Ruminated Tensor Decomposition algorithm for solving inviscid Burgers' equation

In this paper, we propose a Ruminated Tensor Decomposition (RTD) algorithm for solving inviscid Burgers' equation. A discrete loss function is designed from the residual of Godunov's method. While Proper Generalization Decomposition (PGD) suffers from slow convergence in terms of mode number, and Tensor Decomposition (TD) suffers from sensitivity to random initial guesses of minimization iterations, RTD turns out to reproduce Godunov's method with a moderate number of modes. The pretraining of initial guesses for every three additional modes by PGD makes RTD a robust algorithm. Numerical examples justify the effectiveness of RTD. It is readily extended to general nonlinear conservation laws, as illustrated by two examples: the Euler equations for gas dynamics, and an isothermal flow in two space dimensions.