Philipp Schulze

Julius Reiss, Philipp Schulze, Jörn Sesterhenn et al.

Transport-dominated phenomena provide a challenge for common mode-based model reduction approaches. We present a model reduction method, which is suited for these kind of systems. It extends the proper orthogonal decomposition (POD) by introducing time-dependent shifts of the snapshot matrix. The approach, called shifted proper orthogonal decomposition (sPOD), features a determination of the {\it multiple} transport velocities and a separation of these. One- and two-dimensional test examples reveal the good performance of the sPOD for transport-dominated phenomena and its superiority in comparison to the POD.

Robert Altmann, Attila Karsai, Philipp Schulze

We investigate discretization strategies for a recently introduced class of energy-based models. The model class encompasses classical port-Hamiltonian systems, generalized gradient flows, and certain systems with algebraic constraints. Our framework combines existing ideas from the literature and systematically addresses temporal discretization, spatial discretization, and model order reduction, ensuring that all resulting schemes are dissipation-preserving in the sense of a discrete dissipation inequality. For this, we use a Petrov--Galerkin ansatz together with appropriate projections. Numerical results for a nonlinear circuit model, the Cahn--Hilliard equation, and a doubly nonlinear parabolic equation illustrate the effectiveness of the approach.

Philipp Schulze, Benjamin Unger, Christopher Beattie et al.

We present a framework for constructing structured realizations of linear dynamical systems having transfer functions of the form $C(\sum_{k=1}^K h_k(s)A_k)^{-1}B$ where $h_1,h_2,\ldots,h_K$ are prescribed functions that specify the surmised structure of the model. Our construction is data-driven in the sense that an interpolant is derived entirely from measurements of a transfer function. Our approach extends the Loewner realization framework to more general system structure that includes second-order (and higher) systems as well as systems with internal delays. Numerical examples demonstrate the advantages of this approach.

Philipp Schulze, Julius Reiss, Volker Mehrmann

We propose a new algorithm to compute a shifted proper orthogonal decomposition (sPOD) for systems dominated by multiple transport velocities. The sPOD is a recently proposed mode decomposition technique which overcomes the poor performance of classical methods like the proper orthogonal decomposition (POD) for transport-dominated phenomena. This is achieved by identifying the transport directions and velocities and by shifting the modes in space to track the transports. Our new algorithm carries out a residual minimization in which the main computational cost arises from solving a nonlinear optimization problem scaling with the snapshot dimension. We apply the algorithm to snapshot data from the simulation of a pulsed detonation combuster and observe that very few sPOD modes are sufficient to obtain a good approximation. For the same accuracy, the common POD needs ten times as many modes and, in contrast to the sPOD modes, the POD modes do not reflect the moving front profiles properly.

Philipp Schulze, Benjamin Unger

We introduce a novel model order reduction method for large-scale linear switched systems (LSS) where the coefficient matrices are affected by a low-rank switching. The key idea is to replace the LSS by a non-switched system with extended input and output vectors - called the envelope system - which is able to reproduce the dynamical behavior of the original LSS by applying a certain feedback law. The envelope system can be reduced using standard model order reduction schemes and then transformed back to an LSS. Furthermore, we present an upper bound for the output error of the reduced-order LSS and show how to preserve quadratic Lyapunov stability. The approach is tested by means of various numerical examples demonstrating the efficacy of the presented method.

Paul Schwerdtner, Philipp Schulze, Jules Berman et al.

This work focuses on the conservation of quantities such as Hamiltonians, mass, and momentum when solution fields of partial differential equations are approximated with nonlinear parametrizations such as deep networks. The proposed approach builds on Neural Galerkin schemes that are based on the Dirac--Frenkel variational principle to train nonlinear parametrizations sequentially in time. We first show that only adding constraints that aim to conserve quantities in continuous time can be insufficient because the nonlinear dependence on the parameters implies that even quantities that are linear in the solution fields become nonlinear in the parameters and thus are challenging to discretize in time. Instead, we propose Neural Galerkin schemes that compute at each time step an explicit embedding onto the manifold of nonlinearly parametrized solution fields to guarantee conservation of quantities. The embeddings can be combined with standard explicit and implicit time integration schemes. Numerical experiments demonstrate that the proposed approach conserves quantities up to machine precision.