Data-driven Model Reduction For Multiscale Parabolic Problems

In this doctoral dissertation,we mainly study data-driven model reduction for multiscale parabolic problems.For multiscale parabolic problems,when the model information is completely known or partially unknown,we aim to establish a datadriven model reduction method to improve the efficiency of numerical simulation and predict the future state.In physics and engineering applications,many phenomena can be described by multiscale parabolic equations.For example,the flow of an isentropic gas through porous media.Capturing effective features in computational multiscale model is usually done in a fine scale when using the traditional finite element approach,which requires extensive computation resources.So reduced models are necessary to compute the multiscale problems.In this paper,we present a constraint energy minimizing generalized multiscale finite element method(CEM-GMsFEM)for parabolic equations with multiscale coefficients,and get a reduced-order model,which significantly reduces the computational cost.However,there exist a couple of challenges for CEM-GMsFEM solving the nonlinear multiscale parabolic problems.First,it is required to evaluate the nonlinear functionals and compute the residual and the Jacobian on a fine grid.This is computationally expensive.Another challenge is that iterative methods(eg,Piccard iteration and Newton method)are usually used to solve the nonlinear problems.Thus,it is necessary to construct an effective reduced model for nonlinear multiscale parabolic problems.For complex nonlinear multiscale problems,the parameters and structure of the model may be unknown,so it is difficult to directly construct the mathematical model.With the development of science and technology,a large amount of experimental data can be obtained.Therefore,when the specific form of the model is unknown,it is necessary and challenging to discover potential models from the data.For linear multiscale parabolic problems,when the model is known,we use the CEM-GMsFEM to establish its reduced order model,and rigorously analyze its convergence.The convergence rate is characterized by the coarse grid size and the eigenvalue decay of local spectral problems,but is independent of the high contrast of the media.A few numerical results for porous media applications are presented to confirm the theoretical findings and demonstrate the performance of the approach.For nonlinear multiscale parabolic problems,when the model is known,we develop a data-driven model reduction method by integrating CEM-GMsFEM with dynamic mode decomposition.The proposed method can establish a coarse linear surrogate model for nonlinear multiscale parabolic equations,which avoids the nonlinear solver in the fine space.In this paper,we introduce two different observations:fine scale observation and coarse scale observation.To show the performance of the proposed method using the different observations,we present a few numerical results for the nonlinear multiscale parabolic problems in heterogeneous porous media.For complex nonlinear multiscale problems,when the specific form of the model is unknown,we can get a large number of coarse scale data through experiments.In this paper,we suppose that the truth multiscale model is unknown,but multiscale basis functions and some data are available.With the help of Koopman operator theory,we propose a data-driven method to predict the future state.However,it is usually very challenging to choose a set of suitable observation functions spanning the Koopman invariant subspace.We use the deep learning method to learn these observation functions.In order to return to the model’s state space from the observation function space,we also learn a reconstruction operator.A few numerical examples are presented to show that the effectiveness of learning multiscale models and the long-time prediction.