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Martin Ziegler

Publications and source records attributed to Martin Ziegler.

At least 19 recordsLinked to original sources
arXiv2026

Algorithmic Cost in "Exact Real Computation"

Turing completeness of a programming language or system characterizes its expressive power; and the strong Church-Turing hypo-/thesis refines such from qualitative to polynomial-time equivalence. Exact Real Computation (ERC) is a novel numerical programming language paradigm: for the imperative processing of continuous data as entities appearing as exact, i.e. devoid of rounding errors [doi:10.1007/978-3-662-44199-2_107]. ERC has been designed [doi:10.46298/lmcs-20(2:17)2024] as convenient and practical alternative, namely proven qualitatively equivalent, to the Turing machines originally underlying Computable Analysis [doi:10.1007/978-3-642-56999-9,doi:10.1007/978-1-4684-6802-1]. The present work quantitatively strengthens this qualitative Turing-completeness: We assign bit-costs to ERC's operational primitives (including partial/multivalued tests) in such a way that any real function incurring polynomial cost becomes Turing-computable in polynomial time, and vice versa. Runtime measurements on implementations in the iRRAM C++ library confirm our theoretical performance predictions.

math.NA↗
arXiv2026

Thermo-Hydro-Mechanical Modeling of a TBM Tunnel in Opalinus Clay: Thermal Coupling Effects on Long-term Lining Pressures and Excavation Damage Zone Development

Tunnels in argillaceous formations such as Opalinus Clay (OPA) develop a coupled thermal, hydraulic, and mechanical response. This study examines the New Belchen Tunnel (STB), excavated by tunnel boring machine through faulted OPA shale. Field measurements show a strong relation between lining temperature and radial pressure. We use a two-dimensional plane-strain THM model to examine excavation, construction heat, seasonal temperature cycles, pore-pressure transients, and stress redistribution. The model reproduces the phasing and approximate magnitude of the seasonal pressure cycles and shows how the thermal expansion mismatch between the lining, grout, and OPA modifies lining load and near-field pore pressure. It underestimates the monotonic long-term pressure increase at several sensors. Comparison with radial extensometer measurements indicates that moisture-driven swelling, local geological structure, grout or interface damage, and three-dimensional construction effects also contribute. The model therefore supports a thermal interpretation of the cyclic response while defining the processes required for a fuller long-term back-analysis.

physics.geo-ph↗
arXiv2026

Self-organized defect clustering and concentration-dependent vacancy diffusion in MoS$_2$

Sulfur vacancy migration has a crucial impact on electronic transport and the functional behavior of MoS$_2$-based devices such as memristors and memtransistors. According to recent atomistic simulations, vacancy migration proceeds via cooperative, vacancy-assisted sulfur jumps, implying strongly correlated defect dynamics. Here, we investigate the collective behavior of sulfur-vacancy clusters in MoS$_2$ using kinetic Monte-Carlo simulations with transition rates derived from machine learning interatomic potential molecular dynamics simulations. We identify three transport regimes: At low concentrations, vacancies are immobile or confined within small clusters, whereas at high concentrations, classical diffusive transport with a constant diffusion coefficient is observed, and vacancies aggregate into anisotropically extended clusters. A well defined intermediate regime is characterized by clusters merging into a connected, fluctuating network with a concentration-dependent diffusion coefficient. This regime is characterized by a broad distribution of cluster sizes. The strong dependence of the vacancy diffusion coefficient on the average defect concentration provides new insights into the origin of memristive behavior observed in MoS$_2$.

cond-mat.mtrl-sci↗
arXiv2026

Two-branch retention behavior in unsaturated fractured rock driven by fracture-matrix flow partitioning

Upscaling unsaturated flow in fractured rock remains challenging because fractures and matrix often exhibit sharply contrasting hydraulic behaviors across saturation states. Here, we demonstrate that unsaturated flow undergoes a transition between matrix- and fracture-dominated regimes. Three-dimensional direct numerical simulations reveal that both relative permeability and capillary pressure curves display a robust two-branch structure. We analytically derive a generalized retention formulation that identifies a critical saturation marking the transition between the two distinct retention regimes and reproduces the two-branch behavior captured in the numerical simulations. An analytical expression for the critical pressure head is further derived to represent the limiting case of fully connected fracture networks, providing a physical explanation for the retention regime shift and showing good agreement with the numerical results for systems above the percolation threshold. Our results provide a mechanistic framework for understanding and upscaling unsaturated flow in fractured rock, with broad implications for hydrology and geophysics.

physics.geo-ph↗
arXiv2025

Prediction performance of random reservoirs with different topology for nonlinear dynamical systems with different number of degrees of freedom

Reservoir computing (RC) is a powerful framework for predicting nonlinear dynamical systems, yet the role of reservoir topology$-$particularly symmetry in connectivity and weights$-$remains not adequately understood. This work investigates how the structure of the network influences the performance of RC in four systems of increasing complexity: the Mackey-Glass system with delayed-feedback, two low-dimensional thermal convection models, and a three-dimensional shear flow model exhibiting transition to turbulence. Using five reservoir topologies in which connectivity patterns and edge weights are controlled independently, we evaluate both direct- and cross-prediction tasks. The results show that symmetric reservoir networks substantially improve prediction accuracy for the convection-based systems, especially when the input dimension is smaller than the number of degrees of freedom. In contrast, the shear-flow model displays almost no sensitivity to topological symmetry due to its strongly chaotic high-dimensional dynamics. These findings reveal how structural properties of reservoir networks affect their ability to learn complex dynamics and provide guidance for designing more effective RC architectures.

physics.flu-dyn↗
arXiv2025

Large-scale cooperative sulfur vacancy dynamics in two-dimensional MoS2 from machine learning interatomic potentials

The formation of extended sulfur vacancies in MoS2 monolayers is closely associated with catalytic activity and may also be the basis for its memristive behavior. Nanosecond-scale molecular dynamics simulations using machine learning interatomic potentials (MLIPs) reveal key mechanisms of cooperative vacancy transport, including incorporation of vacancies into clusters of arbitrary size. The simulations provide a coherent atomistic explanation for irradiation-induced vacancy patterns observed experimentally, especially the formation of line defects spanning tens of nanometers. Results and performance are compared of two MLIP frameworks: (i) on-the-fly learning with Gaussian approximation potential, and (ii) fine-tuning of an equivariant foundation model.

cond-mat.mtrl-sci↗
arXiv2025

High-resolution spatial memory requires grid-cell-like neural codes

Continuous attractor networks (CANs) are widely used to model how the brain temporarily retains continuous behavioural variables via persistent recurrent activity, such as an animal's position in an environment. However, this memory mechanism is very sensitive to even small imperfections, such as noise or heterogeneity, which are both common in biological systems. Previous work has shown that discretising the continuum into a finite set of discrete attractor states provides robustness to these imperfections, but necessarily reduces the resolution of the represented variable, creating a dilemma between stability and resolution. We show that this stability-resolution dilemma is most severe for CANs using unimodal bump-like codes, as in traditional models. To overcome this, we investigate sparse binary distributed codes based on random feature embeddings, in which neurons have spatially-periodic receptive fields. We demonstrate theoretically and with simulations that such grid-cell-like codes enable CANs to achieve both high stability and high resolution simultaneously. The model extends to embedding arbitrary nonlinear manifolds into a CAN, such as spheres or tori, and generalises linear path integration to integration along freely-programmable on-manifold vector fields. Together, this work provides a theory of how the brain could robustly represent continuous variables with high resolution and perform flexible computations over task-relevant manifolds.

cs.NE↗
arXiv2025

Second-Order Parameterizations for the Complexity Theory of Integrable Functions

We develop a unified second-order parameterized complexity theory for spaces of integrable functions. This generalizes the well-established case of second-order parameterized complexity theory for spaces of continuous functions. Specifically we prove the mutual linear equivalence of three natural parameterizations of the space $\Lrm{p}$ of $p$-integrable complex functions on the real unit interval: (binary) $\Lrm{p}$-modulus, rate of convergence of Fourier series, and rate of approximation by step functions.

cs.CC↗
arXiv2025

Physics-Guided Sequence Modeling for Fast Simulation and Design Exploration of 2D Memristive Devices

Modeling hysteretic switching dynamics in memristive devices is computationally demanding due to coupled ionic and electronic transport processes. This challenge is particularly relevant for emerging two-dimensional (2D) devices, which feature high-dimensional design spaces that remain largely unexplored. We introduce a physics-guided modeling framework that integrates high-fidelity finite-volume (FV) charge transport simulations with a long short-term memory (LSTM) artificial neural network (ANN) to predict dynamic current-voltage behavior. Trained on physically grounded simulation data, the ANN surrogate achieves more than four orders of magnitude speedup compared to the FV model, while maintaining direct access to physically meaningful input parameters and high accuracy with typical normalized errors <1%. This enables iterative tasks that were previously computationally prohibitive, including inverse modeling from experimental data, design space exploration via metric mapping and sensitivity analysis, as well as constrained multi-objective design optimization. Importantly, the framework preserves physical interpretability via access to detailed spatial dynamics, including carrier densities, vacancy distributions, and electrostatic potentials, through a direct link to the underlying FV model. Our approach establishes a scalable framework for efficient exploration, interpretation, and model-driven design of emerging 2D memristive and neuromorphic devices.

cond-mat.mtrl-sci↗
arXiv2024

Numerical analysis and simulation of lateral memristive devices: Schottky, ohmic, and multi-dimensional electrode models

In this paper, we present the numerical analysis and simulations of a multi-dimensional memristive device model. Memristive devices and memtransistors based on two-dimensional (2D) materials have demonstrated promising potential for neuromorphic computing and next-generation memory technologies. Our charge transport model describes the drift-diffusion of electrons, holes, and ionic defects self-consistently in an electric field. We incorporate two types of boundary models: ohmic and Schottky contacts. The coupled drift-diffusion partial differential equations are discretized using a physics-preserving Voronoi finite volume method. It relies on an implicit time-stepping scheme and the excess chemical potential flux approximation. We demonstrate that the fully discrete nonlinear scheme is unconditionally stable, preserving the free-energy structure of the continuous system and ensuring the non-negativity of carrier densities. Novel discrete entropy-dissipation inequalities for both boundary condition types in multiple dimensions allow us to prove the existence of discrete solutions. We perform multi-dimensional simulations to understand the impact of electrode configurations and device geometries, focusing on the hysteresis behavior in lateral 2D memristive devices. Three electrode configurations -- side, top, and mixed contacts -- are compared numerically for different geometries and boundary conditions. These simulations reveal the conditions under which a simplified one-dimensional electrode geometry can well represent the three electrode configurations. This work lays the foundations for developing accurate, efficient simulation tools for 2D memristive devices and memtransistors, offering tools and guidelines for their design and optimization in future applications.

math.NA↗
arXiv2024

On Foundation Models for Dynamical Systems from Purely Synthetic Data

Foundation models have demonstrated remarkable generalization, data efficiency, and robustness properties across various domains. In this paper, we explore the feasibility of foundation models for applications in the control domain. The success of these models is enabled by large-scale pretaining on Internet-scale datasets. These are available in fields like natural language processing and computer vision, but do not exist for dynamical systems. We address this challenge by pretraining a transformer-based foundation model exclusively on synthetic data and propose to sample dynamics functions from a reproducing kernel Hilbert space. Our pretrained model generalizes for prediction tasks across different dynamical systems, which we validate in simulation and hardware experiments, including cart-pole and Furuta pendulum setups. Additionally, the model can be fine-tuned effectively to new systems to increase performance even further. Our results demonstrate the feasibility of foundation models for dynamical systems that outperform specialist models in terms of generalization, data efficiency, and robustness.

cs.LG↗
arXiv2024

TEXEL: A neuromorphic processor with on-chip learning for beyond-CMOS device integration

Recent advances in memory technologies, devices and materials have shown great potential for integration into neuromorphic electronic systems. However, a significant gap remains between the development of these materials and the realization of large-scale, fully functional systems. One key challenge is determining which devices and materials are best suited for specific functions and how they can be paired with CMOS circuitry. To address this, we introduce TEXEL, a mixed-signal neuromorphic architecture designed to explore the integration of on-chip learning circuits and novel two- and three-terminal devices. TEXEL serves as an accessible platform to bridge the gap between CMOS-based neuromorphic computation and the latest advancements in emerging devices. In this paper, we demonstrate the readiness of TEXEL for device integration through comprehensive chip measurements and simulations. TEXEL provides a practical system for testing bio-inspired learning algorithms alongside emerging devices, establishing a tangible link between brain-inspired computation and cutting-edge device research.

cs.NE↗
arXiv2024

Asymmetrically connected reservoir networks learn better

We show that connectivity within the high-dimensional recurrent layer of a reservoir network is crucial for its performance. To this end, we systematically investigate the impact of network connectivity on its performance, i.e., we examine the symmetry and structure of the reservoir in relation to its computational power. Reservoirs with random and asymmetric connections are found to perform better for an exemplary Mackey-Glass time series than all structured reservoirs, including biologically inspired connectivities, such as small-world topologies. This result is quantified by the information processing capacity of the different network topologies which becomes highest for asymmetric and randomly connected networks.

cs.NE↗
arXiv2024

Distributed Representations Enable Robust Multi-Timescale Symbolic Computation in Neuromorphic Hardware

Programming recurrent spiking neural networks (RSNNs) to robustly perform multi-timescale computation remains a difficult challenge. To address this, we describe a single-shot weight learning scheme to embed robust multi-timescale dynamics into attractor-based RSNNs, by exploiting the properties of high-dimensional distributed representations. We embed finite state machines into the RSNN dynamics by superimposing a symmetric autoassociative weight matrix and asymmetric transition terms, which are each formed by the vector binding of an input and heteroassociative outer-products between states. Our approach is validated through simulations with highly nonideal weights; an experimental closed-loop memristive hardware setup; and on Loihi 2, where it scales seamlessly to large state machines. This work introduces a scalable approach to embed robust symbolic computation through recurrent dynamics into neuromorphic hardware, without requiring parameter fine-tuning or significant platform-specific optimisation. Moreover, it demonstrates that distributed symbolic representations serve as a highly capable representation-invariant language for cognitive algorithms in neuromorphic hardware.

cs.NE↗
arXiv2023

Noetherian theories

A first-order theory is Noetherian with respect to the collection of formulae $\mathcal{F}$ if every definable set is a Boolean combination of instances of formulae in $\mathcal{F}$ and the topology whose subbasis of closed sets is the collection of instances of arbitrary formulae in $\mathcal{F}$ is Noetherian. Noetherianity is a strengthening of equationality, which itself implies stability. We show the Noetherianity of the theory of proper pairs of algebraically closed fields in any characteristic.

math.LO↗
arXiv2023

Degrees of Second and Higher-Order Polynomials

Second-order polynomials generalize classical first-order ones in allowing for additional variables that range over functions rather than values. We are motivated by their applications in higher-order computational complexity theory, extending for example classical classes like P or PSPACE to operators in Analysis [doi:10.1137/S0097539794263452, doi:10.1145/2189778.2189780]. The degree subclassifies ordinary polynomial growth into linear, quadratic, cubic etc. In order to similarly classify second-order polynomials, define their degree to be an 'arctic' first-order polynomial (namely a term/expression over variable $D$ and operations $+$ and $\cdot$ and $\max$). This degree turns out to transform as nicely under (now two kinds of) polynomial composition as the ordinary one. We also establish a normal form and semantic uniqueness for second-order polynomials. Then we define the degree of a third-order polynomial to be an arctic second-order polynomial, and establish its transformation under three kinds of composition.

cs.LO↗
arXiv2023

The Role of Mobile Point Defects in Two-Dimensional Memristive Devices

Two-dimensional (2D) layered transition metal dichalcogenides (TMDCs) are promising memristive materials for neuromorphic computing systems as they could solve the problem of the excessively high energy consumption of conventional von Neumann computer architectures. Despite extensive experimental work, the underlying switching mechanisms are still not understood, impeding progress in material and device functionality. This study reveals the dominant role of mobile defects in the switching dynamics of 2D TMDC materials. The switching process is governed by the formation and annihilation dynamics of a local vacancy depletion zone. Moreover, minor changes in the interface potential barriers cause fundamentally different device behavior previously thought to originate from multiple mechanisms. The key mechanisms are identified with a charge transport model for electrons, holes, and ionic point defects, including image-charge-induced Schottky barrier lowering (SBL). The model is validated by comparing simulations to measurements for various 2D MoS$_2$-based devices, strongly corroborating the relevance of vacancies in TMDC devices and offering a new perspective on the switching mechanisms. The insights gained from this study can be used to extend the functional behavior of 2D TMDC memristive devices in future neuromorphic computing applications.

physics.app-ph↗