse spatial mesh size, which can provide significant computational advantage. However, solution part still be expensive, especially nonlinear problems. In paper, we introduce modified machine learning algorithms replace algorithm. These first introduced in ‘HEI: Hybrid Explicit-Implicit Learning For Multiscale Problems’, homogeneous source term considered along with Transformer, neural network predict future dynamics. consider time-dependent terms generalization previous work. Moreover, use whole history train network. equations more complicated solve, design it based on training. Furthermore, compute explicit using our strategy. addition, Proper Orthogonal Decomposition model reduction learning. saving without sacrificing accuracy. We present three numerical examples show stable accurate.