Finite Input Research Articles

A PNP ion channel deep learning solver with local neural network and finite element input data

Abstract This paper presents a deep learning method for solving an improved one-dimensional Poisson–Nernst–Planck ion channel (PNPic) model, called the PNPic deep learning solver. The solver combines a novel local neural network, adapted from the neural network with local converging inputs, with an efficient PNPic finite element solver, developed in this work. In particular, the local neural network is extended to handle the complexities of the PNPic model—a system of nonlinear convection–diffusion and elliptic equations with multiple subdomains connected by interface conditions. The PNPic finite element solver efficiently generates input and reference datasets for fast training the local neural network, as well as input datasets for quickly predicting PNPic solutions with high accuracy for a family of PNPic models. Initial numerical tests, involving perturbations of model parameters and interface locations, demonstrate that the PNPic deep learning solver can generate highly accurate numerical solutions.

Residence time density (RTd) identification through deconvolution of finite time horizon input output data and its Application to a novel analytical dispersion process RTd model

This work presents first an approach to the identification of Residence Time density (RTd) and Residence Time Distribution (RTD) functions, through deconvolution of finite time horizon, input/output data generated by a zero-order hold input, using the novel concept of zero-order hold, average, impulse response coefficients. Subsequently, a novel RTd analytical expression is derived for an open-open axial dispersion process with a finite experimental section in a tube of infinite length, and an explanation is provided for this RTd model’s differences from the literature. This new RTd expression is then employed in two case studies to demonstrate the effectiveness of the proposed identification through a deconvolution conceptual framework in identifying RTd model parameters, and in assisting in the selection of RTd models which are compatible with experimental data.

An epsilon‐near‐zero circuit based on half‐mode SIW with enhanced integrability for applications of narrowband large group delay

AbstractThe tunneling effect of an epsilon‐near‐zero (ENZ) is exploited to fulfill a narrowband large group delay by using a compact half‐mode substrate integrated waveguide structure built on thin substrate. The ENZ channel possesses a finite input impedance at the tunneling frequency point, and the finite input impedance can be matched to the final I/O port simply by a quarter wavelength impedance transformer. Thus, the enhanced integrability is realized in a planar form in the proposed circuit comparing to the conventional waveguide ENZ channel. Meanwhile, in the real ENZ channel, the effective impedance is observed to be close to 50 Ω, so that the ENZ can be connected with 50 Ω microstrip directly, which has been experimentally validated. The final measured net group delay is 1.36 ns at 3.08 GHz with 3.6 dB insertion loss, and the proposed method provides an alternative for the group delay engineering.

Global sensitivity and domain‐selective testing for functional‐valued responses: An application to climate economy models

AbstractUnderstanding the dynamics and evolution of climate change and associated uncertainties is key for designing robust policy actions. Computer models are key tools in this scientific effort, which have now reached a high level of sophistication and complexity. Model auditing is needed in order to better understand their results, and to deal with the fact that such models are increasingly opaque with respect to their inner workings. Current techniques such as Global Sensitivity Analysis (GSA) are limited to dealing either with multivariate outputs, stochastic ones, or finite‐change inputs. This limits their applicability to time‐varying variables such as future pathways of greenhouse gases. To provide additional semantics in the analysis of a model ensemble, we provide an extension of GSA methodologies tackling the case of stochastic functional outputs with finite change inputs. To deal with finite change inputs and functional outputs, we propose an extension of currently available GSA methodologies while we deal with the stochastic part by introducing a novel, domain‐selective inferential technique for sensitivity indices. Our method is explored via a simulation study that shows its robustness and efficacy in detecting sensitivity patterns. We apply it to real‐world data, where its capabilities can provide to practitioners and policymakers additional information about the time dynamics of sensitivity patterns, as well as information about robustness.

A novel method for settlement imputation and monitoring of earth-rockfill dams subjected to large-scale missing data

Various missing values inevitably exist in the settlement monitoring data of prolonged operation of earth-rockfill dams, which delays or even interrupts the dam structure analytical procedure. This study aims to develop a reliable method for addressing the missing data problem and monitoring earth-rockfill dam settlement behavior. First, an imputation model, support vector regression based on the finite element method and time input (FEMTSVR), is proposed and offers superior performance when capturing complex nonlinear mappings from environmental variables to settlement on small sample data. Herein, an improved particle swarm optimization algorithm (IPSO) is developed, realizing the nonlinear adjustment of weights and parameter reduction during hyperparameter optimization. Then, this study establishes a sequential prediction model based on gate recurrent unit (GRU) networks to monitor dam behavior on the imputed and complete dataset. Eventually, the proposed method is evaluated using real-world earth-rockfill dam monitoring data with the help of statistical indicators, demonstrating its efficiency in monitoring settlement data subjected to large-scale missingness. This study provides a robust database for dam structural health monitoring, while also providing a promising framework for the safety assessment of other civil or hydraulic engineering based on raw monitoring data.

A Discrete-Time Queueing Model of a Bottleneck with an Energy-Saving Mechanism Based on Setup and Shutdown Times

Producers are encouraged to reduce their energy consumption of manufacturing systems by applying less-energy-intensive modern technologies and advanced machine tools and operating methods at the system level. In the paper, organizational and analytical solutions are combined to model the sustainable production system. Managers can study the behavior of a production system organized using energy-saving rules by changing key parameters of the input model (arrival intensity, bottleneck service rate, buffer size, setup and shutdown time) to analyze the queue size of the production system and therefore performance. A discrete-time queueing model of a single-bottleneck production line with a finite input buffer capacity is proposed. Jobs occur according to a binomial process and are processed individually, one by one, according to the natural FIFO service discipline, with a general discrete-type cumulative distribution function. The total number of jobs present in the system is bounded by a non-random fixed value N. Every time the system becomes empty, an energy-saving mechanism is started: the processing machine (server) is turned off during a geometrically distributed shutdown time. Similarly, the first job arriving into the empty system initializes a geometrically distributed setup time. Identifying renewal moments in the evolution of the model, a system of difference equations is built for the transient queue-size distribution conditioned by the state of the system at the opening. The solution is obtained explicitly in terms of probability-generating functions. In addition, the Drum-Buffer-Rope concept is proposed to reduce the energy consumption of the production line. The throughput of the production system is maximized by adjusting the time between the order arrivals and the size of the input buffer to the capacity of the bottleneck. Turning off a machine under certain conditions and slowing down non-critical machines are strategies to reduce energy consumption. A detailed illustrating numerical and simulation study of the considered model is attached as well, in which the sensitivity of the queue-size behavior to changes of the key input model parameters is investigated.

Fixed-energy inverse scattering with radial basis function neural networks and its application to neutron–α interactions

Abstract This paper proposes a data-driven method to solve the fixed-energy inverse scattering problem for radially symmetric potentials using radial basis function (RBF) neural networks in an open-loop control system. The method estimates the scattering potentials in the Fourier domain by training an appropriate number of RBF networks, while the control step is carried out in the coordinate space by using the measured phase shifts as control parameters. The system is trained by both finite and singular input potentials, and is capable of modeling a great variety of scattering events. The method is applied to neutron–α scattering at 10 MeV incident neutron energy, where the underlying central part of the potential is estimated by using the measured l = 0, 1, 2 phase shifts as inputs. The obtained potential is physically sensible and the recalculated phase shifts are within a few percent relative error.

Branched Latent Neural Maps

We introduce Branched Latent Neural Maps (BLNMs) to learn finite dimensional input–output maps encoding complex physical processes. A BLNM is defined by a simple and compact feedforward partially-connected neural network that structurally disentangles inputs with different intrinsic roles, such as the time variable from model parameters of a differential equation, while transferring them into a generic field of interest. BLNMs leverage latent outputs to enhance the learned dynamics and break the curse of dimensionality by showing excellent in-distribution generalization properties with small training datasets and short training times on a single processor. Indeed, their in-distribution generalization error remains comparable regardless of the adopted discretization during the testing phase. Moreover, the partial connections, in place of a fully-connected structure, significantly reduce the number of tunable parameters. We show the capabilities of BLNMs in a challenging test case involving biophysically detailed electrophysiology simulations in a biventricular cardiac model of a pediatric patient with hypoplastic left heart syndrome. The model includes a 1D Purkinje network for fast conduction and a 3D heart-torso geometry. Specifically, we trained BLNMs on 150 in silico generated 12-lead electrocardiograms (ECGs) while spanning 7 model parameters, covering cell-scale, organ-level and electrical dyssynchrony. Although the 12-lead ECGs manifest very fast dynamics with sharp gradients, after automatic hyperparameter tuning the optimal BLNM, trained in less than 3 h on a single CPU, retains just 7 hidden layers and 19 neurons per layer. The resulting mean square error is on the order of 10−4 on an independent test dataset comprised of 50 additional electrophysiology simulations. In the online phase, the BLNM allows for 5000x faster real-time simulations of cardiac electrophysiology on a single core standard computer and can be employed to solve inverse problems via global optimization in a few seconds of computational time. This paper provides a novel computational tool to build reliable and efficient reduced-order models for digital twinning in engineering applications. The Julia implementation is publicly available under MIT License at https://github.com/StanfordCBCL/BLNM.jl.

Bit-Interleaved Multiple Access: Improved Fairness, Reliability, and Latency for Massive IoT Networks

Internet-of-Things (IoT) networks require massive connections in dense areas. Therefore, a resource efficient multiple access scheme seems inevitable to enable immense connectivity where multiple devices have to share the same resource block. Non-orthogonal multiple access (NOMA) has been considered as the strongest candidate in recent years. However, in this paper, by considering the practical implementation, we first provide a true power allocation (PA) constraint with finite alphabet inputs for conventional downlink NOMA and demonstrate that it cannot support massive connections in practical systems. To this end, we propose bit-interleaved multiple access (BIMA) scheme in downlink IoT networks. The proposed BIMA scheme implements bitwise multiaccess interleaving and deinterleaving at the transceiver ends and there are no strict PA constraints, unlike conventional NOMA, thus allowing a high number of devices in the same resource block. We provide a comprehensive analytical framework for BIMA by investigating all key performance indicators (KPIs) to present both information-theoretic (i.e., ergodic capacity EC and outage probability OP) and finite alphabet inputs (i.e., bit error rate BER) performance metrics with both instantaneous and statistical channel ordering. In addition, we define Jain’s fairness index and proportional fairness index in terms of all KPIs. Based on the extensive computer simulations, we reveal that BIMA outperforms conventional NOMA significantly, with a performance gain of up to 20-30 dB in terms of KPIs in some scenarios. In other words, compared to conventional NOMA schemes, the same KPIs are met in BIMA with 20-30 dB less transmit power, which is quite promising for energy-limited use cases. Moreover, this performance gain becomes greater when more IoT devices are supported. BIMA provides a full diversity order for all IoT devices and enables the implementation of an arbitrary number of devices and modulation orders, which is crucial for IoT networks where a huge number of devices should be supported in a single resource block in dense areas. In addition to the overall performance gain, BIMA guarantees a fairness system where none of the devices gets a severely degraded performance and the sum-rate is shared in a fair manner among devices. It guarantees QoS satisfaction for all devices. Finally, we provide an intense complexity and latency analysis for BIMA and demonstrate that it provides lower latency compared to conventional NOMA receivers, since it allows parallel computation at the receivers and no iterative operations are required. We show that compared to conventional NOMA receivers, BIMA reduces latency by up to 350% for specific IoT devices and 170% on average.

Permutation Matrix Modulation

We propose a novel scheme that allows MIMO system to modulate a set of permutation matrices to send more information bits, extending our initial work on the topic. This system is called Permutation Matrix Modulation (PMM). The basic idea is to employ a permutation matrix as a precoder and treat it as a modulated symbol. We continue the evolution of index modulation in MIMO by adopting all-antenna activation and obtaining a set of unique symbols from altering the positions of the antenna transmit power. We provide the analysis of the achievable rate of PMM under Gaussian Mixture Model (GMM) distribution \revv{and finite cardinality input (FCI). Numerical results are evaluated by comparing PMM with the other existing systems.} We also present a way to attain the optimal achievable rate of PMM by solving a maximization problem via interior-point method. A low complexity detection scheme based on zero-forcing (ZF) is proposed, and maximum likelihood (ML) detection is discussed. We demonstrate the trade-off between simulation of the symbol error rate (SER) and the computational complexity where ZF performs worse in the SER simulation but requires much less computational complexity than ML.

Biased-Power Allocation and Shared-Antenna Selection Techniques for Spatial Modulation-Based Layer Division Multiplexing Systems

This study proposes two approaches for improving the effectiveness of spatial modulation integrated into layer division multiplexing (SM-LDM) in broadcasting systems: biased-power allocation (Bi-PA) and shared antenna selection (SAS). Even though different data rates are employed in SM-LDM systems, Bi-PA enhances bit error rate (BER) fairness across layers. The ideal power ratios are adaptively determined by balancing signal-to-interference plus noise ratios with a preference for the lower layer (LL) that involves a higher modulation order. SAS alleviates the complexity of successive interference cancellation and enhances spectral and energy efficiencies. Both the LL and upper layer (UL) share the antenna selection decision and transmit using a single antenna. The UL carries a space shift keying signal while the entire power is allocated for the LL. We analyze the spectral efficiency for the SAS-based SM-LDM system with finite alphabet inputs. Numerical results demonstrate the advantages of the proposed approaches. Compared to pre-assigned-PA (Pre-PA), Bi-PA shows nearly identical BERs for both layers and solves the error floor problem. The sharing property and common layer transmission of SAS-based SM-LDM yield a significant BER reduction relative to conventional SM-LDM. It provides gains ranging from 7 to 15 dB for LL at BER equal to 10−3, while UL performance ranges from slight gain to minor loss. Furthermore, both Bi-PA and SAS techniques enhance the achievable LL rate and sum-rate at low and intermediate signal-to-noise ratio values. They can achieve an improvement of up to two bits in LL rate and less than one bit in sum-rate at a signal-to-noise ratio of −0.5 dB. These findings show that both proposed techniques have a considerable impact on enhancing the fairness, BER performance, and feasible rates of SM-LDM systems, making them promise for broadcast system designs.

Performance Analysis and Power Allocation for Spatial Modulation-Aided MIMO-LDM With Finite Alphabet Inputs

In this paper, we study a spatial modulation (SM) aided multiple-input multiple-output (MIMO) layered division multiplexing (LDM) system for broadcast/multicast service delivery in future broadcast/multicast systems. Comprehensive performance analysis and the injection level (IL) optimization are investigated for the SM-aided MIMO-LDM system over correlated Rayleigh fading channels. First, the symbol detection pairwise error probability (PEP) and the average symbol error rate (SER) union bound under joint maximum-likelihood (ML) detection are derived. Second, the spectral efficiency (SE) of the SM-aided MIMO-LDM system with finite alphabet inputs is analyzed. Since the theoretical SE lacks closed-form expression and involves prohibitive computational complexity, we then derive the closed-form lower bounds and tight approximations for the theoretical SE, in which the computational complexity is relieved by several orders of magnitude compared to direct calculation of the theoretical SE. Third, based on the SE analysis, a weighted sum (WS) SE maximization problem with quality of service (QoS) constraints is formulated to optimize IL for the SM-aided MIMO-LDM system. We first demonstrate that WS SE is a unimodal function of IL, and then develop an efficient golden section search (GSS) based algorithm. Simulation results are provided to validate the theoretical analysis. It is shown that our derived SER union bound well matches the simulated SER in the high SNR region. The tightness of the derived closed-form lower bounds and approximations for theoretical SE and the effectiveness of the developed IL optimization algorithm are also verified by simulation results.

Approximation bounds for convolutional neural networks in operator learning

Recently, deep Convolutional Neural Networks (CNNs) have proven to be successful when employed in areas such as reduced order modeling of parametrized PDEs. Despite their accuracy and efficiency, the approaches available in the literature still lack a rigorous justification on their mathematical foundations. Motivated by this fact, in this paper we derive rigorous error bounds for the approximation of nonlinear operators by means of CNN models. More precisely, we address the case in which an operator maps a finite dimensional input μ∈Rp onto a functional output uμ:[0,1]d→R, and a neural network model is used to approximate a discretized version of the input-to-output map. The resulting error estimates provide a clear interpretation of the hyperparameters defining the neural network architecture. All the proofs are constructive, and they ultimately reveal a deep connection between CNNs and the Fourier transform. Finally, we complement the derived error bounds by numerical experiments that illustrate their application.

Comments on the H2 Norm of Discrete-Time Stochastic Jump Parameter Linear Systems

In this note, we consider a general class of stochastic, jump parameter systems and show that the so-called H2 norm is the Frobenius norm of the impulse response matrix of the system. We present a necessary and sufficient condition for the H2 norm to be the induced norm from the input to the output for a particular type of deterministic input signal. As expected in the context of time-varying systems, we cannot extend this result to more general deterministic input signals. Moreover, finite H2 norm do not prevent infinite norm output in response to a finite norm input, unless additional conditions are imposed on the system. Here, we give relatively mild conditions -mean square stability and jumps driven by a finite-dimensional Markov chain starting in equilibrium -and we show that finite H2 norm implies a bounded ratio between the norms of the output and the input, for certain suitable norms.

A Function Space Analysis of Finite Neural Networks With Insights From Sampling Theory.

This work suggests using sampling theory to analyze the function space represented by interpolating mappings. While the analysis in this paper is general, we focus it on neural networks with bounded weights that are known for their ability to interpolate (fit) the training data. First, we show, under the assumption of a finite input domain, which is the common case in training neural networks, that the function space generated by multi-layer networks with bounded weights, and non-expansive activation functions are smooth. This extends over previous works that show results for the case of infinite width ReLU networks. Then, under the assumption that the input is band-limited, we provide novel error bounds for univariate neural networks. We analyze both deterministic uniform and random sampling showing the advantage of the former.

Data-Driven Synthesis of Safety Controllers via Multiple Control Barrier Certificates

This work proposes a data-driven framework to synthesize safety controllers for nonlinear systems with finite input sets and unknown mathematical models. The proposed scheme leverages new notions of multiple control barrier certificates (M-CBC) and provides controllers ensuring the safety of systems with confidence 1. While there may not exist a common control barrier certificate with a fixed template, our proposed technique adaptively partitions the state set to potentially find M-CBC of the same template for different regions. In the proposed data-driven framework, we first cast our proposed conditions of M-CBC as a robust optimization program (ROP). Given that the unknown model appears in some of the constraints of the ROP, we propose a sampling approach for collecting data and provide a scenario optimization program (SOP) associated with the proposed ROP. We solve the resulted SOP and construct M-CBC together with safety controllers for the unknown system with 100% correctness guarantee. We apply our results to a nonlinear jet engine compressor with unknown dynamics to illustrate the efficacy of our data-driven approach. In the case study, we show that while there exists no common polynomial-type control barrier certificate of a given degree, there exist polynomial-type M-CBC of the same degree by partitioning the state set to different regions.

Optimal finite alphabet scheme for NOMA uplink channels

AbstractThe design of an optimal non‐orthogonal multiple access (NOMA) transmission scheme with finite alphabet inputs for a typical two‐user uplink wireless communication system is investigated in which each terminal is equipped with a single antenna. Each of the two users utilises a four quadrature amplitude modulation (4‐QAM) constellation to transmit information data to a common base station, and the receiver employs a maximum likelihood (ML) detector to jointly estimate both transmitted signals. Assuming the availability of channel state information at both the transmitters and the receiver, it is aimed to design a pair of scalar beamformers for the two users such that the minimum Euclidean distance between elements of the received sum‐constellation is maximised subject to the power constraints on the users. A thorough consideration of all the different conditions results in the derivation of a closed‐form optimal beamformer design. As well, examination of the construction of sum‐constellation resulted from the optimum design directly leads to the unique decoding of the original transmitted signal of each user. To facilitate practical implementation, a fast decoding procedure of the optimum NOMA scheme is further developed. The corresponding theoretic probability of ML detection error is also derived. The theoretical development of an optimum sum‐constellation for the basic 2‐user 4‐QAM system provides a solid platform for the derivation of an optimum sum‐constellation for a K‐user and/or M‐QAM system. Indeed, a simple development of the basic sum‐constellation map facilitates such extensions. Numerical simulations not only demonstrate that the performance of the fast decoder agrees closely with the theoretical analysis, but also verify that it is superior in performance to other existing NOMA designs for the same system under high signal‐to‐noise ratio.

Ergodic Rate Analysis of Massive Spatial Modulation MIMO System With NOMA in Cooperative Relay Networks

In this article, by introducing nonorthogonal multiple access (NOMA) scheme into the massive spatial modulation (SM) MIMO in cooperative relay networks, a massive cooperative NOMA-aided SM (NOMA-SM) system is presented. The ergodic rate performance of uplink massive cooperative NOMA-SM with finite alphabet inputs is investigated over Rayleigh fading channels. According to the performance analysis, the mutual information between the SM signals transmitted by users and the signal received at the base station (BS) is derived. Based on this, the theoretical lower bound and tight approximation of the ergodic rates are deduced for performance evaluation. With the aforementioned results, simple expressions of the ergodic rates are derived for the system with single-antenna relay, which can simplify the calculation of theoretical ergodic rate. Besides, when the number of receive antennas at the BS is very large, the asymptotic ergodic rate analysis is presented for massive cooperative NOMA-SM system, and the corresponding asymptotic lower bounds of the ergodic rate are derived. Simulation results indicate that the presented theoretical analysis is valid and can match the corresponding simulation well. Thus, the ergodic rate performance can be well assessed in theory.

The judicious use of finite marine resources can sustain Atlantic salmon (salmo salar) aquaculture to 2100 and beyond.

The production of farmed Atlantic salmon (Salmo salar) is currently linked to finite marine resources, particularly fish oil (FO) and fish meal (FM). Understanding this relationship in a quantitative manner is imperative if this critical balance is to be maintained within sustainable limits as the industry grows. Here we project the potential production and associated growth of the Atlantic salmon aquaculture industry on the basis of a variety of FO and FM utilization scenarios in aquafeed. Reducing FO and FM dietary inclusion to 3% each could permit production growth of 2% per year until the turn of the century (2097 and beyond 2100, respectively), independent of a host of alternatives now being utilized-with three portions of salmon per week providing almost all the recommended weekly long-chain omega-3 fatty acids for human intake. The Atlantic salmon industry's positive annual growth can continue in an era of finite marine resource availability-without the need for additional finite marine resource inputs.

Complex Interval-Valued q-Rung Orthopair Fuzzy Hamy Mean Operators and Their Application in Decision-Making Strategy

This paper deals with uncertainty, asymmetric information, and risk modelling in a complex power system. The uncertainty is managed by using probability and decision theory methods. Multi-attribute decision-making (MADM) technique is a very effective and well-known tool to investigate fuzzy information more effectively. However, the selection of houses cannot be carried out by utilizing symmetry information, because enterprises does not have complete information, so asymmetric information should be used when selecting enterprises. Hamy mean (HM) operator is a feasible tool to handle strategic decision-making problems because it can capture the order between the finite input terms. Additionally, the complex interval-valued q-rung orthopair fuzzy (CIVq-ROF) setting is a broadly flexible and massively dominant technique to operate problematic and awkward data in actual life problems. The major contribution of this analysis is how to aggregate the collection of alternatives into a singleton set, for this we analyzed the technique of CIVq-ROF Hamy mean (CIVq-ROFHM) operator and CIVq-ROF weighted Hamy mean (Cq-ROFWHM) operator and some well-known results are deliberated. Keeping the advantages of the parameters in HM operators, we discussed the specific cases of the invented operators. To investigate the decision-making problems based on CIVq-ROF information, we suggested the following multi-attribute decision-making (MADM) technique to determine the beneficial term from the finite group of alternatives with the help of evaluating several examples. This manuscript showed how to make decisions when there is asymmetric information about enterprises. Finally, based on the evaluating examples, we try to discover the sensitive analysis and supremacy of the invented operators to find the flexibility and dominancy of the diagnosed approaches.