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    On dissipativity of cross-entropy loss in training ResNets — A turnpike towards architecture search
    (Elsevier, 2026-04-01)
    Püttschneider, Jens
    ;
    Faulwasser, Timm
    The training of ResNets and neural ODEs can be formulated and analyzed from the perspective of optimal control. This paper proposes a dissipative formulation of the training of ResNets and neural ODEs for classification problems. Specifically, we consider a variant of the cross-entropy (label smoothing) as a loss function and as a regularization in the stage cost. Based on our dissipative formulation of the training, we prove that the training OCPs for ResNets and neural ODEs alike exhibit the turnpike phenomenon. We illustrate this finding with numerical results for the two spirals and MNIST datasets. Crucially, our training formulation ensures that the transformation of the data from input to output is achieved in the first layers. In the following layers, which constitute the turnpike, the data remains at an equilibrium state and therefore these layers do not contribute to the transformation learned. In principle, these layers can be pruned after training, resulting in a network with only the necessary number of layers thus simplifying tuning of hyperparameters.
    Publicationtype: Journal Article
      34  17
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    Optimal control of port-Hamiltonian systems: energy, entropy, and exergy
    (Elsevier, 2024-10-26)
    Philipp, Friedrich
    ;
    Schaller, Manuel
    ;
    Worthmann, Karl
    ;
    Faulwasser, Timm
    ;
    Maschke, Bernhard
    We consider irreversible and coupled reversible–irreversible nonlinear port-Hamiltonian systems and the respective sets of thermodynamic equilibria. In particular, we are concerned with optimal state transitions and output stabilization on finite-time horizons. We analyze a class of optimal control problems, where the performance functional can be interpreted as a linear combination of energy supply, entropy generation, or exergy supply. Our results establish the integral turnpike property towards the set of thermodynamic equilibria providing a rigorous connection of optimal system trajectories to optimal steady states. Throughout the paper, we illustrate our findings by means of two examples: a network of heat exchangers and a gas-piston system.
    Publicationtype: Journal Article
      45  33
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    Manifold turnpikes, trims, and symmetries
    (Springer, 2022-12)
    Faulwasser, Timm
    ;
    Flaßkamp, Kathrin
    ;
    Ober-Blöbaum, Sina
    ;
    Schaller, Manuel
    ;
    Worthmann, Karl
    Classical turnpikes correspond to optimal steady states which are attractors of infinite-horizon optimal control problems. In this paper, motivated by mechanical systems with symmetries, we generalize this concept to manifold turnpikes. Specifically, the necessary optimality conditions projected onto a symmetry-induced manifold coincide with those of a reduced-order problem defined on the manifold under certain conditions. We also propose sufficient conditions for the existence of manifold turnpikes based on a tailored notion of dissipativity with respect to manifolds. Furthermore, we show how the classical Legendre transformation between Euler–Lagrange and Hamilton formalisms can be extended to the adjoint variables. Finally, we draw upon the Kepler problem to illustrate our findings.
    Publicationtype: Journal Article
    Scopus© Citations 7  21  47
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    Model compendium, data, and optimization benchmarks for sector-coupled energy systems
    (Elsevier, 2020-04-06)
    Sass, Susanne
    ;
    Faulwasser, Timm
    ;
    Hollermann, Dinah Elena
    ;
    Kappatou, Chrysoula Dimitra
    ;
    Sauer, Dominique
    Decarbonization and defossilization of energy supply as well as increasing decentralization of energy generation necessitate the development of efficient strategies for design and operation of sector-coupled energy systems. Today, design and operation of process and energy systems rely on powerful numerical methods, in particular, optimization methods. The development of such methods benefits from reproducible benchmarks including transparent model equations and complete input data sets. However, to the authors’ best knowledge and with respect to design and optimal control of sector-coupled energy systems, there is a lack of available benchmarks. Hence, this article provides a model compendium, exemplary realistic data sets, as well as two case studies (i.e., optimization benchmarks) for an industrial/research campus in an open-source description. The compendium includes stationary, quasi-stationary, and dynamic models for typical components as well as linearization schemes relevant for optimization of design, operation, and control of sector-coupled energy systems.
    Publicationtype: Journal Article
    Scopus© Citations 22  8  60