CAREER: Formulations, Theory, and Algorithms for Nonlinear Model Reduction in Transport-Dominated Systems

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Often in science and engineering, systems need to be numerically simulated with multiple different parameter configurations, inputs, and coefficients to gain a deeper understanding of the underlying physical phenomena. Model reduction (complexity reduction) has become a ubiquitous tool that makes these repeated numerical simulations computationally tractable by exploiting intrinsic low-rank structures in the system dynamics. However, most of today's model reduction methods exploit linear structures that are present only in a limited class of problems, which excludes, for example, important problems with transport phenomena that describe waves and other moving coherent structures. This severely limits the use of today's model reduction methods in a wide range of applications from multi-phase problems in materials science to combustion in engineering to pattern formation in biology. The goal of this project is to develop new types of model reduction methods at the intersection of machine learning, numerical analysis, and scientific computing that exploit nonlinear structures to make tractable repeated simulations of transport phenomena to support scientists and engineers.The nonlinear model reduction techniques proposed in this project will derive reduced solutions on manifolds as compositions of low-rank approximations. This means that the traditional linear vector-space structure of today's model reduction methods is given up in favor of increased expressiveness ("breaking the Kolmogorov barrier"). The work packages are as follows: (i) New approximation theory concepts will be developed to show for prototypical transport-dominated partial differential equations that the proposed reduced models break the Kolmogorov barrier of today's linear model reduction methods, meaning the best-approximation error of the proposed nonlinear models decays exponentially faster than the best-approximation error of linear models. (ii) Numerical methods will be proposed that exploit the nonlinear low rankness of the composed approximations of the proposed nonlinear reduced models to achieve speedups compared to traditional models. The nonlinear solutions will be numerically computed by integrating in time the governing equations to ensure that the nonlinear reduced models generalize well to new inputs/parameters, in contrast to purely relying on data-fitted maps without going back to the governing equations. (iii) Two real-world applications in collaborations with domain scientists in storm-surge forecasting and additive manufacturing will demonstrate that the proposed techniques have the potential to impact a wide range of applications that currently are limited by prohibitively expensive simulations of transport phenomena. (iv) Integrated education components such as interactive online courses via flipped classrooms, tutorials for engineers, and outreach activities building on the two real-world applications in storm-surge forecasting and additive manufacturing will educate a broad audience and ensure relevance in practice.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.