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We show how random feature maps can be used to forecast dynamical systems with excellent forecasting skill. We consider the tanh activation function and judiciously choose the internal weights in a data-driven manner such that the resulting features explore the nonlinear, non-saturated regions of the activation function. We introduce skip connections and construct a deep variant of random feature maps by combining several units. To mitigate the curse of dimensionality, we introduce localization where we learn local maps, employing conditional independence. Our modified random feature maps provide excellent forecasting skill for both single trajectory forecasts as well as long-time estimates of statistical properties, for a range of chaotic dynamical systems with dimensions up to 512. In contrast to other methods such as reservoir computers which require extensive hyperparameter tuning, we effectively need to tune only a single hyperparameter, and are able to achieve state-of-the-art forecast skill with much smaller networks.

In the past decade, photoemission orbital tomography (POT) has evolved into a powerful tool to investigate the electronic structure of organic molecules adsorbed on surfaces. Here we show that POT allows for the comprehensive experimental identification of all molecular orbitals in a substantial binding energy range, in the present case more than 10 eV. Making use of the angular distribution of photoelectrons as a function of binding energy, we exemplify this by extracting orbital-resolved partial densities of states (pDOS) for 15 $\pi$ and 23 $\sigma$ orbitals from the experimental photoemission intensities of the prototypical organic molecule bisanthene (C$_{28}$H$_{14}$) on a Cu(110) surface. In their entirety, these experimentally measured orbital-resolved pDOS for an essentially complete set of orbitals serve as a stringent benchmark for electronic structure methods, which we illustrate by performing density functional theory (DFT) calculations employing four frequently-used exchange-correlation functionals. By computing the respective molecular-orbital-projected densities of states of the bisanthene/Cu(110) interface, a one-to-one comparison with experimental data for an unprecedented number of 38 orbital energies becomes possible. The quantitative analysis of our data reveals that the range-separated hybrid functional HSE performs best for the investigated organic/metal interface. At a more fundamental level, the remarkable agreement between the experimental and the Kohn-Sham orbital energies over a binding energy range larger than 10\,eV suggests that -- perhaps unexpectedly -- Kohn-Sham orbitals approximate Dyson orbitals, which would rigorously account for the electron extraction process in photoemission spectroscopy but are notoriously difficult to compute, in a much better way than previously thought., Comment: arXiv admin note: text overlap with arXiv:2209.11516

Higher-point functions in N = 4 super Yang-Mills theory can be constructed using integrability by triangulating the surfaces on which Feynman graphs would be drawn. It remains hard to analytically compute the necessary re-gluing of the tiles by virtual particles. We propose a new approach to study a series of residues encountered in the two-particle gluing of the planar one-loop five-point function of stress tensor multiplets. After exposing the twisted period nature of the integral functions, we employ intersection theory to derive canonical differential equations and present a solution., Comment: 11 pages, 1 figure, LaTeX

The Hegselmann-Krause model is a prototypical model for opinion dynamics. It models the stochastic time evolution of an agent's or voter's opinion in response to the opinion of other like-minded agents. The Hegselmann-Krause model only considers the opinions of voters; we extend it here by incorporating the dynamics of political parties which influence and are influenced by the voters. We show in numerical simulations for $1$- and $2$-dimensional opinion spaces that, as for the original Hegselmann-Krause model, the modified model exhibits opinion cluster formation as well as a phase transition from disagreement to consensus. We provide an analytical sufficient condition for the formation of unanimous consensus in which voters and parties collapse to the same point in opinion space in the deterministic case. Using mean-field theory, we further derive an approximation for the critical noise strength delineating consensus from non-consensus in the stochastically driven modified Hegselmann-Krause model. We compare our analytical findings with simulations of the modified Hegselmann-Krause model.

Shannon information and Shannon entropy are undoubtedly the most commonly used quantitative measures of information, cropping up in the literature across a broad variety of disciplines, often in contexts unrelated to coding theory. Here, we generalize the original idea behind Shannon entropy as the cost of encoding a sample of a random variable in terms of the required codeword length, to arbitrary loss functions by considering the optimally achievable loss given a certain level of knowledge about the random variable. By formalizing knowledge in terms of the measure-theoretic notion of sub-$\sigma$-algebras, we arrive at a general notion of uncertainty reduction that includes entropy and information as special cases: entropy is the reduction of uncertainty from no (or partial) knowledge to full knowledge about a random variable, whereas information is uncertainty reduction from no (or partial) knowledge to partial knowledge. As examples, we get Shannon information and entropy when measuring loss in terms of message length, variance for square error loss, and more generally, for the Bregman loss, we get Bregman information. Moreover, we show that appealing properties of Shannon entropy and information extend to the general case, including well-known relations involving the KL divergence, which are extended to divergences of proper scoring rules.

We consider the problem of sampling from an unknown distribution for which only a sufficiently large number of training samples are available. In this paper, we build on previous work combining Schr\"odinger bridges and plug & play Langevin samplers. A key bottleneck of these approaches is the exponential dependence of the required training samples on the dimension, $d$, of the ambient state space. We propose a localization strategy which exploits conditional independence of conditional expectation values. Localization thus replaces a single high-dimensional Schr\"odinger bridge problem by $d$ low-dimensional Schr\"odinger bridge problems over the available training samples. In this context, a connection to multi-head self attention transformer architectures is established. As for the original Schr\"odinger bridge sampling approach, the localized sampler is stable and geometric ergodic. The sampler also naturally extends to conditional sampling and to Bayesian inference. We demonstrate the performance of our proposed scheme through experiments on a high-dimensional Gaussian problem, on a temporal stochastic process, and on a stochastic subgrid-scale parametrization conditional sampling problem. We also extend the idea of localization to plug & play Langevin samplers using kernel-based denoising in combination with Tweedie's formula.

The computationally cheap machine learning architecture of random feature maps can be viewed as a single-layer feedforward network in which the weights of the hidden layer are random but fixed and only the outer weights are learned via linear regression. The internal weights are typically chosen from a prescribed distribution. The choice of the internal weights significantly impacts the accuracy of random feature maps. We address here the task of how to best select the internal weights. In particular, we consider the forecasting problem whereby random feature maps are used to learn a one-step propagator map for a dynamical system. We provide a computationally cheap hit-and-run algorithm to select good internal weights which lead to good forecasting skill. We show that the number of good features is the main factor controlling the forecasting skill of random feature maps and acts as an effective feature dimension. Lastly, we compare random feature maps with single-layer feedforward neural networks in which the internal weights are now learned using gradient descent. We find that random feature maps have superior forecasting capabilities whilst having several orders of magnitude lower computational cost.

We consider the problem of sampling from an unknown distribution for which only a sufficiently large number of training samples are available. Such settings have recently drawn considerable interest in the context of generative modelling and Bayesian inference. In this paper, we propose a generative model combining Schr\"odinger bridges and Langevin dynamics. Schr\"odinger bridges over an appropriate reversible reference process are used to approximate the conditional transition probability from the available training samples, which is then implemented in a discrete-time reversible Langevin sampler to generate new samples. By setting the kernel bandwidth in the reference process to match the time step size used in the unadjusted Langevin algorithm, our method effectively circumvents any stability issues typically associated with the time-stepping of stiff stochastic differential equations. Moreover, we introduce a novel split-step scheme, ensuring that the generated samples remain within the convex hull of the training samples. Our framework can be naturally extended to generate conditional samples and to Bayesian inference problems. We demonstrate the performance of our proposed scheme through experiments on synthetic datasets with increasing dimensions and on a stochastic subgrid-scale parametrization conditional sampling problem as well as generating sample trajectories of a dynamical system using conditional sampling.

We perform a stochastic model reduction of the Kuramoto-Sakaguchi model for finitely many coupled phase oscillators with phase frustration. Whereas in the thermodynamic limit coupled oscillators exhibit stationary states and a constant order parameter, finite-size networks exhibit persistent temporal fluctuations of the order parameter. These fluctuations are caused by the interaction of the synchronized oscillators with the non-entrained oscillators. We present numerical results suggesting that the collective effect of the non-entrained oscillators on the synchronized cluster can be approximated by a Gaussian process. This allows for an effective closed evolution equation for the synchronized oscillators driven by a Gaussian process which we approximate by a two-dimensional Ornstein-Uhlenbeck process. Our reduction reproduces the stochastic fluctuations of the order parameter and leads to a simple stochastic differential equation for the order parameter.

We consider the correlator $\langle \mathcal{L} \mathcal{K} \tilde{ \mathcal{K}} \rangle $ of the Lagrange operator of $\mathcal{N}=4$ super Yang-Mills theory and two conjugate two-excitation operators in an $su(2)$ sector. We recover the planar one-loop anomalous dimension of the renormalised operators from this hexagon computation., Comment: 6 pages, 1 figure, 3 tables; v2: minor corrections, one reference added

Automatically identifying bat species from their echolocation calls is a difficult but important task for monitoring bats and the ecosystem they live in. Major challenges in automatic bat call identification are high call variability, similarities between species, interfering calls and lack of annotated data. Many currently available models suffer from relatively poor performance on real-life data due to being trained on single call datasets and, moreover, are often too slow for real-time classification. Here, we propose a Transformer architecture for multi-label classification with potential applications in real-time classification scenarios. We train our model on synthetically generated multi-species recordings by merging multiple bats calls into a single recording with multiple simultaneous calls. Our approach achieves a single species accuracy of 88.92% (F1-score of 84.23%) and a multi species macro F1-score of 74.40% on our test set. In comparison to three other tools on the independent and publicly available dataset ChiroVox, our model achieves at least 25.82% better accuracy for single species classification and at least 6.9% better macro F1-score for multi species classification., Comment: Volume 78, December 2023, 102288

Inferring the state and unknown parameters of a network of coupled oscillators, such as neurons in the brain, is of utmost importance. This task is made harder when only partial and noisy observations are available, which is a typical scenario in realistic high-dimensional systems. The general task of inference falls under data assimilation, and a commonly used assimilation method is the Ensemble Kalman Filter. Employing network-specific localization of the forecast covariance, an Ensemble Kalman Filter with state space augmentation is shown to yield highly accurate estimates of both the oscillator phases and unknown model parameters in the case where only a subset of oscillator phases are observed. In contrast, standard data assimilation methods yield poor results. We demonstrate the effectiveness of our approach for Kuramoto oscillators and for networks of theta neurons, using a variety of network topologies.

The $\textrm{HRI}_\textrm{EUV}$ telescope was calibrated on ground at the Physikalisch-Technische Bundesanstalt (PTB), Germany's national metrology institute, using the Metrology Light Source (MLS) synchrotron in April 2017 during the calibration campaign of the Extreme Ultraviolet Imager (EUI) instrument onboard the Solar Orbiter mission. We use the pre-flight end-to-end calibration and component-level (mirror multilayer coatings, filters, detector) characterization results to establish the beginning-of-life performance of the $\textrm{HRI}_\textrm{EUV}$ telescope which shall serve as a reference for radiometric analysis and monitoring of the telescope in-flight degradation. Calibration activities at component level and end-to-end calibration of the instrument were performed at PTB/MLS synchrotron light source (Berlin, Germany) and the SOLEIL synchrotron. Each component optical property is measured and compared to its semi-empirical model. This pre-flight characterization is used to estimate the parameters of the semi-empirical models. The end-to-end response is measured and validated by comparison with calibration measurements, as well as with its main design performance requirements. The telescope spectral response semi-empirical model is validated by the pre-flight end-to-end ground calibration of the instrument. It is found that $\textrm{HRI}_\textrm{EUV}$ is a highly efficient solar EUV telescope with a peak efficiency superior to 1 e$^-$.ph$^{-1}$), low detector noise ($\approx$ 3 e- rms), low dark current at operating temperature, and a pixel saturation above 120 ke- in low-gain or combined image mode. The ground calibration also confirms a well-modeled spectral selectivity and rejection, and low stray light. The EUI instrument achieves state-of-the-art performance in terms of signal-to-noise and image spatial resolution.

Canards are a well-studied phenomenon in fast-slow ordinary differential equations implying the delayed loss of stability after the slow passage through a singularity. Recent studies have shown that the corresponding maps stemming from explicit Runge-Kutta discretizations, in particular the forward Euler scheme, exhibit significant distinctions to the continuous-time behavior: for folds, the delay in loss of stability is typically shortened whereas, for transcritical singularities, it is arbitrarily prolonged. We employ the method of modified equations, which correspond with the fixed discretization schemes up to higher order, to understand and quantify these effects directly from a fast-slow ODE, yielding consistent results with the discrete-time behavior and opening a new perspective on the wide range of (de-)stabilization phenomena along canards.

The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming framework SCIP. The focus of this paper is on the role of the SCIP Optimization Suite in supporting research. SCIP's main design principles are discussed, followed by a presentation of the latest performance improvements and developments in version 8.0, which serve both as examples of SCIP's application as a research tool and as a platform for further developments. Further, the paper gives an overview of interfaces to other programming and modeling languages, new features that expand the possibilities for user interaction with the framework, and the latest developments in several extensions built upon SCIP., Comment: arXiv admin note: substantial text overlap with arXiv:2112.08872

Paleoclimate proxies reveal abrupt transitions of the North Atlantic climate during past glacial intervals known as Dansgaard--Oeschger (DO) events. A central feature of DO events is a sudden warming of about 10$^{\circ}$C in Greenland marking the beginning of relatively mild phases termed interstadials. These exhibit gradual cooling over several hundred to a few thousand years until a final abrupt decline brings the temperatures back to cold stadial levels. As of now, the exact mechanism behind this millennia-scale variability remains inconclusive. Here, we propose an excitable model to explain Dansgaard-Oeschger cycles, where interstadials occur as noise-induced state space excursions. Our model comprises the mutual multi-scale interactions between four dynamical variables representing Arctic atmospheric temperatures, Nordic Seas' temperatures and sea ice cover, and the Atlantic Meridional Overturning Circulation. The model's atmosphere-ocean heat flux is moderated by the sea ice, which in turn is subject to large perturbations dynamically generated by fast-evolving intermittent noise. If supercritical, perturbations trigger interstadial-like state space excursions during which all four model variables undergo qualitative changes that consistently resemble the signature of interstadials in corresponding proxy records. As a physical intermittent process generating the noise we propose convective events in the ocean or atmospheric blocking events. Our model accurately reproduces the DO cycle shape, return times and the dependence of the interstadial and stadial durations on the background conditions. In contrast to the prevailing understanding that DO variability is based on bistability in the underlying dynamics, we show that multi-scale, monostable excitable dynamics provide a promising alternative to explain millennial-scale climate variability associated with DO events.

Absorption losses and laser-induced damage threshold (LIDT) are considered as the major constraint for the development of optical coatings for high-power laser optics. Such coatings require paramount properties like low losses due to optical absorption, high mechanical stability, and enhanced damage resistance to withstand high-intensity laser pulses. In this work, heterostructure films were developed by the intermixing of SiO$_2$ and HfO$_2$ using plasma-enhanced atomic layer deposition (PEALD) technique. Thin film characterization techniques such as spectroscopic ellipsometry, spectrophotometry, substrate curvature measurements, x-ray reflectivity, and Fourier transform infrared spectroscopy were employed for extracting optical constants, spectral inter-pretation, residual stress, layer formation, and functional groups present in the heterostructures, respectively. These heterostructures demonstrate tunable refractive index, bandgap, and improved optical losses and LIDT properties. The films were incorporated into antireflection coatings (multilayer stacks and graded index coatings) and the LIDT was determined at 355 nm wavelength by the R-on-1 method. Optical absorptions at the reported wavelengths were characterized using photothermal common-path interferometry and laser-induced deflection techniques.

We give a complete description and clarification of the structure of the Levy area correction to Ito/Stratonovich stochastic integrals arising as limits of time-reversible deterministic dynamical systems. In particular, we show that time-reversibility forces the Levy area to vanish only in very specific situations that are easily classified., Comment: Clarified/improved some of the statements. Added Sections 5 and 6

In the past decade, photoemission orbital tomography (POT) has evolved into a powerful tool to investigate the electronic structure of organic molecules adsorbed on (metallic) surfaces. By measuring the angular distribution of photoelectrons as a function of binding energy and making use of the momentum-space signature of molecular orbitals, POT leads to an orbital-resolved picture of the electronic density of states at the organic/metal interface. In this combined experimental and theoretical work, we apply POT to the prototypical organic $\pi$-conjugated molecule bisanthene (C$_{28}$H$_{14}$) which forms a highly oriented monolayer on a Cu(110) surface. Experimentally, we identify an unprecedented number of 13 $\pi$ and 12 $\sigma$ orbitals of bisanthene and measure their respective binding energies and spectral lineshapes at the bisanthene/Cu(110) interface. Theoretically, we perform density functional calculations for this interface employing four widely used exchange-correlation functionals from the families of the generalized gradient approximations as well as global and range-separated hybrid functionals. By analyzing the electronic structure in terms of orbital-projected density of states, we arrive at a detailed orbital-by-orbital assessment of theory vs. experiment. This allows us to benchmark the performance of the investigated functionals with regards to their capability of accounting for the orbital energy alignment at organic/metal interfaces.

Anomalous diffusion and L\'evy flights, which are characterized by the occurrence of random discrete jumps of all scales, have been observed in a plethora of natural and engineered systems, ranging from the motion of molecules to climate signals. Mathematicians have recently unveiled mechanisms to generate anomalous diffusion, both stochastically and deterministically. However, there exists to the best of our knowledge no explicit example of a spatially extended system which exhibits anomalous diffusion without being explicitly driven by L\'evy noise. We show here that the Landau-Lifshitz-Gilbert equation, a stochastic partial differential equation (SPDE), despite only driven by Gaussian white noise, exhibits superdiffusive behaviour. The anomalous diffusion is an entirely emergent behaviour and manifests itself in jumps in the location of its travelling front solution. Using a collective coordinate approach we reduce the SPDE to a set of stochastic differential equations (SDEs) driven by Gaussian white noise. This allows us to identify the mechanism giving rise to the anomalous diffusion as random widening events of the front interface.

The application of thermodynamic reasoning in the study of learning systems has a long tradition. Recently, new tools relating perfect thermodynamic adaptation to the adaptation process have been developed. These results, known as fluctuation theorems, have been tested experimentally in several physical scenarios and, moreover, they have been shown to be valid under broad mathematical conditions. Hence, although not experimentally challenged yet, they are presumed to apply to learning systems as well. Here we address this challenge by testing the applicability of fluctuation theorems in learning systems, more specifically, in human sensorimotor learning. In particular, we relate adaptive movement trajectories in a changing visuomotor rotation task to fully adapted steady-state behavior of individual participants. We find that human adaptive behavior in our task is generally consistent with fluctuation theorem predictions and discuss the merits and limitations of the approach.

We extend the data-driven method of Sparse Identification of Nonlinear Dynamics (SINDy) developed by Brunton et al, Proc. Natl. Acad. Sci USA 113 (2016) to the case of delay differential equations (DDEs). This is achieved in a bilevel optimization procedure by first applying SINDy for fixed delay and then subsequently optimizing the error of the reconstructed SINDy model over delay times. We test the SINDy-delay method on a noisy short data set from a toy delay differential equation and show excellent agreement. We then apply the method to experimental data of gene expressions in the bacterium {\it Pseudomonas aeruginosa} subject to the influence of zinc. The derived SINDy model suggests that the increase of zinc concentration mainly affects the time delay and not the strengths of the interactions between the different agents controlling the zinc export mechanism., Comment: 21 pages. To appear in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Majorization is a fundamental model of uncertainty with several applications in areas ranging from thermodynamics to entanglement theory, and constitutes one of the pillars of the resource-theoretic approach to physics. Here, we improve on its relation to measurement apparatuses. In particular, after discussing what the proper notion of second law in this scenario is, we show that, for a sufficiently large state space, any family of entropy-like functions constituting a second law must be countably infinite. Moreover, we provide an analogous result for a variation of majorization known as thermo-majorization which, in fact, does not require any constraint on the state space provided the equilibrium distribution is not uniform. Lastly, we discuss the applicability of our results to molecular diffusion and catalytic majorization. In this regard, we consider a variation of majorization used in plasma physics as a model of molecular diffusion and show that no finite family of entropy-like functions constituting a second law of molecular diffusion exists. Moreover, we show how our results are useful when dealing with a conjecture regarding catalytic majorization (i.e. trumping). In particular, we show that the sort of characterizations of trumping that have been considered before require an infinite family of real-valued functions., Comment: Accepted for publication at Proceedings of the Royal Society A

Phase transitions in equilibrium and nonequilibrium systems play a major role in the natural sciences. In dynamical networks, phase transitions organize qualitative changes in the collective behavior of coupled dynamical units. Adaptive dynamical networks feature a connectivity structure that changes over time, co-evolving with the nodes' dynamical state. In this Letter, we show the emergence of two distinct first-order nonequilibrium phase transitions in a finite size adaptive network of heterogeneous phase oscillators. Depending on the nature of defects in the internal frequency distribution, we observe either an abrupt single-step transition to full synchronization or a more gradual multi-step transition. This observation has a striking resemblance to heterogeneous nucleation. We develop a mean-field approach to study the interplay between adaptivity and nodal heterogeneity and describe the dynamics of multicluster states and their role in determining the character of the phase transition. Our work provides a theoretical framework for studying the interplay between adaptivity and nodal heterogeneity.

Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate such notions from Turing machines to uncountable spaces. Since these machines are used as a baseline for computability in these approaches, countability restrictions on the ordered structures are fundamental. Here, we show several relations between the usual countability restrictions in order-theoretic theories of computability and some more common order-theoretic countability constraints, like order density properties and functional characterizations of the order structure in terms of multi-utilities. As a result, we show how computability can be introduced in some order structures via countability order density and multi-utility constraints., Comment: Accepted in Mathematical Structures in Computer Science (Cambridge University Press)

Although there is a somewhat standard formalization of computability on countable sets given by Turing machines, the same cannot be said about uncountable sets. Among the approaches to define computability in these sets, order-theoretic structures have proven to be useful. Here, we discuss the mathematical structure needed to define computability using order-theoretic concepts. In particular, we introduce a more general framework and discuss its limitations compared to the previous one in domain theory. We expose four features in which the stronger requirements in the domain-theoretic structure allow to improve upon the more general framework: computable elements, computable functions, model dependence of computability and complexity theory. Crucially, we show computability of elements in uncountable spaces can be defined in this new setup, and argue why this is not the case for computable functions. Moreover, we show the stronger setup diminishes the dependence of computability on the chosen order-theoretic structure and that, although a suitable complexity theory can be defined in the stronger framework and the more general one posesses a notion of computable elements, there appears to be no proper notion of element complexity in the latter.

Presolving has become an essential component of modern MIP solvers both in terms of computational performance and numerical robustness. In this paper, we present PaPILO, a new C++ header-only library that provides a large set of presolving routines for MIP and LP problems from the literature. The creation of PaPILO was motivated by the current lack of (a) solver-independent implementations that (b) exploit parallel hardware, and (c) support multiprecision arithmetic. Traditionally, presolving is designed to be fast. Whenever necessary, its low computational overhead is usually achieved by strict working limits. PaPILO's parallelization framework aims at reducing the computational overhead also when presolving is executed more aggressively or is applied to large-scale problems. To rule out conflicts between parallel presolve reductions, PaPILO uses a transaction-based design. This helps to avoid both the memory-intensive allocation of multiple copies of the problem and special synchronization between presolvers. Additionally, the use of Intel's TBB library aids PaPILO to efficiently exploit recursive parallelism within expensive presolving routines such as probing, dominated columns, or constraint sparsification. We provide an overview of PaPILO's capabilities and insights into important design choices.