Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism

Data-Driven Model Reduction for Stochastic Burgers Equations

“…In particular, the term approximates F based on information up to time n (e.g., the conditional expectation), and the noise aims to statistically represent the residual of the approximation. Since the truncation error depends on the past history of the low wavenumber modes, and as suggested by the Mori–Zwanzig formalism [ 6 , 7 ], we make depend on the trajectory of the state process, as well as the trajectories and : For simplicity, we assume the noise to be iid Gaussian, and the resulted time series model in ( 10 ) is a nonlinear autoregression moving average model (NARMA) [ 13 , 45 , 46 ].…”

“…These criteria are by no means exhaustive. Other methods include Bayesian information criterion (BIC, see, e.g., [ 50 ]), and the error reduction ratio [ 51 ] may be applied, but in our experience, they provide limited help for the selection of reduced models [ 7 , 14 , 46 ].…”

“…We have the following observations from the previous sections: Space dimension reduction, memory length of the reduced model and the stochastic force are closely related. As suggested by the discrete Mori–Zwanzig formalism for random dynamics (see e.g., [ 7 ]), space dimension reduction would lead to non-Markovian closure models. Figure 1 suggests that a proper medium length of the memory leads to best NAR model.…”

“…Consisting of low-dimensional resolved variables, the closure model must take into account the non-negligible effects of unresolved variables so as to capture both the short-time dynamics and large-time statistics. As suggested by the Mori–Zwanzig formalism [ 5 , 6 , 7 ], trajectory-wise approximation is no longer appropriate, and the approximation is in a statistical sense. That is, the reduced model aims to generate a process that approximates the target process in distribution, or at least, reproduce the key statistics and dynamics for the quantities of interest.…”

“…The past decades witness revolutionary developments of data-driven strategies, ranging from parametric models (see, e.g., [ 8 , 9 , 10 , 11 , 12 , 13 , 14 ] and the references therein) to nonparametric and machine learning methods (see, e.g., [ 15 , 16 , 17 , 18 ]). These developments demand a systematic understanding of model reduction from the perspectives of dynamical systems (see, e.g., [ 7 , 19 , 20 ]), numerical approximation [ 21 , 22 ], and statistical learning [ 17 , 23 ].…”