Physics-Aware Reduced-Order Modeling of Nonautonomous Advection-Dominated Problems

We present a variant of dynamic mode decomposition (DMD) for constructing a reduced-order model (ROM) of advection-dominated problems with time-dependent coefficients. Existing DMD strategies, such as the physics-aware DMD and the time-varying DMD, struggle to tackle such problems due to their inherent assumptions of time-invariance and locality. To overcome the compounded difficulty, we propose to learn the evolution of characteristic lines as a nonautonomous system. A piecewise locally time-invariant approximation to the infinite-dimensional Koopman operator is then constructed. We test the accuracy of time-dependent DMD operator on 2d Navier-Stokes equations, and test the Lagrangian-based method on 1- and 2-dimensional advection-diffusion with variable coefficients. Finally, we provide predictive accuracy and perturbation error upper bounds to guide the selection of rank truncation and subinterval sizes.Comment: 27 pages, 21 figure

Learning Nonautonomous Systems via Dynamic Mode Decomposition

We present a data-driven learning approach for unknown nonautonomous dynamical systems with time-dependent inputs based on dynamic mode decomposition (DMD). To circumvent the difficulty of approximating the time-dependent Koopman operators for nonautonomous systems, a modified system derived from local parameterization of the external time-dependent inputs is employed as an approximation to the original nonautonomous system. The modified system comprises a sequence of local parametric systems, which can be well approximated by a parametric surrogate model using our previously proposed framework for dimension reduction and interpolation in parameter space (DRIPS). The offline step of DRIPS relies on DMD to build a linear surrogate model, endowed with reduced-order bases (ROBs), for the observables mapped from training data. Then the offline step constructs a sequence of iterative parametric surrogate models from interpolations on suitable manifolds, where the target/test parameter points are specified by the local parameterization of the test external time-dependent inputs. We present a number of numerical examples to demonstrate the robustness of our method and compare its performance with deep neural networks in the same settings.Comment: arXiv admin note: text overlap with arXiv:2006.02392 by other author