9 From the POD-Galerkin method to sparse manifold models [chapter]

2020 Applications  
Reduced-order models are essential for the accurate and efficient prediction, estimation, and control of complex systems. This is especially true in fluid dynamics, where the fully resolved state space may easily contain millions or billions of degrees of freedom. Because these systems typically evolve on a low-dimensional attractor, model reduction is defined by two essential steps: (1) identifying a good state space for the attractor and (2) identifying the dynamics on this attractor. The
more » ... ing method for model reduction in fluids is Galerkin projection of the Navier-Stokes equations onto a linear subspace of modes obtained via proper orthogonal decomposition (POD). However, there are serious challenges in this approach, including truncation errors, stability issues, difficulty handling transients, and mode deformation with changing boundaries and operating conditions. Many of these challenges result from the choice of a linear POD subspace in which to represent the dynamics. In this chapter, we describe an alternative approach, feature-based manifold modeling (FeMM), in which the low-dimensional attractor and nonlinear dynamics are characterized from typical experimental data: time-resolved sensor data and optional nontime-resolved particle image velocimetry (PIV) snapshots. FeMM consists of three steps: First, the sensor signals are lifted to a dynamic feature space. Second, we identify a sparse humaninterpretable nonlinear dynamical system for the feature state based on the sparse identification of nonlinear dynamics (SINDy). Third, if PIV snapshots are available, a local linear mapping from the feature state to the velocity field is performed to reconstruct the full state of the system. We demonstrate this approach, and compare with POD-Galerkin modeling, on the incompressible two-dimensional flow around a circular cylinder. Best practices and perspectives for future research are also included, along with open-source code for this example. 9 Sparse feature-based manifold modeling | 281 Figure 9.1: Illustration of reduced-order modeling. Starting from a direct numerical simulation of the Navier-Stokes equations (left), the dominant spatio-temporal coherent structures are extracted from a set of velocity snapshots (center). The temporal evolution of these structures then provides a simplified representation of the system's dynamics (right) amenable to modeling. turbulence, almost three decades ago. Subsequent POD models have been developed for the transitional boundary layer [83], the mixing layer [111, 114] , the cylinder wake [33, 42] , and the Ahmed body wake [80] , to name only a few. POD-Galerkin modeling is challenging for changing domains [18], changing boundary conditions [45] , and slow deformation of the modal basis [5] . Standard Galerkin projection can also be expected to suffer from stability issues [82, 90, 29] , although including energy-preserving constraints may improve the long-time stability and performance of nonlinear models [7, 31] . POD-Galerkin models tend to be valid for a narrow range of operating conditions, near those of the data set used to generate the POD modes. Transients also pose a challenge to POD modeling. Refs. [77] and [106] demonstrate the ability of a low-dimensional model to reproduce nonlinear transients of the von Kármán vortex shedding past a two-dimensional cylinder, provided the projection basis includes a shift mode quantifying the distortion between the linearly unstable base flow and marginally stable mean flow. These techniques have been extended to include the effect of wall actuation [45, 81] . In addition to the physics-informed Galerkin projection, data-driven modeling approaches are prevalent in fluid dynamics [21, 85] . For example, dynamic mode decomposition (DMD) [50, 86, 55] , the eigensystem realization algorithm (ERA) [51], Koopman analysis [72, 73, 109, 116] , cluster-based reduced-order models [53], NAR-MAX models [15, 95, 120, 44] , and network analysis [76] have all been used to identify dynamical systems models from fluids data, without relying on prior knowledge of the underlying Navier-Stokes equations. DMD models are readily obtained directly from data, and they provide interpretability in terms of flow structures, but the resulting models are linear, and the connection to nonlinear systems is tenuous unless DMD is enriched with nonlinear functions of the data [116, 55] . Neural networks have long been used for flow modeling and control [74, 122, 56, 54] , and recently deep neural networks have been used for Reynolds-averaged turbulence modeling [59] . However, many machine learning methods may be prone to overfitting, have limited
doi:10.1515/9783110499001-009 fatcat:ubjntebufbculkiayjaog344om