Lift & Learn: A Scientific Machine Learning Framework for Learning Low-Dimensional Models for Nonlinear PDEs

Elizabeth Qian, Massachusetts Institute of Technology

Many systems of engineering interest exhibit complex nonlinear dynamics for which high-fidelity simulation is expensive and the development of accurate low-cost approximations is challenging. Lift & Learn is a new scientific machine learning framework that incorporates physical knowledge to learn accurate, inexpensive reduced models for complex nonlinear systems. The method first ‘lifts’ the known system governing equations to quadratic form; i.e., we transform and augment the system state representation so that the governing equations in the new lifted state variables contain only quadratic nonlinearities in the lifted states. The method then obtains lifted snapshot data by running the original nonlinear high-fidelity simulation and applying the lifting transformations to the resulting data. Proper orthogonal decomposition (POD) is used to reduce the dimension of the lifted snapshot data, and a low-dimensional quadratic model is fit to the reduced lifted data via a least-squares operator inference procedure. Unlike black box methods such as neural nets, our Lift & Learn models respect the structure of the physics in the transformed equations and can be analyzed. We prove that the Lift & Learn models capture the system physics at least as accurately as traditional intrusive model reduction approaches, providing a bridge to the interpretability and analyzability of traditional model reduction. Numerical experiments demonstrate the accuracy and generalizability of the Lift & Learn models.

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