Dynamic mode decomposition (DMD) is a model reduction algorithm developed by Schmid (Schmid 2010). Since then has emerged as a powerful tool for analyzing the dynamics of nonlinear systems. It is used for a data-driven model simplification based on spatiotemporal coherent structures. DMD relies only on the high-fidelity measurements, like experimental data and numerical simulations, so it is an equation-free algorithm. Its popularity is also due to the fact that it does not make any assumptions about the underlying system. See (Kutz et al. 2016) for a comprehensive overview of the algorithm and its connections to the Koopman-operator analysis, initiated in (Koopman 1931), along with examples in computational fluid dynamics. In the last years many variants arose, such as multiresolution DMD, compressed DMD, forward backward DMD, and higher order DMD among others, in order to deal with noisy data, big dataset, or spurius data for example.

In the field of parametric partial differential equations, shape optimization represents a challenging problem due to the required computational resources. In this contribution, a data-driven framework involving multiple reduction techniques is proposed to reduce such computational burden. Proper orthogonal decomposition (POD) and active subspace genetic algorithm (ASGA) are applied for a dimensional reduction of the original (high fidelity) model and for an efficient genetic optimization based on active subspace property. The parameterization of the shape is applied directly to the computational mesh, propagating the generic deformation map applied to the surface (of the object to optimize) to the mesh nodes using a radial basis function (RBF) interpolation. Thus, topology and quality of the original mesh are preserved, enabling application of POD-based reduced order modeling techniques, and avoiding the necessity of additional meshing steps. Model order reduction is performed coupling POD and Gaussian process regression (GPR) in a data-driven fashion. The framework is validated on a benchmark ship.

Model Order Reduction, roughly speaking, can be summarized in two parts: an offline and an online part. In the first, one constructs the database of solutions, or snapshots, for proper selected parameters. In the latter the database is used for a fast evaluation of the quatity of interest for a new parameter (see for example Schilders, Van der Vorst, and Rommes 2008; Hesthaven et al. 2016). Choices can be made either for the selection of the parameters in the construction of the database or on how the database is used to approximate the manifold of the snapshots.

Dynamic mode decomposition (DMD) has recently become a popular tool for the nonintrusive analysis of dynamical systems. Exploiting proper orthogonal decomposition (POD) as a dimensionality reduction technique, DMD is able to approximate a dynamical system as a sum of spatial bases evolving linearly in time, thus enabling a better understanding of the physical phenomena and forecasting of future time instants. In this work we propose an extension of DMD to parameterized dynamical systems, focusing on the future forecasting of the output of interest in a parametric context. Initially all the snapshots—for different parameters and different time instants—are projected to a reduced space; then DMD, or one of its variants, is employed to approximate reduced snapshots for future time instants. Exploiting the low dimension of the reduced space, we then combine the predicted reduced snapshots using regression techniques, thus enabling the possibility of approximating any untested parametric configuration in the future. This paper depicts in detail the algorithmic core of this method; we also present and discuss three test cases for our algorithm: a simple dynamical system with a linear parameter dependency, a heat problem with nonlinear parameter dependency, and a fluid dynamics problem with nonlinear parameter dependency.

This work describes the implementation of a data-driven approach for the reduction of the complexity of parametrical partial differential equations (PDEs) employing Proper Orthogonal Decomposition (POD) and Gaussian Process Regression (GPR). This approach is applied initially to a literature case, the simulation of the Stokes problem, and in the following to a real-world industrial problem, within a shape optimization pipeline for a naval engineering problem.