Constructing reduced model for complex physical systems via interpolation and neural networks

The work studies model reduction method for nonlinear systems based on proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM). Instead of using the classical DEIM to directly approximate the nonlinear term of a system, our approach extracts the main part of the nonlinear term with a linear approximation before approximating the residual with the DEIM. We construct the linear term by Taylor series expansion and dynamic mode decomposition (DMD), respectively, so as to obtain a more accurate reconstruction of the nonlinear term. In addition, a novel error prediction model is devised for the POD-DEIM reduced systems by employing neural networks with the aid of error data. The error model is cheaply computable and can be adopted as a remedy model to enhance the reduction accuracy. Finally, numerical experiments are performed on two nonlinear problems to show the performance of the proposed method.