arXiv subjects

Benjamin Unger

Publications and source records attributed to Benjamin Unger.

At least 37 records · Page 2Linked to original sources

arXiv2023

Port-Hamiltonian Dynamic Mode Decomposition

We present a novel physics-informed system identification method to construct a passive linear time-invariant system. In more detail, for a given quadratic energy functional, measurements of the input, state, and output of a system in the time domain, we find a realization that approximates the data well while guaranteeing that the energy functional satisfies a dissipation inequality. To this end, we use the framework of port-Hamiltonian (pH) systems and modify the dynamic mode decomposition, respectively operator inference, to be feasible for continuous-time pH systems. We propose an iterative numerical method to solve the corresponding least-squares minimization problem. We construct an effective initialization of the algorithm by studying the least-squares problem in a weighted norm, for which we present the analytical minimum-norm solution. The efficiency of the proposed method is demonstrated with several numerical examples.

math.DS↗

arXiv2022

Robust Recurrent Neural Network to Identify Ship Motion in Open Water with Performance Guarantees -- Technical Report

Recurrent neural networks are capable of learning the dynamics of an unknown nonlinear system purely from input-output measurements. However, the resulting models do not provide any stability guarantees on the input-output mapping. In this work, we represent a recurrent neural network as a linear time-invariant system with nonlinear disturbances. By introducing constraints on the parameters, we can guarantee finite gain stability and incremental finite gain stability. We apply this identification method to learn the motion of a four-degrees-of-freedom ship that is moving in open water and compare it against other purely learning-based approaches with unconstrained parameters. Our analysis shows that the constrained recurrent neural network has a lower prediction accuracy on the test set, but it achieves comparable results on an out-of-distribution set and respects stability conditions.

cs.LG↗

arXiv2022

Towards a modeling class for port-Hamiltonian systems with time-delay

The framework of port-Hamiltonian (pH) systems is a powerful and broadly applicable modeling paradigm. In this paper, we extend the scope of pH systems to time-delay systems. Our definition of a delay pH system is motivated by investigating the Kalman-Yakubovich-Popov inequality on the corresponding infinite-dimensional operator equation. Moreover, we show that delay pH systems are passive and closed under interconnection. We describe an explicit way to construct a Lyapunov-Krasovskii functional and discuss implications for delayed feedback.

math.DS↗

arXiv2022

Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNs

Prediction error quantification in machine learning has been left out of most methodological investigations of neural networks, for both purely data-driven and physics-informed approaches. Beyond statistical investigations and generic results on the approximation capabilities of neural networks, we present a rigorous upper bound on the prediction error of physics-informed neural networks. This bound can be calculated without the knowledge of the true solution and only with a priori available information about the characteristics of the underlying dynamical system governed by a partial differential equation. We apply this a posteriori error bound exemplarily to four problems: the transport equation, the heat equation, the Navier-Stokes equation and the Klein-Gordon equation.

cs.LG↗

arXiv2022

Certified machine learning: A posteriori error estimation for physics-informed neural networks

Physics-informed neural networks (PINNs) are one popular approach to incorporate a priori knowledge about physical systems into the learning framework. PINNs are known to be robust for smaller training sets, derive better generalization problems, and are faster to train. In this paper, we show that using PINNs in comparison with purely data-driven neural networks is not only favorable for training performance but allows us to extract significant information on the quality of the approximated solution. Assuming that the underlying differential equation for the PINN training is an ordinary differential equation, we derive a rigorous upper limit on the PINN prediction error. This bound is applicable even for input data not included in the training phase and without any prior knowledge about the true solution. Therefore, our a posteriori error estimation is an essential step to certify the PINN. We apply our error estimator exemplarily to two academic toy problems, whereof one falls in the category of model-predictive control and thereby shows the practical use of the derived results.

cs.LG↗

arXiv2022

Control of port-Hamiltonian differential-algebraic systems and applications

The modeling framework of port-Hamiltonian descriptor systems and their use in numerical simulation and control are discussed. The structure is ideal for automated network-based modeling since it is invariant under power-conserving interconnection, congruence transformations, and Galerkin projection. Moreover, stability and passivity properties are easily shown. Condensed forms under orthogonal transformations present easy analysis tools for existence, uniqueness, regularity, and numerical methods to check these properties. After recalling the concepts for general linear and nonlinear descriptor systems, we demonstrate that many difficulties that arise in general descriptor systems can be easily overcome within the port-Hamiltonian framework. The properties of port-Hamiltonian descriptor systems are analyzed, time-discretization, and numerical linear algebra techniques are discussed. Structure-preserving regularization procedures for descriptor systems are presented to make them suitable for simulation and control. Model reduction techniques that preserve the structure and stabilization and optimal control techniques are discussed. The properties of port-Hamiltonian descriptor systems and their use in modeling simulation and control methods are illustrated with several examples from different physical domains. The survey concludes with open problems and research topics that deserve further attention.

math.DS↗

arXiv2021

Physics-informed Neural Networks-based Model Predictive Control for Multi-link Manipulators

We discuss nonlinear model predictive control (NMPC) for multi-body dynamics via physics-informed machine learning methods. Physics-informed neural networks (PINNs) are a promising tool to approximate (partial) differential equations. PINNs are not suited for control tasks in their original form since they are not designed to handle variable control actions or variable initial values. We thus present the idea of enhancing PINNs by adding control actions and initial conditions as additional network inputs. The high-dimensional input space is subsequently reduced via a sampling strategy and a zero-hold assumption. This strategy enables the controller design based on a PINN as an approximation of the underlying system dynamics. The additional benefit is that the sensitivities are easily computed via automatic differentiation, thus leading to efficient gradient-based algorithms. Finally, we present our results using our PINN-based MPC to solve a tracking problem for a complex mechanical system, a multi-link manipulator.

math.OC↗

arXiv2021

Identification of linear time-invariant systems with Dynamic Mode Decomposition

Dynamic mode decomposition (DMD) is a popular data-driven framework to extract linear dynamics from complex high-dimensional systems. In this work, we study the system identification properties of DMD. We first show that DMD is invariant under linear transformations in the image of the data matrix. If, in addition, the data is constructed from a linear time-invariant system, then we prove that DMD can recover the original dynamics under mild conditions. If the linear dynamics are discretized with a Runge-Kutta method, then we further classify the error of the DMD approximation and detail that for one-stage Runge-Kutta methods even the continuous dynamics can be recovered with DMD. A numerical example illustrates the theoretical findings.

math.NA↗

arXiv2021

Passivity preserving model reduction via spectral factorization

We present a novel model-order reduction (MOR) method for linear time-invariant systems that preserves passivity and is thus suited for structure-preserving MOR for port-Hamiltonian (pH) systems. Our algorithm exploits the well-known spectral factorization of the Popov function by a solution of the Kalman-Yakubovich-Popov (KYP) inequality. It performs MOR directly on the spectral factor inheriting the original system's sparsity enabling MOR in a large-scale context. Our analysis reveals that the spectral factorization corresponding to the minimal solution of an associated algebraic Riccati equation is preferable from a model reduction perspective and benefits pH-preserving MOR methods such as a modified version of the iterative rational Krylov algorithm (IRKA). Numerical examples demonstrate that our approach can produce high-fidelity reduced-order models close to (unstructured) $\mathcal{H}_2$-optimal reduced-order models.

math.DS↗

arXiv2021

Decomposition of flow data via gradient-based transport optimization

We study an optimization problem related to the approximation of given data by a linear combination of transformed modes. In the simplest case, the optimization problem reduces to a minimization problem well-studied in the context of proper orthogonal decomposition. Allowing transformed modes in the approximation renders this approach particularly useful to compress data with transported quantities, which are prevalent in many flow applications. We prove the existence of a solution to the infinite-dimensional optimization problem. Towards a numerical implementation, we compute the gradient of the cost functional and derive a suitable discretization in time and space. We demonstrate the theoretical findings with three challenging numerical examples.

math.OC↗

arXiv2021

Efficient Wildland Fire Simulation via Nonlinear Model Order Reduction

We propose a new hyper-reduction method for a recently introduced nonlinear model reduction framework based on dynamically transformed basis functions and especially well-suited for advection-dominated systems. Furthermore, we discuss applying this new method to a wildland fire model whose dynamics feature traveling combustion waves and local ignition and is thus challenging for classical model reduction schemes based on linear subspaces. The new hyper-reduction framework allows us to construct parameter-dependent reduced-order models (ROMs) with efficient offline/online decomposition. The numerical experiments demonstrate that the ROMs obtained by the novel method outperform those obtained by a classical approach using the proper orthogonal decomposition and the discrete empirical interpolation method in terms of run time and accuracy.

math.NA↗

arXiv2021

Port-Hamiltonian formulations of poroelastic network models

We investigate an energy-based formulation of the two-field poroelasticity model and the related multiple-network model as they appear in geosciences or medical applications. We propose a port-Hamiltonian formulation of the system equations, which is beneficial for preserving important system properties after discretization or model-order reduction. For this, we include the commonly omitted second-order term and consider the corresponding first-order formulation. The port-Hamiltonian formulation of the quasi-static case is then obtained by (formally) setting the second-order term zero. Further, we interpret the poroelastic equations as an interconnection of a network of submodels with internal energies, adding a control-theoretic understanding of the poroelastic equations.

math.DS↗

arXiv2021

Parametric model reduction via rational interpolation along parameters

We present a novel projection-based model reduction framework for parametric linear time-invariant systems that allows interpolating the transfer function at a given frequency point along parameter-dependent curves as opposed to the standard approach where transfer function interpolation is achieved for a discrete set of parameter and frequency samples. We accomplish this goal by using parameter-dependent projection spaces. Our main result shows that for holomorphic system matrices, the corresponding interpolatory projection spaces are also holomorphic. The coefficients of the power series representation of the projection spaces can be computed iteratively using standard methods. We illustrate the analysis on three numerical examples.

math.NA↗

arXiv2020

Projection-Based Model Reduction with Dynamically Transformed Modes

We propose a new model reduction framework for problems that exhibit transport phenomena. As in the moving finite element method (MFEM), our method employs time-dependent transformation operators and, especially, generalizes MFEM to arbitrary basis functions. The new framework is suitable to obtain a low-dimensional approximation with small errors even in situations where classical model order reduction techniques require much higher dimensions for a similar approximation quality. Analogously to the MFEM framework, the reduced model is designed to minimize the residual, which is also the basis for an a-posteriori error bound. Moreover, since the dependence of the transformation operators on the reduced state is nonlinear, the resulting reduced order model is obtained by projecting the original evolution equation onto a nonlinear manifold. Furthermore, for a special case, we show a connection between our approach and the method of freezing, which is also known as symmetry reduction. Besides the construction of the reduced order model, we also analyze the problem of finding optimal basis functions based on given data of the full order solution. Especially, we show that the corresponding minimization problem has a solution and reduces to the proper orthogonal decomposition of transformed data in a special case. Finally, we demonstrate the effectiveness of our method with several analytical and numerical examples.

math.NA↗

arXiv2020

The Pantelides algorithm for delay differential-algebraic equations

We present a graph-theoretical approach that can detect which equations of a delay differential-algebraic equation (DDAE) need to be differentiated or shifted to construct a solution of the DDAE. Our approach exploits the observation that differentiation and shifting are very similar from a structural point of view, which allows us to generalize the Pantelides algorithm for differential-algebraic equations to the DDAE setting. The primary tool for the extension is the introduction of equivalence classes in the graph of the DDAE, which also allows us to derive a necessary and sufficient criterion for the termination of the new algorithm.

math.DS↗

arXiv2020

Delay differential-algebraic equations in real-time dynamic substructuring

Hybrid numerical-experimental testing is a standard approach for complex dynamical structures that are, on the one hand, not easy to model due to complexity and parameter uncertainty and, on the other hand, too expensive for full-scale experiments. The main idea is to subdivide the structure in a part that can be accurately simulated with numerical methods and an experimental component. The numerical simulation and the experiment are coupled in real-time by a so-called transfer system, which induces a time-delay into the system. In this paper, we study the solvability of the resulting hybrid numerical-experimental system, which is typically described by a set of nonlinear delay differential-algebraic equations, and extend existing results from the literature to this case.

math.DS↗

arXiv2019

Semi-explicit discretization schemes for weakly-coupled elliptic-parabolic problems

We prove first-order convergence of the semi-explicit Euler scheme combined with a finite element discretization in space for elliptic-parabolic problems which are weakly coupled. This setting includes poroelasticity, thermoelasticity, as well as multiple-network models used in medical applications. The semi-explicit approach decouples the system such that each time step requires the solution of two small and well-structured linear systems rather than the solution of one large system. The decoupling improves the computational efficiency without decreasing the convergence rates. The presented convergence proof is based on an interpretation of the scheme as an implicit method applied to a constrained partial differential equation with delay term. Here, the delay time equals the used step size. This connection also allows a deeper understanding of the weak coupling condition, which we accomplish to quantify explicitly.

math.NA↗

arXiv2019

From Time-Domain Data to Low-Dimensional Structured Models

We present a framework for constructing a structured realization of a linear time-invariant dynamical system solely from a discrete sampling of an input and output trajectory of the system. We estimate the transfer function of the original model at selected frequencies using a modification of the empirical transfer function estimation that was recently presented in [Peherstorfer, Gugercin, Willcox, SIAM J. Sci. Comput., 39(5):2152-2178, 2017]. Our realization interpolates the transfer function estimates and can be seen as a generalization of the Loewner framework to structured systems. We demonstrate the presented framework by means of a delay example.

math.OC↗