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Soft robots can revolutionize several applications with high demands on dexterity and safety. When operating these systems, real-time estimation and control require fast and accurate models. However, prediction with first-principles (FP) models is slow, and learned black-box models have poor generalizability. Physics-informed machine learning offers excellent advantages here, but it is currently limited to simple, often simulated systems without considering changes after training. We propose physics-informed neural networks (PINNs) for articulated soft robots (ASRs) with a focus on data efficiency. The amount of expensive real-world training data is reduced to a minimum - one dataset in one system domain. Two hours of data in different domains are used for a comparison against two gold-standard approaches: In contrast to a recurrent neural network, the PINN provides a high generalizability. The prediction speed of an accurate FP model is improved with the PINN by up to a factor of 466 at slightly reduced accuracy. This enables nonlinear model predictive control (MPC) of the pneumatic ASR. In nine dynamic MPC experiments, an average joint-tracking error of 1.3{\deg} is achieved.

We present HoloGraphs, a novel approach for physically representing, explaining, exploring, and interacting with dynamic networks. HoloGraphs addresses the challenges of visualizing and understanding evolving network structures by providing an engaging method of interacting and exploring dynamic network structures using physicalization techniques. In contrast to traditional digital interfaces, our approach leverages tangible artifacts made from transparent materials to provide an intuitive way for people with low visualization literacy to explore network data. The process involves printing network embeddings on transparent media and assembling them to create a 3D representation of dynamic networks, maintaining spatial perception and allowing the examination of each timeslice individually. Interactivity is envisioned using optional Focus+Context layers and overlays for node trajectories and labels. Focus layers highlight nodes of interest, context layers provide an overview of the network structure, and global overlays show node trajectories over time. In this paper, we outline the design principles and implementation of HoloGraphs and present how elementary digital interactions can be mapped to physical interactions to manipulate the elements of a network and temporal dimension in an engaging matter. We demonstrate the capabilities of our concept in a case study. Using a dynamic network of character interactions from a popular book series, we showcase how it represents and supports understanding complex concepts such as dynamic networks.

QCD jets are considered important probes for quark gluon plasma created in collisions of nuclei at high energies. Their parton showers are significantly altered if they develop inside of a deconfined medium. Hadronization of jets is also thought to be affected by the presence of quarks and gluons. We present a systematic study of the effects of a thermal bath of partons on the hadronization of parton showers. We use the JETSCAPE framework to create parton showers both in vacuum and in a brick of quark gluon plasma. The brick setup allows important parameters, like the size of the plasma as well as the collective flow of partons, to be varied systematically. We hadronize the parton showers using Hybrid Hadronization, which permits shower partons to form strings with thermal partons, or to recombine directly with thermal partons as well as with each other. We find a sizeable amount of interaction of shower partons with thermal partons during hadronization, indicating a natural continuation of the interaction of jet and medium during this stage. The observed effects grow with the size of the medium. Collective flow easily transfers from the thermal partons onto the emerging jet hadrons. We also see a significant change in hadron chemistry as expected in the presence of quark recombination processes., Comment: 12 pages, 6 figures

The assimilation and prediction of phase-resolved surface gravity waves are critical challenges in ocean science and engineering. Potential flow theory (PFT) has been widely employed to develop wave models and numerical techniques for wave prediction. However, traditional wave prediction methods are often limited. For example, most simplified wave models have a limited ability to capture strong wave nonlinearity, while fully nonlinear PFT solvers often fail to meet the speed requirements of engineering applications. This computational inefficiency also hinders the development of effective data assimilation techniques, which are required to reconstruct spatial wave information from sparse measurements to initialize the wave prediction. To address these challenges, we propose a novel solver method that leverages physics-informed neural networks (PINNs) that parameterize PFT solutions as neural networks. This provides a computationally inexpensive way to assimilate and predict wave data. The proposed PINN framework is validated through comparisons with analytical linear PFT solutions and experimental data collected in a laboratory wave flume. The results demonstrate that our approach accurately captures and predicts irregular, nonlinear, and dispersive wave surface dynamics. Moreover, the PINN can infer the fully nonlinear velocity potential throughout the entire fluid volume solely from surface elevation measurements, enabling the calculation of fluid velocities that are difficult to measure experimentally., Comment: 22 pages, 12 Figures, preprint

This work introduces a novel application for predicting the macroscopic intrinsic permeability tensor in deformable porous media, using a limited set of micro-CT images of real microgeometries. The primary goal is to develop an efficient, machine-learning (ML)-based method that overcomes the limitations of traditional permeability estimation techniques, which often rely on time-consuming experiments or computationally expensive fluid dynamics simulations. The novelty of this work lies in leveraging Convolutional Neural Networks (CNN) to predict pore-fluid flow behavior under deformation and anisotropic flow conditions. Particularly, the described approach employs binarized CT images of porous micro-structure as inputs to predict the symmetric second-order permeability tensor, a critical parameter in continuum porous media flow modeling. The methodology comprises four key steps: (1) constructing a dataset of CT images from Bentheim sandstone at different volumetric strain levels; (2) performing pore-scale simulations of single-phase flow using the lattice Boltzmann method (LBM) to generate permeability data; (3) training the CNN model with the processed CT images as inputs and permeability tensors as outputs; and (4) exploring techniques to improve model generalization, including data augmentation and alternative CNN architectures. Examples are provided to demonstrate the CNN's capability to accurately predict the permeability tensor, a crucial parameter in various disciplines such as geotechnical engineering, hydrology, and material science. An exemplary source code is made available for interested readers., Comment: 21 pages, 12 figures, research paper

An investigation of high transverse momentum (high-$p_T$) photon triggered jets in proton-proton ($p$-$p$) and ion-ion ($A$-$A$) collisions at $\sqrt{s_{NN}} = 0.2$ and $5.02~\mathrm{TeV}$ is carried out, using the multistage description of in-medium jet evolution. Monte Carlo simulations of hard scattering and energy loss in heavy-ion collisions are performed using parameters tuned in a previous study of the nuclear modification factor ($R_{AA}$) for inclusive jets and high-$p_T$ hadrons. We obtain a good reproduction of the experimental data for photon triggered jet $R_{AA}$, as measured by the ATLAS detector, the distribution of the ratio of jet to photon $p_T$ ($X_{\rm J \gamma}$), measured by both CMS and ATLAS, and the photon-jet azimuthal correlation as measured by CMS. We obtain a moderate description of the photon triggered jet $I_{AA}$, as measured by STAR. A noticeable improvement in the comparison is observed when one goes beyond prompt photons and includes bremsstrahlung and decay photons, revealing their significance in certain kinematic regions, particularly at $X_{J\gamma} > 1$. Moreover, azimuthal angle correlations demonstrate a notable impact of non-prompt photons on the distribution, emphasizing their role in accurately describing experimental results. This work highlights the success of the multistage model of jet modification to straightforwardly predict (this set of) photon triggered jet observables. This comparison, along with the role played by non-prompt photons, has important consequences on the inclusion of such observables in a future Bayesian analysis., Comment: 16 pages, 10 figures

Quantum Reservoir Computing (QRC) leverages quantum systems to perform complex computational tasks with exceptional efficiency and reduced energy consumption. We introduce a minimalistic QRC framework utilizing only a few two-level atoms in a single-mode optical cavity, combined with continuous quantum measurements. To achieve high computational expressivity with minimal hardware, we include two critical elements: reservoir feedback and polynomial regression. Reservoir feedback modifies the reservoir's dynamics without altering its hardware, while polynomial regression enhances output resolution by nonlinearly extending expressions. We evaluate QRC's memory retention and nonlinear data processing through two tasks: predicting chaotic time-series data via the Mackey-Glass task and classifying sine-square waveforms. Our results demonstrate significant QRC performance with minimal reservoirs containing as few as five atoms, further enhanced by feedback mechanisms and polynomial regression. This framework fulfills QRC's objectives to minimize hardware size and energy consumption, marking a significant advancement in integrating quantum physics with machine learning technology., Comment: 10 pages and 6 figures in the main text; 5 pages and 3 figures in the supplementary information. arXiv admin note: substantial text overlap with arXiv:2405.04799

Soft robots pose difficulties in terms of control, requiring novel strategies to effectively manipulate their compliant structures. Model-based approaches face challenges due to the high dimensionality and nonlinearities such as hysteresis effects. In contrast, learning-based approaches provide nonlinear models of different soft robots based only on measured data. In this paper, recurrent neural networks (RNNs) predict the behavior of an articulated soft robot (ASR) with five degrees of freedom (DoF). RNNs based on gated recurrent units (GRUs) are compared to the more commonly used long short-term memory (LSTM) networks and show better accuracy. The recurrence enables the capture of hysteresis effects that are inherent in soft robots due to viscoelasticity or friction but cannot be captured by simple feedforward networks. The data-driven model is used within a nonlinear model predictive control (NMPC), whereby the correct handling of the RNN's hidden states is focused. A training approach is presented that allows measured values to be utilized in each control cycle. This enables accurate predictions of short horizons based on sensor data, which is crucial for closed-loop NMPC. The proposed learning-based NMPC enables trajectory tracking with an average error of 1.2deg in experiments with the pneumatic five-DoF ASR., Comment: Accepted for publication in IEEE Robotics and Automation Letters (RA-L) 2024

Chains of co-Buechi automata (COCOA) have recently been introduced as a new canonical representation of omega-regular languages. The co-Buechi automata in a chain assign to each omega-word its natural color, which depends only on the language itself and not on its automaton representation. The automata in such a chain can be minimized in polynomial time and are good-for-games, making the representation attractive for verification and reactive synthesis applications. However, since in such applications, a specification is usually given in linear temporal logic (LTL), to make COCOA useful, the specification first has to be translated into such a chain of automata. Currently, the only known translation procedure involves a detour through deterministic parity automata (LTL to DPW to COCOA), where the first step neglects the natural colors and requires intricate constructions by Safra or Esparza et al. This observation raises the question whether, by leveraging the definition of the natural color of words, these complex constructions can be avoided, leading to a more direct translation from LTL to COCOA. In this paper, we present a surprisingly simple translation from LTL to COCOA, along with a variant that translates from LTL to DPW. Our procedure relies on standard operations on weak alternating automata, Miyano-Hayashi's breakpoint construction, an augmented subset construction, and simple graph algorithms. With weak alternating automata as a starting point, the procedure can also handle specifications in linear dynamic logic. Although the translation procedure runs in sub-optimal triply-exponential time (vs. doubly-exponential optimal), it constitutes a novel path for translating from LTL to DPW, avoiding the aforementioned intricate constructions entirely., Comment: there was a bug in complexity estimate; 2exp -> 3exp

Refrigerated truck trailers are currently mainly operated with environmentally harmful diesel units; an alternative is to operate the refrigeration unit with electrical energy. However, this requires a battery, the size of which can be reduced by using a recuperation axle, which recovers energy during braking. Current systems work purely reactively and often in so-called towing mode, in which a generator torque is provided without a braking request from the driver in order to secure the energy supply. However, this drag leads to additional consumption in the truck. This work quantifies the potential of predictive energy management that uses route and environmental data to minimize CO2 emissions. This was done using simulation data obtained with the help of VECTO. It was shown that there is still considerable potential for savings, so this paper provides an important basis for the later development of predictive energy management and, thus, for the electrification of refrigerated truck transports.

Intelligent real-world systems critically depend on expressive information about their system state and changing operation conditions, e.g., due to variation in temperature, location, wear, or aging. To provide this information, online inference and learning attempts to perform state estimation and (partial) system identification simultaneously. Current works combine tailored estimation schemes with flexible learning-based models but suffer from convergence problems and computational complexity due to many degrees of freedom in the inference problem (i.e., parameters to determine). To resolve these issues, we propose a procedure for data-driven offline conditioning of a highly flexible Gaussian Process (GP) formulation such that online learning is restricted to a subspace, spanned by expressive basis functions. Due to the simplicity of the transformed problem, a standard particle filter can be employed for Bayesian inference. In contrast to most existing works, the proposed method enables online learning of target functions that are nested nonlinearly inside a first-principles model. Moreover, we provide a theoretical quantification of the error, introduced by restricting learning to a subspace. A Monte-Carlo simulation study with a nonlinear battery model shows that the proposed approach enables rapid convergence with significantly fewer particles compared to a baseline and a state-of-the-art method., Comment: 8 pages, 4 figures, accepted for presentation at 2024 Conf. on Decision and Control

The JETSCAPE Collaboration reports a new determination of the jet transport parameter $\hat{q}$ in the Quark-Gluon Plasma (QGP) using Bayesian Inference, incorporating all available inclusive hadron and jet yield suppression data measured in heavy-ion collisions at RHIC and the LHC. This multi-observable analysis extends the previously published JETSCAPE Bayesian Inference determination of $\hat{q}$, which was based solely on a selection of inclusive hadron suppression data. JETSCAPE is a modular framework incorporating detailed dynamical models of QGP formation and evolution, and jet propagation and interaction in the QGP. Virtuality-dependent partonic energy loss in the QGP is modeled as a thermalized weakly-coupled plasma, with parameters determined from Bayesian calibration using soft-sector observables. This Bayesian calibration of $\hat{q}$ utilizes Active Learning, a machine--learning approach, for efficient exploitation of computing resources. The experimental data included in this analysis span a broad range in collision energy and centrality, and in transverse momentum. In order to explore the systematic dependence of the extracted parameter posterior distributions, several different calibrations are reported, based on combined jet and hadron data; on jet or hadron data separately; and on restricted kinematic or centrality ranges of the jet and hadron data. Tension is observed in comparison of these variations, providing new insights into the physics of jet transport in the QGP and its theoretical formulation., Comment: 20 pages, 10 figures, 2 tables, submitted to PRC; updated acknowledgements

A new framework, called x-scape, for the combined study of both hard and soft transverse momentum sectors in high energy proton-proton ($p$-$p$) and proton-nucleus ($p$-$A$) collisions is set up. A dynamical initial state is set up using the 3d-Glauber model with transverse locations of hotspots within each incoming nucleon. A hard scattering that emanates from two colliding hotspots is carried out using the Pythia generator. Initial state radiation from the incoming hard partons is carried out in a new module called I-matter, which includes the longitudinal location of initial splits. The energy-momentum of both the initial hard partons and their associated beam remnants is removed from the hot spots, depleting the energy-momentum available for the formation of the bulk medium. Outgoing showers are simulated using the matter generator, and results are presented for both cases, allowing for and not allowing for energy loss. First comparisons between this hard-soft model and single inclusive hadron and jet data from $p$-$p$ and minimum bias $p$-$Pb$ collisions are presented. Single hadron spectra in $p$-$p$ are used to carry out a limited (in number of parameters) Bayesian calibration of the model. Fair comparisons with data are indicative of the utility of this new framework. Theoretical studies of the correlation between jet $p_T$ and event activity at mid and forward rapidity are carried out., Comment: 18 pages, 15 figures

As part of the automation of commercial vehicles, the number of assistance systems in this field is continuously increasing. The semitrailer plays an important role for the vehicles driving dynamics due to its highly varying loads and the large proportion to the total mass of the truck-semitrailer, especially when it is fully loaded. To create a basis for further development of assistance systems for the semitrailer, this paper presents a two-track model which includes the lateral and roll dynamics of the semitrailer. This enables the future development of observer and filter-based estimation of vehicle states and parameters, which are impossible or very difficult to measure. For offline identification of the unknown model parameters, a Particle-Swarm-Optimization (PSO) algorithm will be used. The validation of the model is based on measurements from a test vehicle. The focus is on the lateral and vertical tire forces of the semitrailer, which are measured at the test vehicle using strain gauges., Comment: in German language. VDI Mechatroniktagung 2021

Advanced driver assistance systems are critically dependent on reliable and accurate information regarding a vehicles' driving state. For estimation of unknown quantities, model-based and learning-based methods exist, but both suffer from individual limitations. On the one hand, model-based estimation performance is often limited by the models' accuracy. On the other hand, learning-based estimators usually do not perform well in "unknown" conditions (bad generalization), which is particularly critical for semitrailers as their payload changes significantly in operation. To the best of the authors' knowledge, this work is the first to analyze the capability of state-of-the-art estimators for semitrailers to generalize across "unknown" loading states. Moreover, a novel hybrid Extended Kalman Filter (H-EKF) that takes advantage of accurate Artificial Neural Network (ANN) estimates while preserving reliable generalization capability is presented. It estimates the articulation angle between truck and semitrailer, lateral tire forces and the truck steering angle utilizing sensor data of a standard semitrailer only. An experimental comparison based on a full-scale truck-semitrailer combination indicates the superiority of the H-EKF compared to a state-of-the-art extended Kalman filter and an ANN estimator., Comment: 8 pages, 3 figures, Presented at 2024 European Control Conference (ECC)

Reservoir computing is a form of machine learning that utilizes nonlinear dynamical systems to perform complex tasks in a cost-effective manner when compared to typical neural networks. Many recent advancements in reservoir computing, in particular quantum reservoir computing, make use of reservoirs that are inherently stochastic. However, the theoretical justification for using these systems has not yet been well established. In this paper, we investigate the universality of stochastic reservoir computers, in which we use a stochastic system for reservoir computing using the probabilities of each reservoir state as the readout instead of the states themselves. In stochastic reservoir computing, the number of distinct states of the entire reservoir computer can potentially scale exponentially with the size of the reservoir hardware, offering the advantage of compact device size. We prove that classes of stochastic echo state networks, and therefore the class of all stochastic reservoir computers, are universal approximating classes. We also investigate the performance of two practical examples of stochastic reservoir computers in classification and chaotic time series prediction. While shot noise is a limiting factor in the performance of stochastic reservoir computing, we show significantly improved performance compared to a deterministic reservoir computer with similar hardware in cases where the effects of noise are small., Comment: 30 pages, 6 figures

We conducted a study to systematically investigate the communication of complex dynamic processes along a two-dimensional design space, where the axes represent a representation's manifestation (physical or virtual) and operation (manual or automatic). We exemplify the design space on a model embodying cardiovascular pathologies, represented by a mechanism where a liquid is pumped into a draining vessel, with complications illustrated through modifications to the model. The results of a mixed-methods lab study with 28 participants show that both physical manifestation and manual operation have a strong positive impact on the audience's engagement. The study does not show a measurable knowledge increase with respect to cardiovascular pathologies using manually operated physical representations. However, subjectively, participants report a better understanding of the process-mainly through non-visual cues like haptics, but also auditory cues. The study also indicates an increased task load when interacting with the process, which, however, seems to play a minor role for the participants. Overall, the study shows a clear potential of physicalization for the communication of complex dynamic processes, which only fully unfold if observers have to chance to interact with the process.

Quantum Reservoir Computing (QRC) harnesses quantum systems to tackle intricate computational problems with exceptional efficiency and minimized energy usage. This paper presents a QRC framework that utilizes a minimalistic quantum reservoir, consisting of only a few two-level atoms within an optical cavity. The system is inherently scalable, as newly added atoms automatically couple with the existing ones through the shared cavity field. We demonstrate that the quantum reservoir outperforms traditional classical reservoir computing in both memory retention and nonlinear data processing through two tasks, namely the prediction of time-series data using the Mackey-Glass task and the classification of sine-square waveforms. Our results show significant performance improvements with an increasing number of atoms, facilitated by non-destructive, continuous quantum measurements and polynomial regression techniques. These findings confirm the potential of QRC as a practical and efficient solution to addressing complex computational challenges in quantum machine learning., Comment: 9 pages, 8 figures

Generalized Reactivity(1) (GR(1)) synthesis is a reactive synthesis approach in which the specification is split into two parts: a symbolic game graph, describing the safe transitions of a system, a liveness specification in a subset of Linear Temporal Logic (LTL) on top of it. Many specifications can naturally be written in this restricted form, and the restriction gives rise to a scalable synthesis procedure -- the reasons for the high popularity of the approach. For specifications even slightly beyond GR(1), however, the approach is inapplicable. This necessitates a transition to synthesizers for full LTL specifications, introducing a huge efficiency drop. This paper proposes a synthesis approach that smoothly bridges the efficiency gap from GR(1) to LTL by unifying synthesis for both classes of specifications. The approach leverages a recently introduced canonical representation of omega-regular languages based on a chain of good-for-games co-B\"uchi automata (COCOA). By constructing COCOA for the liveness part of a specification, we can then build a fixpoint formula that can be efficiently evaluated on the symbolic game graph. The COCOA-based synthesis approach outperforms standard approaches and retains the efficiency of GR(1) synthesis for specifications in GR(1) form and those with few non-GR(1) specification parts., Comment: This is an extended version of the paper accepted at TACAS'24

Prompt photons are created in the early stages of heavy ion collisions and traverse the QGP medium without any interaction. Therefore, photon-triggered jets can be used to study the jet quenching in the QGP medium. In this work, photon-triggered jets are studied through different jet and jet substructure observables for different collision systems and energies using the JETSCAPE framework. Since the multistage evolution used in the JETSCAPE framework is adequate to describe a wide range of experimental observables simultaneously using the same parameter tune, we use the same parameters tuned for jet and leading hadron studies. The same isolation criteria used in the experimental analysis are used to identify prompt photons for better comparison. For the first time, high-accuracy JETSCAPE results are compared with multi-energy LHC and RHIC measurements to better understand the deviations observed in prior studies. This study highlights the importance of multistage evolution for the simultaneous description of experimental observables through different collision systems and energies using a single parameter tune., Comment: 4 pages, 5 figures. Proceedings of Quark Matter 2023 - XXXth International Conference on Ultra-relativistic Nucleus-Nucleus Collisions, Houston, TX, 3-9 September 2023

The measurement of deep water gravity wave elevations using in-situ devices, such as wave gauges, typically yields spatially sparse data. This sparsity arises from the deployment of a limited number of gauges due to their installation effort and high operational costs. The reconstruction of the spatio-temporal extent of surface elevation poses an ill-posed data assimilation problem, challenging to solve with conventional numerical techniques. To address this issue, we propose the application of a physics-informed neural network (PINN), aiming to reconstruct physically consistent wave fields between two designated measurement locations several meters apart. Our method ensures this physical consistency by integrating residuals of the hydrodynamic nonlinear Schr\"{o}dinger equation (NLSE) into the PINN's loss function. Using synthetic wave elevation time series from distinct locations within a wave tank, we initially achieve successful reconstruction quality by employing constant, predetermined NLSE coefficients. However, the reconstruction quality is further improved by introducing NLSE coefficients as additional identifiable variables during PINN training. The results not only showcase a technically relevant application of the PINN method but also represent a pioneering step towards improving the initialization of deterministic wave prediction methods., Comment: 16 pages with 11 figures

The JETSCAPE Collaboration reports the first multi-messenger study of the QGP jet transport parameter $\hat{q}$ using Bayesian inference, incorporating all available hadron and jet inclusive yield and jet substructure data from RHIC and the LHC. The theoretical model utilizes virtuality-dependent in-medium partonic energy loss coupled to a detailed dynamical model of QGP evolution. Tension is observed when constraining $\hat{q}$ for different kinematic cuts of the inclusive hadron data. The addition of substructure data is shown to improve the constraint on $\hat{q}$, without inducing tension with the constraint due to inclusive observables. These studies provide new insight into the mechanisms of jet interactions in matter, and point to next steps in the field for comprehensive understanding of jet quenching as a probe of the QGP., Comment: 4 pages, 2 figures. Proceedings of Quark Matter 2023 - XXXth International Conference on Ultra-relativistic Nucleus-Nucleus Collisions, Houston, TX, 3-9 September 2023

Considerable information about the early-stage dynamics of heavy-ion collisions is encoded in the rapidity dependence of measurements. To leverage the large amount of experimental data, we perform a systematic analysis using three-dimensional hydrodynamic simulations of multiple collision systems -- large and small, symmetric and asymmetric. Specifically, we perform fully 3D multi-stage hydrodynamic simulations initialized by a parameterized model for rapidity-dependent energy deposition, which we calibrate on the hadron multiplicity and anisotropic flow coefficients. We utilize Bayesian inference to constrain properties of the early- and late- time dynamics of the system, and highlight the impact of enforcing global energy conservation in our 3D model.

Reservoir computing, using nonlinear dynamical systems, offers a cost-effective alternative to neural networks for complex tasks involving processing of sequential data, time series modeling, and system identification. Echo state networks (ESNs), a type of reservoir computer, mirror neural networks but simplify training. They apply fixed, random linear transformations to the internal state, followed by nonlinear changes. This process, guided by input signals and linear regression, adapts the system to match target characteristics, reducing computational demands. A potential drawback of ESNs is that the fixed reservoir may not offer the complexity needed for specific problems. While directly altering (training) the internal ESN would reintroduce the computational burden, an indirect modification can be achieved by redirecting some output as input. This feedback can influence the internal reservoir state, yielding ESNs with enhanced complexity suitable for broader challenges. In this paper, we demonstrate that by feeding some component of the reservoir state back into the network through the input, we can drastically improve upon the performance of a given ESN. We rigorously prove that, for any given ESN, feedback will almost always improve the accuracy of the output. For a set of three tasks, each representing different problem classes, we find that with feedback the average error measures are reduced by $30\%-60\%$. Remarkably, feedback provides at least an equivalent performance boost to doubling the initial number of computational nodes, a computationally expensive and technologically challenging alternative. These results demonstrate the broad applicability and substantial usefulness of this feedback scheme., Comment: 36 pages, 6 figures