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This letter addresses the uncertainty quantification of a power network and is based on surrogate models built via Machine Learning techniques. Specifically, the least-square support vector machine regression is combined with the principal component analysis to generate a compressed surrogate model capable of predicting all the nodal voltages of the network as a function of the uncertain electrical parameters of the transmission lines. The surrogate model is built from a limited number of system responses provided by the computational model. The power flow analysis of the benchmark IEEE-118 bus system with 250 parameters is considered as a test case. The performance of the proposed modeling strategy in terms of accuracy, efficiency and convergence, are assessed and compared with those of an alternative surrogate model based on a sparse implementation of the polynomial chaos expansion. The results of a Monte Carlo simulation are used as reference in the above comparison.

This article presents an innovative technique for the black-box modeling of the conductive emissions of switching circuits from external observations. The proposed methodology relies on the robust and elegant theory of periodically switched linear systems, providing the user with a tool for the generation of frequency-domain augmented linear time-invariant equivalents. The obtained models offer both a remarkable accuracy for different supply conditions and a deep understanding of the inherent switching mechanisms of this class of devices. The proposed framework also enables the derivation of a simplified enhanced Norton representation which outperforms the classical state-of-the-art models based on linear time-invariant approximations. Model parameters are computed via a blind procedure based only on two measurements. The feasibility and strength of the proposed approach are demonstrated on a dc–dc boost power converter and on a dc electrical motor via the prediction of their conducted disturbances.

This paper deals with the development of a Machine Learning (ML)-based regression for the construction of complex-valued surrogate models for the analysis of the frequency-domain responses of electromagnetic (EM) structures. The proposed approach relies on the combination of two-techniques: (i) the principal component analysis (PCA) and (ii) an unusual complex-valued formulation of the Least Squares Support Vector Machine (LS-SVM) regression. First, the training and test dataset is obtained from a set of parametric electromagnetic simulations. The spectra collected in the training set are compressed via the PCA by exploring the correlation among the available data. In the next step, the compressed dataset is used for the training of compact set of complex-valued surrogate models and their accuracy is evaluated on the test samples. The effectiveness and the performance of the complex-valued LS-SVM regression with three kernel functions are investigated on two application examples consisting of a serpentine delay structure with three parameters and a high-speed link with four parameters. Moreover, for the last example, the performance of the proposed approach is also compared with those provided by a real-valued multi-output feedforward Neural Network model.

This paper presents an innovative modeling strategy for the construction of efficient and compact surrogate models for the uncertainty quantification of time-domain responses of digital links. The proposed approach relies on a two-step methodology. First, the initial dataset of available training responses is compressed via principal component analysis (PCA). Then, the compressed dataset is used to train compact surrogate models for the reduced PCA variables using advanced techniques for uncertainty quantification and parametric macromodeling. Specifically, in this work sparse polynomial chaos expansion and least-square support-vector machine regression are used, although the proposed methodology is general and applicable to any surrogate modeling strategy. The preliminary compression allows limiting the number and complexity of the surrogate models, thus leading to a substantial improvement in the efficiency. The feasibility and performance of the proposed approach are investigated by means of two digital link designs with 54 and 115 uncertain parameters, respectively.

Most of the simulation studies on energy networks, including gas grids, derive their results from a limited number of network models. The findings of these works are therefore affected by a substantial case-specificity, which partially limits their validity and prevents their generalisation. To overcome this limitation, the present work proposes a novel statistical-based approach for studying distribution gas networks, enabled by a generator of random gas grids with accurate technical designs and structural features. Ten-thousand random and unique networks are produced in three different tests, where increasingly tight constraints are applied to the synthetisation process for a higher control over the generated grids. The experiments verify the accuracy of the tool and highlight that substantial variations can be found in the hydraulic behaviour (pressures and gas velocities) and structural properties (pipe diameters and network volumes) of real-world gas networks. The observed 10,000 gas grids evidence the information gain offered by statistical-based approaches with respect to traditional case-specific studies. The tool opens a broad range of applications which include, but are not limited to, statistical analyses on the distributed injection of alternative gases, like hydrogen, in integrated, low-carbon, energy systems.

This paper presents an automatic procedure for the optimization of the hyperparameters of a delay rational model approximating the frequency-domain behavior of high-speed interconnects. The proposed model is built via a kernel-based regression, such as the Least-Square Support Vector Machine (LS-SVM), by considering an ad-hoc kernel with two hyperparameters related to the propagation delays introduced by the system. Such hyperparameters, along with the Tikhonov regularizer used by the LS-SVM regression, are carefully tuned via an automatic approach based on a k-fold cross-validation and Bayesian optimization. The feasibility of the effectiveness of the proposed modeling approach are investigated on a high-speed link.

This paper introduces a probabilistic nonparametric surrogate model based on Gaussian process regression to perform uncertainty quantification tasks with the inclusion of confidence bounds on the predicted statistics. The performance of the proposed method is compared against two state-of-the-art techniques, namely the parametric sparse polynomial chaos expansion and the nonparametric least-square support vector machine regression.

International audience; This paper deals with the application of the partial least squares (PLS) regression to the uncertainty quantification of an integrated wireless power transfer with 30 random variables. It considers the development of surrogate models using a limited set of training samples in order to estimate statistical quantities of the converter efficiency with a relatively low computational cost compared to the standard brute force Monte Carlo (MC) simulation. The strength, the performance and the features of the proposed modeling approach are then compared with the ones of an advanced probabilistic surrogate model combining the Least-squares Support Vector Machine (LS-SVM) and the Gaussian Process (GP) regression. The MC simulation is considered as reference.

International audience; This paper focuses on the application of the partial least squares (PLS) regression to the uncertainty quantification of the responses of complex stochastic systems. It considers the development of a surrogate model using a limited set of training samples in order to estimate statistical quantities of the system output with relatively low computational cost compared to the standard brute force Monte Carlo (MC) simulation. The performance and the strength of the proposed modeling scheme is investigated for an integrated voltage regulator (IVR) with 8 random variables. The results highlight the ability of the PLS regression to deals with complex nonlinear problems with very few principal components, also providing important insights about the input variables. © 2020 IEEE.

This paper deals with the application of the support vector machine (SVM) and the least-squares SVM regressions to the uncertainty quantification of complex systems with a high-dimensional parameter space. The above regression techniques are used to build accurate and compact surrogate models of the system responses from a limited set of training samples. The accuracy and the feasibility of the proposed modeling techniques are then investigated by comparing their results with the ones predicted by a sparse polynomial chaos expansion by considering two real-life problems with 8 and 30 random variables, respectively.

This paper investigates the application of support vector machine to the modeling of high-speed interconnects with largely varying and/or highly uncertain design parameters. The proposed method relies on a robust and well-established mathematical framework, yielding accurate surrogates of complex dynamical systems. An identification procedure based on the observation of a small set of system responses allows generating compact parametric relations, which can be used for design optimization and/or stochastic analysis. The feasibility and strength of the method are demonstrated based on a benchmark function and on the statistical assessment of a realistic printed circuit board interconnect, highlighting the main features and benefits of this technique over state-of-the-art solutions. Emphasis is given to the effects of the initial sample size and of input noise on the model estimation.

This brief presents a novel approach to predict the bounds of the time-domain response of a linear system subject to multiple bounded uncertain input parameters. The method leverages the framework of Taylor models in conjunction with the numerical inversion of Laplace transform (NILT). Different formulations of the NILT are reviewed, and their advantages and limitations are discussed. An implementation relying on an inverse fast Fourier transform turns out to be the most efficient and accurate alternative. The feasibility of the technique is validated based on several diverse application examples, namely a control loop, a lossy transmission-line network, and an active low-pass filter.

Simultaneous PET/MR/EEG (Positron Emission Tomography – Magnetic Resonance – Electroencephalography), a new tool for the investigation of neuronal networks in the human brain, is presented here within the framework of the European Union Project TRIMAGE. The trimodal, cost-effective PET/MR/EEG imaging tool makes use of cutting edge technology both in PET and in MR fields. A novel type of magnet (1.5T, non-cryogenic) has been built together with a PET scanner that makes use of the most advanced photodetectors (i.e., SiPM matrices), scintillators matrices (LYSO) and digital electronics. The combined PET/MR/EEG system is dedicated to brain imaging and has an inner diameter of 260 mm and an axial Field-of-View of 160 mm.It enables the acquisition and assessment of molecular metabolic information with high spatial and temporal resolution in a given brain simultaneously. The dopaminergic system and the glutamatergic system in schizophrenic patients are investigated via PET, the same physiological/pathophysiological conditions with regard to functional connectivity, via fMRI, and its electrophysiological signature via EEG. In addition to basic neuroscience questions addressing neurovascular-metabolic coupling, this new methodology lays the foundation for individual physiological and pathological fingerprints for a wide research field addressing healthy aging, gender effects, plasticity and different psychiatric and neurological diseases.The preliminary performances of two components of the imaging tool (PET and MR) are discussed. Initial results of the search of possible candidates for suitable schizophrenia biomarkers are also presented as obtained with PET/MR systems available to the collaboration.

We present a complete workflow for the extraction of behavioral reduced-order models of wired interconnect links, including an explicit dependence on geometrical or material parameters describing internal discontinuities that may affect the quality of signal transmission. Thanks to the adopted structure, the models are easily identified from sampled frequency responses at discrete points in the parameter space. Such responses are obtained from off-the-shelf full-wave solvers. A novel algorithm is used for checking and enforcing model stability and passivity, two fundamental requirements for reliably running stable transient simulations. Finally, an ad hoc procedure is devised to synthesize the models as parameterized circuit equivalents, compatible with any SPICE solver. Several examples illustrate and validate the workflow, confirming the suitability of the proposed approach for what-if, parameter sweep, design centering, and optimization through time-domain simulations, possibly including nonlinear devices and terminations.

This paper focuses on the design and the characterization of a frequency-selective shield for positron emission tomography (PET) detector modules of hybrid magnetic resonance-PET scanners, where the shielding of the PET cassettes is located close to the observed object. The proposed shielding configuration is designed and optimized to guarantee a high shielding effectiveness (SE) of up to 60 dB for ${B} _{1}$ -fields at the Larmor frequency of 64 MHz, thus preventing interactions between the radio-frequency (RF) coil and PET electronics. On the other hand, the shield is transparent to the gradient fields with the consequence that eddy-current artifacts in the acquired EPI images are significantly reduced with respect to the standard solid-shield configuration. The frequency-selective behavior of the shield is characterized and validated via simulation studies with CST MICROWAVE STUDIO in the megahertz and kilohertz range. Bench measurements with an RF coil built in-house demonstrated the high SE at the Larmor frequency. Moreover, measurements on a 4-T human scanner confirmed the abolishment of eddy current artifact and also provided an understanding of where the eddy currents occur with respect to the sequence parameters. Simulations and measurements for the proposed shielding concept were compared with a solid copper shielding configuration.

This paper focuses on the application of the Least-Square Support Vector Machine (LS-SVM) regression for the modeling of frequency responses of complex interconnect structures. The goal is to obtain a delayed-rational model (DRM) for the structure accounting for multiple time-delays generated by wave propagation and reflections along the channel. A novel approach for the time-delays estimation based on the LS-SVM regression is introduced. The delays are estimated using the dual space formulation of the LS-SVM with an ad-hoc kernel that considers a possible delay interval. The results highlight the lower order of DRMs obtained using the delays identified through this method when companng to the vector fitting approach by applying it to a high-speed cable link.

This letter presents a black-box modeling approach for the prediction of the spectrum of the maximum currents induced on a generic linear load in an automotive radiated immunity test. The proposed approach relies on a parametric Thevenin-based circuit equivalent built from a limited set of measured or simulated data. The frequency-domain behavior of the equivalent voltage source is provided via a metamodel by combining the support vector machine (SVM) regression with a regularized Fourier kernel with a simple adaptive algorithm. The latter allows defining the minimum number of training samples needed to accurately predict the maximum values of the currents induced on a generic linear load for different azimuth directions of the excitation field. The accuracy and the strength of the proposed approach are demonstrated for an example, by comparing the model predictions with the results of a parametric full-wave electromagnetic simulation.

This paper presents a preliminary version of an active learning (AL) scheme for the sample selection aimed at the development of a surrogate model for the uncertainty quantification based on the Gaussian process regression. The proposed AL strategy iteratively searches for new candidate points to be included within the training set by trying to minimize the relative posterior standard deviation provided by the Gaussian process regression surrogate. The above scheme has been applied for the construction of a surrogate model for the statistical analysis of the efficiency of a switching buck converter as a function of seven uncertain parameters. The performance of the surrogate model constructed via the proposed active learning method is compared with that provided by an equivalent model built via a Latin hypercube sampling. The results of a Monte Carlo simulation with the computational model are used as reference.

This paper presents an iterative and decoupled perturbative stochastic Galerkin (SG) method for the variability analysis of stochastic linear circuits with a large number of uncertain parameters. State-of-the-art implementations of polynomial chaos expansion and SG projection produce a large deterministic circuit that is fully coupled, thus becoming cumbersome to implement and inefficient to solve when the number of random parameters is large. In a perturbative approach, component variability is interpreted as a perturbation of its nominal value. The relaxation of the resulting equations and the application of a SG method lead to a decoupled system of equations, corresponding to a modified equivalent circuit in which each stochastic component is replaced by the nominal element equipped with a parallel current source accounting for the effect of variability. The solution of the perturbation problem is carried out in an iterative manner by suitably updating the equivalent current sources by means of Jacobi- or Gauss-Seidel strategies, until convergence is reached. A sparse implementation allows avoiding the refinement of negligible coefficients, yielding further efficiency improvement. Moreover, for time-invariant circuits, the iterations are effectively performed in post-processing after characterizing the circuit in time or frequency domain by means of a limited number of simulations. Several application examples are used to illustrate the proposed technique and highlight its performance and computational advantages.

In this paper, the performance of four machine learning regressions like Support Vector Machine (SVM), Least Square-Support Vector Machine (LS-SVM), Gaussian Process Regression (GPR) and Random Forest method (RF) are investigated by means of an illustrative example referring to the characteristic impedance of a microstrip line in terms of electrical and geometrical parameters. The required dataset for training is obtained from a set of parametric electromagnetic simulations. The performance comparison of the four methods is done in the presence and absence of numerical noise and inaccuracies affecting the training samples. The results of our comparison provide a guidance for the proper method selection to model the electromagnetic characteristics of interconnects for high-speed signals: advantages and drawbacks of each of the proposed techniques clearly emerge from this analysis.

This article deals with the development of a probabilistic surrogate model for the uncertainty quantification of the voltage output spectral envelope of a power converter with several stochastic parameters. The proposed approach relies on the combination of the least-squares support vector machine (LS-SVM) regression with the Gaussian process regression (GPR), but it can suitably be applied to any deterministic regression techniques. As a first step, the LS-SVM regression is used to build an accurate and fast-to-evaluate deterministic model of the system responses starting from a limited set of training samples provided by the full-computational model. Then the GPR is used to provide a probabilistic model of the regression error. The resulting LS-SVM+GPR probabilistic model not only approximates the system responses for any configuration of its input parameters, but also provides an estimation of its prediction uncertainty, such as the confidence intervals (CIs). The above technique has been applied to qualify the uncertainty of the spectral envelope of the output voltage of a buck converter with 17 independent Gaussian parameters. The feasibility and the accuracy of the resulting model have been investigated by comparing its predictions and CI with the ones obtained by five different surrogate models based on state-of-the-art techniques and by the reference Monte Carlo results.

This paper presents an alternative approach for the design of high-speed link based on a preliminary version of a surrogate model for the inverse problem. Specifically, given the overall structure of the link, our goal is to build an accurate and fast-to-evaluate model for the estimation of the geometrical parameters of its interconnect starting from the desired eye diagram characteristics. The modeling scheme proposed in this paper relies on a powerful machine learning regression technique such as the least-squares support vector machine (LS-SVM) which is used to provide an accurate relationship among the desired eye features and the geometrical parameters of the link interconnect. The proposed model is built from a set of training samples generated by a parametric simulation of the link through the full-computational model. The feasibility and the accuracy of the proposed modeling scheme are then investigated by comparing its predictions with the corresponding results provided by the full-computational model on 250 unseen samples.

Design of microwave structures and tuning parameters have mostly relied on the domain expertise of circuit designers by doing many simulations, which can be prohibitively time consuming. An inverse problem approach suggests going in the opposite direction to determine design parameters from characteristics of the desired output. In this work, we propose a novel machine learning architecture that circumvents usual design method for given quality of eye characteristics by means of a Lifelong Learning Architecture. Our proposed machine learning architecture is a large-scale coupled training system in which multiple predictions and classifications are done jointly for inverse mapping of transmission line geometry from eye characteristics. Our model is trained in a guided manner by using intra-tasks results, common Knowledge Base (KB), and coupling constraints. Our method of inverse design is general and can be applied to other applications.

This paper presents a probabilistic model for the prediction of the transmission performance of a miscrostrip link in presence of a ground discontinuity. The proposed model is based on a machine learning approach. Specifically, it combines the least-squares support vector machine regression with a Gaussian process regression with the aim of predicting the first resonance frequency of a miscrostrip structure for different geometries and positions of a rectangular slit in the ground plane. The model is trained from a small set of costly electromagnetic simulations generated via a latin hypercube sampling scheme and provides also the confidence intervals of its predictions. The accuracy and the capability of the proposed modeling approach are demonstrated by comparing the model predictions and their relative confidence intervals with the results provided by a parametric full-wave electromagnetic simulation.

This paper provides a quick overview on three machine learning regression techniques for the uncertainty quantification and the parametric modeling of the responses of electronic systems. Specifically, in this work support vector machine, least-squares support vector machine and Gaussian process regressions are adopted to build accurate and fast-to-evaluate metamodels for the prediction of the behaviour of the output of interest in stochastic systems as a function of the uncertain parameters. The above regressions techniques are trained from a limited set of training pairs provided by either measurements or simulations of the full-computational model. The resulting metamodels can be suitably adopted for both uncertainty quantification and optimization purposes, thus providing the user with a set of helpful tools for the design of complex electrical systems. The feasibility and the accuracy of the considered machine learning regression techniques are investigated by considering a realistic printed circuit board interconnect structure.

This paper proposes a novel paradigm to generate a parameterized model of the response of linear circuits with the inclusion of worst case bounds. The methodology leverages the so-called Taylor models and represents parameter-dependent responses in terms of a multivariate Taylor polynomial, in conjunction with an interval remainder accounting for the approximation error. The Taylor model representation is propagated from input parameters to circuit responses through a suitable redefinition of the basic operations, such as addition, multiplication or matrix inversion, that are involved in the circuit solution. Specifically, the remainder is propagated in a conservative way based on the theory of interval analysis. While the polynomial part provides an accurate, analytical and parametric representation of the response as a function of the selected design parameters, the complementary information on the remainder error yields a conservative, yet tight, estimation of the worst case bounds. Specific and novel solutions are proposed to implement complex-valued matrix operations and to overcome well-known issues in the state-of-the-art Taylor model theory, like the determination of the upper and lower bound of the multivariate polynomial part. The proposed framework is applied to the frequency-domain analysis of linear circuits. An in-depth discussion of the fundamental theory is complemented by a selection of relevant examples aimed at illustrating the technique and demonstrating its feasibility and strength.

This paper presents a preliminary version of a probabilistic model for the uncertainty quantification of complex electronic systems resulting from the combination of the leastsquares support vector machine (LS-SVM) and the Gaussian process (GP) regression. The proposed model, trained with a limited set of training pairs provided by a set of full-wave expensive simulations, is adopted for the prediction of the efficiency of an integrated voltage regulator (IVR) with 8 uniformly distributed random parameters. The accuracy and the feasibility of the proposed model have been investigated by comparing the model predictions and its confidence intervals with the results of a Monte Carlo (MC) full-wave simulation of the device.

Polynomial chaos-based techniques recently became popular tools for signal integrity investigations that include the effects of parameter variability. Most of the available approaches are limited by the "curse of dimensionality", apply only to Gaussian correlations, or they are hindered by the lack of explicit parametrization or knowledge of the input random parameters. This paper presents a hierarchical approach for transmission line analysis, according to which line voltages and currents are mod-eled as polynomial chaos expansions that are function of the per-unit-length parameters, rather than of the underlying geometrical and material parameters. This new approach exhibits some useful advantages such as non-parametricity (with respect to physical parameters), higher accuracy for low expansion orders, and a potential for dimensionality reduction. An application example involving the transient analysis of a stripline interconnect is used to illustrate the feasibility of the advocated approach and discuss its performance.

This paper presents an example of use of the support vector machine regression for the design of a passive equalizer network in a SpaceWire communication channel. The support vector machine is a machine learning technique, here applied to generate a compact surrogate model of the eye diagram heights at the end of the differential communication channel. The surrogate model is estimated from a limited set of randomly selected training samples. It provides the designer with a powerful and accurate tool for the optimization of the equalizer network, in presence of different lengths of the SpaceWire cable. The benefits of the proposed equalizer solution, combined with the SVM-based modeling tool, are investigated on a realistic SpaceWire link.

This paper proposes a decoupled and iterative circuit implementation of the stochastic Galerkin method (SGM) for the variability analysis of electrical circuits via polynomial chaos (PC). The method is based on a perturbative reformulation of the SGM, resulting in a decoupled system with equivalent sources that is solved iteratively by suitably updating the sources at each step. This alternative approach is easier to implement and its computational cost scales linearly with the number of unknown PC coefficients. This contribution addresses the case of stochastic resistors. As a validation, the technique is applied to the variability analysis of a boost converter.

This paper presents a preliminary application of the support vector machine regression to the modeling of the eye diagram heights in high speed links for design and optimization purposes. The support vector machine regression is applied to generate a compact surrogate model of the eye diagram heights at a specific node of the link from a set of randomly selected training samples. The surrogate can be suitably adopted both for design optimization and for stochastic analysis. The feasibility and accuracy of the surrogate model calculated via the support vector machine regression are investigated on a realistic high-speed communication channel.