arXiv subjects

Dirk Lebiedz

Publications and source records attributed to Dirk Lebiedz.

At least 19 recordsLinked to original sources

arXiv2025

Riemannian Optimization on Tree Tensor Networks with Application in Machine Learning

Tree tensor networks (TTNs) are widely used in low-rank approximation and quantum many-body simulation. In this work, we present a formal analysis of the differential geometry underlying TTNs. Building on this foundation, we develop efficient first- and second-order optimization algorithms that exploit the intrinsic quotient structure of TTNs. Additionally, we devise a backpropagation algorithm for training TTNs in a kernel learning setting. We validate our methods through numerical experiments on a representative machine learning task.

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arXiv2025

Separatrix configurations in holomorphic flows

We investigate properties of boundary orbits (separatrices) of canonical regions (basins/neighbourhoods of equilibria) in holomorphic flows with real-valued time. We establish the continuity of transit times along these boundary orbits and classify possible path components of the boundary of flow-invariant domains. Thus, we provide central tools for topological and geometric constructions aimed at examining the role of blow-up scenarios in separatrix configurations of basins of simple equilibria and global elliptic sectors: First, we prove that the separatrices of basins of centers is entirely composed of double-sided separatrices with a blow-up in finite positive and finite negative time. Second, we show that the separatrices of node and focus basins (sinks and sources) exhibit a finite-time blow-up in the same time direction in which the orbits within the basin tend towards the equilibrium. Additionally, we propose a counterexample to the claim in Theorem 4.3 (3) in ["The structure of sectors of zeros of entire flows", K. Broughan (2003)], demonstrating that a blow-up does not necessarily have to occur in both time directions. Third, we describe the boundary structure of global elliptic sectors. It consists of the multiple equilibrium, one incoming and one outgoing separatrix attached to it, and at most countably many double-sided separatrices.

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arXiv2024

Sensitivities in complex-time flows: phase transitions, Hamiltonian structure and differential geometry

Reminiscent of physical phase transitions separatrices divide the phase space of dynamical systems with multiple equilibria into regions of distinct flow behavior and asymptotics. We introduce complex time in order to study corresponding Riemann surface solutions of holomorphic and meromorphic flows, explicitly solve their sensitivity differential equation and identify a related Hamiltonian structure and an associated geometry in order to study separatrix properties. As an application we analyze complex-time Newton flow of Riemann's $\xi$-function on the basis of a compactly convergent polynomial approximation of its Riemann surface solution defined as zero set of polynomials, e.g. algebraic curves over $\mathbb{C}$ (in the complex projective plane respectively), that is closely related to a complex-valued Hamiltonian system. Its geometric properties might contain information on the global separatrix structure and the root location of $\xi$ and $\xi'$.

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arXiv2024

Basins of Equilibria and geometry of Global Sectors in Holomorphic Flows

In this follow-up paper, we investigate the global geometry and topology of dynamical systems $\dot{x} = F(x)$ with entire vector field $F$, building on and constructively extending the local structure of simple and higher-order equilibria. We provide a step-by-step analysis to reveal topological properties of the basins of centers, nodes, and foci, while excluding isolated equilibria at the boundaries of the latter two. We propose a definition of global elliptic sectors and introduce the concept of sector-forming orbits based on the geometry within a finite elliptic decomposition of multiple equilibria. Finally, we characterize the structure of heteroclinic regions connecting two equilibria.

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arXiv2024

Local geometry of Equilibria and a Poincaré-Bendixson-type Theorem for Holomorphic Flows

In this paper, we explore the local geometry of dynamical systems $\dot{x}=F(x)$ with real time parameterization, where $F$ is holomorphic on connected open subsets of $\mathbb{C}\stackrel{\sim}{=}\mathbb{R}^2$. We describe the geometry of first-order equilibria. For equilibria of higher orders, we establish an equivalent condition for "definite directions", allowing us to reverse the implication in Theorem 2 of Chapter 2.10 in [Differential equations and dynamical systems, Lawrence Perko (1990)] under the additional condition of holomorphy. This enables the geometric construction of a finite elliptic decomposition. We derive a holomorphic Poincaré-Bendixson-type theorem, leading to the conclusion that bounded non-periodic orbits are always homoclinic or heteroclinic.

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arXiv2023

Benchmarking of Flatness-based Control of the Heat Equation

Flatness-based control design is a well established method to generate open-loop control signals. Several articles discuss the application of flatness-based control design of (reaction-) diffusion problems for various scenarios. Beside the pure analytical derivation also the numerical computation of the input signal is crucial to yield a reliable trajectory planning. Therefore, we derive the input signal step-by-step and describe the influence of system and controller parameters on the computation of the input signal. In particular, we benchmark the control design of the one-dimensional heat equation with Neumann-type boundary actuation for pure aluminum and steel 38Si7, and discuss the applicability of the found input signals for realistic scenarios.

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arXiv2022

Approximating normally attracting invariant manifolds using trajectory-based optimization

The numerical simulation of realistic reactive flows is a major challenge due to the stiffness and high dimension of the corresponding kinetic differential equations. Manifold-based model reduction techniques address this problem by projecting the full phase space onto manifolds of slow motion, which capture the system's long-term behavior. In this article we study the trajectory-based optimization approach by Lebiedz (2004), which determines these manifolds as minimizers of an appropriate entropy functional. Similar to other methods in this field, this approach is based on physical and geometric intuition and was tested on several models. This article provides a rigorous explanation for its effectiveness, showing how it approximates nonuniformly normally attracting orbits. It also outlines how the method can be utilized to approximate nonuniformly normally attracting invariant manifolds of higher dimension. Throughout the article we use a coordinate-free formulation on a Riemannian manifold. This is especially useful for systems subject to nonlinear constraints, e.g., adiabatic constraints.

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arXiv2022

On Differential Geometric Formulations of Slow Invariant Manifold Computation: Geodesic Stretching and Flow Curvature

The theory of slow invariant manifolds (SIMs) is the foundation of various model-order reduction techniques for dissipative dynamical systems with multiple time-scales, e.g. in chemical kinetic models. The construction of SIMs and many approximation methods exploit the restrictive requirement of an explicit time-scale separation parameter. Most of those methods are also not formulated covariantly, i.e. in terms of tensorial constructions. We propose an intrinsically coordinate-free differential geometric approximation criterion approximating normally attracting invariant manifolds (NAIMs). We translate some ideas behind existing approximation approaches, the stretching based diagnostics (SBD) and the flow curvature method (FCM) to tensors of Riemannian geometry, specifically to spacetime curvature in extended phase space. For that purpose we derive from flow-generating smooth vector fields a metric tensor such that the original dynamical system is a geodesic flow on a Riemannian manifold. We apply the resulting method to test models.

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arXiv2021

Generalized Liénard systems, singularly perturbed systems, Flow Curvature Method

In his famous book entitled \textit{Theory of Oscillations}, Nicolas Minorsky wrote: "\textit{each time the system absorbs energy the curvature of its trajectory decreases} and \textit{vice versa}". According to the \textit{Flow Curvature Method}, the location of the points where the \textit{curvature of trajectory curve}, integral of such planar \textit{singularly dynamical systems}, vanishes directly provides a first order approximation in $\varepsilon$ of its \textit{slow invariant manifold} equation. By using this method, we prove that, in the $\varepsilon$-vicinity of the \textit{slow invariant manifold} of generalized Liénard systems, the \textit{curvature of trajectory curve} increases while the \textit{energy} of such systems decreases. Hence, we prove Minorsky's statement for the generalized Liénard systems. Then, we establish a relationship between \textit{curvature} and \textit{energy} for such systems. These results are then exemplified with the classical Van der Pol and generalized Liénard \textit{singularly perturbed systems}.

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arXiv2020

Holomorphic Hamiltonian $ξ$-Flow and Riemann Zeros

With a view on the formal analogy between Riemann-von-Mangoldts explicit formula and semiclassical quantum mechanics in terms of the Gutzwiller trace formula we construct a complex-valued Hamiltonian $H(q,p)=ξ(q)p$ from the holomorphic flow $\dot{q}=ξ(q)$ and its variational differential equation. The Hamiltonian phase portrait $q(p)$ is a Riemann surface equivalent to reparameterized $ξ$-Newton flow solutions in complex-time, its flow map differential is determined by all Riemann zeros and reminiscent of a 'spectral sum' in trace formulas. Canonical quantization for particle quantum mechanics on a circle leads to a Dirac-type momentum operator with discrete spectrum given by classical closed orbit periods determined by derivatives $ξ'(ρ_n)$ at Riemann zeros.

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arXiv2019

On Analytical and Topological Properties of Separatrices in 1-D Holomorphic Dynamical Systems and Complex-Time Newton Flows

Separatrices divide the phase space of some holomorphic dynamical systems into separate basins of attraction or 'stability regions' for distinct fixed points. 'Bundling' (high density) and mutual 'repulsion' of trajectories are often observed at separatrices in phase portraits, but their global mathematical characterisation is a difficult problem. For 1-D complex polynomial dynamical systems we prove the existence of a separatrix for each critical point at infinity via transformation to the Poincaré sphere. We show that introduction of complex time allows a significantly extended view with the study of corresponding Riemann surface solutions, their topology, geometry and their bifurcations/ramifications related to separatrices. We build a bridge to the Riemann $ξ$-function and present a polynomial approximation of its Newton flow solution manifold with precision depending on the polynomial degree.

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arXiv2019

A Spectral View on Slow Invariant Manifolds in Complex-Time Dynamical Systems

Many real-analytic flows, e.g. in chemical kinetics, share a multiple time scale spectral structure. The trajectories of the corresponding dynamical systems are observed to bundle near so-called slow invariant manifolds (SIMs), which are usually addressed in a singular perturbation context. This work exploits the analytic structure of the involved vector fields and presents observations that connect one dimensional slow invariant manifolds to the imaginary-time spectral structure of Riemann surfaces in analytic continuations of dynamical systems to complex time.

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arXiv2019

Considering Slow Manifold Based Model Reduction for Multiscale Chemical Optimal Control Problems

Finite-dimensional dissipative dynamical systems with multiple time-scales are obtained when modeling chemical reaction kinetics with ordinary differential equations. Such stiff systems are computationally hard to solve and therefore, optimal control problems which contain ordinary differential equations as infinitesimal constraints are even more difficult to handle. Model reduction might offer an approach to improve numerical efficiency as well as avoiding stiffness of such models. We show in this paper in benchmark fashion how attracting manifold computation methods could be exploited to solve optimal control more efficiently while having in mind the ambitious long-term goal to apply them to real-time control problems in chemical kinetics.

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arXiv2019

Stretching-Based Diagnostics in a Differential Geometry Setting

The identification of slow invariant manifolds (SIMs) is an essential part in model-order reduction for reactive systems. The mathematical definition of the SIM by Fenichel can be considered unsatisfactory, because it is only applicable to so-called slow-fast system and does not provide the uniqueness of the SIM. Observing the phase space of the dynamical system (not necessarily a slow-fast system), the SIM becomes a geometric object which attracts trajectories, resulting in a bundling behavior. We aim to find a more general definition of the SIM, guided by the prior observations in phase space within the field of differential geometry. This setting provides one major benefit: All quantities are formulated covariantly, i.e. they are independent of the coordinate choice. A recent work by Heiter and Lebiedz \cite{heiter} translates the invariance property to vanishing sectional curvatures in the extended phase space.

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arXiv2019

Characterization of Separatrices in Holomorphic Dynamical Systems

Multiple time scales in dynamical systems lead to a bundling of trajectories onto slow invariant manifolds (SIMs). Although they are absent in two-dimensional holomorphic dynamical systems, a bundling of orbits is often observed as well. They bundle onto special trajectories called separatrices. We apply numerical methods for the approximation of SIMs to holomorphic flows and show how a separatrix between two regions of periodic orbits can be characterized topologically. Complex time reveals a new perspective on holomorphic dynamical systems.

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arXiv2019

Analytic continuation and differential geometry views on slow manifolds and separatrices

We start from a mechano-chemical analogy considering the time evolution of a homogeneous chemical reaction modeled by a nonlinear dynamical system (ordinary differential equation, ODE) as the movement of a phase space point on the solution manifold such as the movement of a mass point in curved spacetime. Based on our variational problem formulation \cite{Lebiedz2011} for slow invariant manifold (SIM) computation and ideas from general relativity theory we argue for a coordinate free analysis treatment \cite{Heiter2018} and a differential geometry formulation in terms of geodesic flows \cite{Poppe2019}. In particular, we propose analytic continuation of the dynamical system to the complex time domain to reveal deeper structures and allow the application of the rich toolbox of Fourier and complex analysis to the SIM problem.

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arXiv2017

Towards differential geometric characterization of slow invariant manifolds in extended state space: Sectional Curvature and Flow Invariance

Some model reduction techniques for multiple time-scale dynamical systems make use of the identification of low dimensional slow invariant attracting manifolds (SIAM) in order to reduce the dimensionality of the phase space by restriction to the slow flow. The focus of this work is on a proposition and discussion of a general viewpoint using differential geometric concepts for submanifolds to deal with slow invariant manifolds in an extended phase space. The motivation is a coordinate independent formulation of the manifold properties and its characterization problem treating the manifold as intrinsic geometric object. We formulate a computationally verifiable necessary condition for the slow invariant manifold graph stated in terms of a differential geometric view on the invariance property. Its application to example systems is illustrated. In addition, we present some ideas and investigations concerning the search for sufficient, differential geometric conditions characterizing slow invariant manifolds based on our previously developed variational principle.

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arXiv2017

Covariant geometric characterization of slow invariant manifolds: New concepts and viewpoints

We point out a new view on slow invariant manifolds (SIM) in dynamical systems which departs from a purely geometric covariant characterization implying coordinate independency. The fundamental idea is to treat the SIM as a well-defined geometric object in phase space and elucidate characterizing geometric properties that can be evaluated as point-wise analytic criteria. For that purpose, we exploit curvature concepts and formulate our recent variational approach in terms of coordinate-independent Hamiltonian mechanics.Finally, we combine both approaches and conjecture a differential geometric definition of slow invariant manifolds. For the Davis-Skodje model the latter can be completely expatiated.

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