Simplicial Dirac structures as finite analogues of the canonical Stokes-Dirac structure, capturing the topological laws of the system, are defined on simplicial manifolds in terms of primal and dual cochains related by the coboundary operators. These finite-dimensional Dirac structures offer a framework for the formulation of standard input-output finite-dimensional portHamiltonian systems that emulate the behavior of distributed-parameter port-Hamiltonian systems. This paper elaborates on the matrix representations of simplicial Dirac structures and the resulting port-Hamiltonian systems on simplicial manifolds. Employing these representations, we consider the existence of structural invariants and demonstrate how they pertain to the energy shaping of port-Hamiltonian systems on simplicial manifolds.

Discrete exterior geometry approach to structure-preserving discretization of distributedparameter port-Hamiltonian systems Seslija, Marko; van der Schaft, Abraham; Scherpen, Jacquelien M.A. Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Seslija, M., van der Schaft, A., & Scherpen, J. M. A. (2012). Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems. Journal of geometry and physics, 62(6), 1509 -1531 . DOI: 10.1016 /j.geomphys.2012 Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. This paper addresses the issue of structure-preserving discretization of open distributedparameter systems with Hamiltonian dynamics. Employing the formalism of discrete exterior calculus, we introduce a simplicial Dirac structure as a discrete analogue of the Stokes-Dirac structure and demonstrate that it provides a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. The spatial domain, in the continuous theory represented by a finitedimensional smooth manifold with boundary, is replaced by a homological manifold-like simplicial complex and its augmented circumcentric dual. The smooth differential forms, in discrete setting, are mirrored by cochains on the primal and dual complexes, while the discrete exterior derivative is defined to be the coboundary operator. This approach of discrete differential geometry, rather than discretizing the partial differential equations, allows to first discretize the underlying Stokes-Dirac structure and then to impose the corresponding finite-dimensional port-Hamiltonian dynamics. In this manner, a number of important intrinsically topological and geometrical properties of the system are preserved.© 2012 Elsevier B.V. All rights reserved. IntroductionThe purpose of this paper is to propose a sound geometric framework for structure-preserving discretization of distributed-parameter port-Hamiltonian systems. Our approach to time-continuous spatially-discrete port-Hamiltonian theory is based on discrete exterior geometry and as such proceeds ab initio by mirroring the continuous setting. The theory is not merely tied to the goal of discretization but rather aims to offer a sound and consistent framework for defining portHamiltonian dynamics on a discrete manifold which is usually, but not necessarily, o...

We apply deep kernel learning (DKL), which can be viewed as a combination of a Gaussian process (GP) and a deep neural network (DNN), to compression ignition engine emissions and compare its performance to a selection of other surrogate models on the same dataset. Surrogate models are a class of computationally cheaper alternatives to physics-based models. High-dimensional model representation (HDMR) is also briefly discussed and acts as a benchmark model for comparison. We apply the considered methods to a dataset, which was obtained from a compression ignition engine and includes as outputs soot and NO x emissions as functions of 14 engine operating condition variables. We combine a quasi-random global search with a conventional grid-optimization method in order to identify suitable values for several DKL hyperparameters, which include network architecture, kernel, and learning parameters. The performance of DKL, HDMR, plain GPs, and plain DNNs is compared in terms of the root mean squared error (RMSE) of the predictions as well as computational expense of training and evaluation. It is shown that DKL performs best in terms of RMSE in the predictions whilst maintaining the computational cost at a reasonable level, and DKL predictions are in good agreement with the experimental emissions data.

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Inspired by the recent developments in modeling and analysis of reaction networks, we provide a geometric formulation of the reversible reaction networks under the influence of diffusion. Using the graph knowledge of the underlying reaction network, the obtained reaction-diffusion system is a distributed-parameter port-Hamiltonian system on a compact spatial domain. Motivated by the need for computer based design, we offer a spatially consistent discretization of the PDE system and, in a systematic manner, recover a compartmental ODE model on a simplicial triangulation of the spatial domain. Exploring the properties of a balanced weighted Laplacian matrix of the reaction network and the Laplacian of the simplicial complex, we characterize the space of equilibrium points and provide a simple stability analysis on the state space modulo the space of equilibrium points. The paper rules out the possibility of the persistence of spatial patterns for the compartmental balanced reaction-diffusion networks.