Learning low-dimensional models: Nonlinear model reduction via lifting transformations and proper orthogonal decomposition

Karen Willcox
University of Texas at Austin

Analysis of the physical governing equations of a system can reveal variable transformations that transform a general nonlinear model into a model with more structure. In particular, the introduction of auxiliary variables can convert a general nonlinear model to a model with polynomial nonlinearities, a so-called "lifted" system. The lifted model is equivalent to the original model; it uses a change of variables, but introduces no approximations. We present an approach that combines lifting with proper orthogonal decomposition model reduction. The approach uses a data-driven formulation to learn the low-dimensional model from high-fidelity simulation data, but a key aspect of the approach is that the state-space in which the learning is achieved is derived using the problem physics. A key benefit of the approach is that there is no need for additional approximations of the nonlinear terms, in contrast with existing nonlinear model reduction methods that require sparse sampling or hyper-reduction. A second benefit is that the lifted problem structure opens new pathways for rigorous analysis and input-independent model reduction. The method is demonstrated for nonlinear systems of partial differential equations arising in rocket combustion applications.

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