Projection-Based Model Reduction with Dynamically Transformed Modes

Abstract

We propose a new model reduction framework for problems that exhibit transport phenomena. As in the moving finite element method (MFEM), our method employs time-dependent transformation operators and, especially, generalizes MFEM to arbitrary basis functions. The new framework is suitable to obtain a low-dimensional approximation with small errors even in situations where classical model order reduction techniques require much higher dimensions for a similar approximation quality. Analogously to the MFEM framework, the reduced model is designed to minimize the residual, which is also the basis for an a-posteriori error bound. Moreover, since the dependence of the transformation operators on the reduced state is nonlinear, the resulting reduced order model is obtained by projecting the original evolution equation onto a nonlinear manifold. Furthermore, for a special case, we show a connection between our approach and the method of freezing, which is also known as symmetry reduction. Besides the construction of the reduced order model, we also analyze the problem of finding optimal basis functions based on given data of the full order solution. Especially, we show that the corresponding minimization problem has a solution and reduces to the proper orthogonal decomposition of transformed data in a special case. Finally, we demonstrate the effectiveness of our method with several analytical and numerical examples.

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• Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations
• Manifold Approximations via Transported Subspaces: Model reduction for transport-dominated problems
• Efficient Wildland Fire Simulation via Nonlinear Model Order Reduction
• Depth separation for reduced deep networks in nonlinear model reduction: Distilling shock waves in nonlinear hyperbolic problems
• A robust shifted proper orthogonal decomposition: Proximal methods for decomposing flows with multiple transports
• Decomposition of flow data via gradient-based transport optimization
• Model order reduction via Lie groups