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Developing accurate, efficient, and robust closure models is essential in theconstruction of reduced order models (ROMs) for realistic nonlinear systems,which generally require drastic ROM mode truncations. We propose a deepresidual neural network (ResNet) closure learning framework for ROMs ofnonlinear systems. The novel ResNet-ROM framework consists of two steps: (i) Inthe first step, we use ROM projection to filter the given nonlinear PDE andconstruct a filtered ROM. This filtered ROM is low-dimensional, but is notclosed (because of the PDE nonlinearity). (ii) In the second step, we useResNet to close the filtered ROM, i.e., to model the interaction between theresolved and unresolved ROM modes. We emphasize that in the new ResNet-ROMframework, data is used only to complement classical physical modeling (i.e.,only in the closure modeling component), not to completely replace it. We alsonote that the new ResNet-ROM is built on general ideas of spatial filtering anddeep learning and is independent of (restrictive) phenomenological arguments,e.g., of eddy viscosity type. The numerical experiments for the 1D Burgersequation show that the ResNet-ROM is significantly more accurate than thestandard projection ROM. The new ResNet-ROM is also more accurate andsignificantly more efficient than other modern ROM closure models.

Closure Learning for Nonlinear Model Reduction Using Deep Residual Neural Network