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Projection-based model reduction has become a popular approach to reduce thecost associated with integrating large-scale dynamical systems so they can beused in many-query settings such as optimization and uncertaintyquantification. For nonlinear systems, significant cost reduction is onlypossible with an additional layer of approximation to reduce the computationalbottleneck of evaluating the projected nonlinear terms. Prevailing methods toapproximate the nonlinear terms are code intrusive, potentially requiring yearsof development time to integrate into an existing codebase, and have been knownto lack parametric robustness.  This work develops a non-intrusive method to efficiently and accuratelyapproximate the expensive nonlinear terms that arise in reduced nonlineardynamical system using deep neural networks. The neural network is trainedusing only the simulation data used to construct the reduced basis andevaluations of the nonlinear terms at these snapshots. Once trained, the neuralnetwork-based reduced-order model only requires forward and backwardpropagation through the network to evaluate the nonlinear term and itsderivative, which are used to integrate the reduced dynamical system at a newparameter configuration. We provide two numerical experiments---the dynamicalsystems result from the semi-discretization of parametrized, nonlinear,hyperbolic partial differential equations---that show, in addition tonon-intrusivity, the proposed approach provides more stable and accurateapproximations to each dynamical system across a large number of training andtesting points than the popular empirical interpolation method.

Non-intrusive model reduction of large-scale, nonlinear dynamical systems using deep learning