Nonlinear Model Reduction Based on the Mori-Zwanzig Scheme and Partial Least Squares

Abstract

The proper orthogonal decomposition and the Galerkin projection are widely used methods for solving the model reduction problems of complex nonlinear systems. However, only a part of basis function modes are extracted with these methods to construct the reduced systems, which usually makes the reduced systems inaccurate. For this issue a method was proposed to efficiently correct the errors of the reduced systems. First, the Mori-Zwanzig scheme was employed to analyze the errors of the reduced systems, with the theoretical form of the error model and the effective predictive variables obtained. Then, the error prediction model was built by means of the partial least square method to construct the multiple regression model between the predictive variables and the system errors. The constructed error prediction model was directly embedded into the original reduced system, to get a modified reduced system formally equivalent to the model obtained with the Petrov-Galerkin projection on the right side of the original model. The error estimation of the modified reduced system was given. Numerical results illustrate that, the proposed method can improve the stability and accuracy of the reduced systems effectively, and has high computation efficiency.