Convolutional Neural Networks for Very Low-Dimensional LPV Approximations of Incompressible Navier-Stokes Equations

Abstract

The control of general nonlinear systems is a challenging task in particular for large-scale models as they occur in the semi-discretization of partial differential equations (PDEs) of, say, fluid flow. In order to employ powerful methods from linear numerical algebra and linear control theory, one may embed the nonlinear system in the class of linear parameter varying (LPV) systems. In this work, we show how convolutional neural networks can be used to design LPV approximations of incompressible Navier-Stokes equations. In view of a possibly low-dimensional approximation of the parametrization, we discuss the use of deep neural networks (DNNs) in a semi-discrete PDE context and compare their performance to an approach based on proper orthogonal decomposition (POD). For a streamlined training of DNNs directed to the PDEs in a Finite Element (FEM) framework, we also discuss algorithmical details of implementing the proper norms in general loss functions.

Convolutional Autoencoders, Clustering and POD for Low-dimensional Parametrization of Navier-Stokes Equations

“…To achieve that make use of corresponding interpolation matrix I c such that the CNN input v CNN (t) can be generated by the linear transformation v CNN (t) = I c v(t). We consider the CNN architecture introduced as introduced in [12] that makes use of POD modes for the reconstruction. Thus, the CNN is defined as a model consisting of (1) a nonlinear convolutional encoder µ:…”

“…In this section, we apply the aforementioned models of Section 3 to define low-dimensional LPV approximations of the incompressible Navier-Stokes equation in the ODE form of (2). Where applicable, i.e., where the decoding is linear, we lay out the affine-linear structure of the LPV approximation that see also [12]. We note that we are mostly interested in low-dimensional parameterized approximations of the nonlinear term.…”

“…Accordingly, if a very low-dimensional parametrization of still satisfactory accuracy is wanted, one may need to resort to nonlinear approaches; cp. [12,18,21].…”

A quadratic decoder approach to nonintrusive reduced‐order modeling of nonlinear dynamical systems

“…We consider the flow past a cylinder in 2D at Reynolds number Re = 30 in the startup phase from the associated steady-state Stokes solution and in the fully developed transient regime. The geometrical setup and a snapshot of the developed flow is presented in Figure 1; for a detailed description see [6,Sec. 5].…”

“…In the very low-dimensional regime, techniques that use nonlinear relations between the actual and the reduced coordinates seem to outperform POD; see, e.g., [5,6,9]. While nonlinearities may lead to fewer dimensional coordinates, their inclusion in simulation schemes comes with extra computational efforts.…”