We study small-time central limit theorems for stochastic Volterra integral equations with Hölder continuous coefficients and general locally square integrable Volterra kernels. We prove the convergence of the finite-dimensional distributions, a functional CLT, and limit theorems for smooth transformations of the process, which covers a large class of Volterra kernels that includes rough models based on Riemann-Liouville kernels with short- and long-range dependencies. To illustrate our results, we derive asymptotic pricing formulae for digital calls on the realized variance in three different regimes. The latter provides a robust and model-independent pricing method for small maturities in rough volatility models. Finally, for the case of completely monotone kernels, we introduce a flexible framework of Hilbert space-valued Markovian lifts and derive analogous limit theorems for such lifts.

The brain operates across multiple spatial and temporal scales, necessitating computationally efficient models that link micro-scale mechanisms to meso- and macro-scale dynamics. Here, we introduce a novel convolutional neural mass model (CNMM) that computes the meso-scale activity of spatially discretized neural populations (“neural masses”) in the rat hippocampal CA3 subregion. The CNMM employs a kernel-based architecture, leveraging first-order Volterra expansions with Laguerre (temporal) and Chebyshev (spatial) basis functions to transform input spike densities from entorhinal cortex (EC), dentate gyrus (DG), and neighboring CA3 masses into output CA3 spike density. The model was trained and validated using data from a biophysically detailed large-scale mechanistic model (LSM) simulating exploratory behavior. The CNMM achieved high predictive accuracy for spike density across 32 neural masses spanning the entire extent of CA3 (mean correlation coefficient ) and replicated theta and beta oscillations consistent with experimental findings. When extended for forward modeling, the CNMM accurately predicted local field potentials (LFPs) at a single neural mass (R = 0.952). Kernel analysis revealed topographic gradients in afferent integration, with DG inputs dominating proximally (CA3c) and associational connections distally (CA3a), aligning with anatomical gradients. Compared to the LSM, the CNMM provided a 658-fold speedup in simulation time, 322-fold reduction in memory usage, and 183-fold less disk space for LFP predictions. This framework offers a scalable, efficient approach for meso-scale modeling of neural tissue, bridging detailed simulations with empirical data for insights into normal and pathological function.

Nonlinear identification and waveform prediction of blast-induced ground vibrations using regularized Volterra series

A two-stage aeromagnetic compensation method incorporating sparsity-pruned volterra series for nonlinear magnetic interference

The synapse is the fundamental unit of communication in the nervous system. Determining how information is transferred across the synaptic interface is one of the most complex endeavors in neuroscience, owing to the large number of contributing factors and events. An approach to solving this problem involves collapsing across these complexities to derive concise mathematical formulas that fully capture the governing dynamics of synaptic transmission. We investigated the feasibility of deriving such a formula - an input-output transformation function for the CA3 to CA1 node of the hippocampus - using the Volterra expansion technique for non-linear system identification. The timecourse of the fEPSP in the apical dendrites of mouse brain slices was described with >94% accuracy by a 2nd order equation that captured the linear and non-linear influence of past inputs on current outputs. This function generalized to cases not included in its derivation and uncovered previously undetected timing rules. The basal dendrites expressed a substantially different transfer function and evidence was obtained that, unlike the apical system, a 3rd order system or higher will be needed for complete characterization. At scale, the approach will also provide information needed for the construction of biologically realistic models of brain networks.

This study focuses on modeling and dynamic identification of a reflux condenser in a batch reactor system. The model uses data from real industrial conditions, along with the Volterra series and Genocchi orthogonal polynomials, to capture the condenser’s nonlinear behavior. Identifying the dynamic behavior of the reflux condenser is essential for the safe and efficient production of phenolic resole resin in batch reactors. The condenser plays a key role in controlling the process temperature during exothermic polymerization by cooling and returning reflux material to the reactor. The model was validated with data from a 3500 kg industrial reactor, achieving a thermal energy prediction error of less than 2.5% during the critical polymerization phase. The results show that the model accurately reflects the condenser’s behavior, supporting its application in advanced control strategies for monitoring and regulating process temperature. Using these strategies can prevent uncontrolled reactions and improve operational safety and the quality of resole phenolic resin production.

This paper introduces a fractional generalization of the classical Legendre differential equation based on the Atangana–Baleanu–Caputo (ABC) derivative. A novel fractional Legendre-type operator is rigorously defined within a functional framework of continuously differentiable functions with absolutely continuous derivatives. The associated initial value problem is reformulated as an equivalent Volterra integral equation, and existence and uniqueness of classical solutions are established via the Banach fixed-point theorem, supported by a proved Lipschitz estimate for the ABC derivative. A constructive solution representation is obtained through a Volterra–Neumann series, explicitly revealing the role of Mittag–Leffler functions. We prove that the fractional solutions converge uniformly to the classical Legendre polynomials as the fractional order approaches unity, with a quantitative convergence rate of order O(1−α) under mild regularity assumptions on the Volterra kernel. A fully reproducible quadrature-based numerical scheme is developed, with explicit kernel formulas and implementation algorithms provided in appendices. Numerical experiments for the quadratic Legendre mode confirm the theoretical convergence and illustrate the smooth interpolation between fractional and classical regimes. An application to time-fractional diffusion in spherical coordinates demonstrates that the operator arises naturally in physical models, providing a mathematically consistent tool for extending classical angular analysis to fractional settings with memory.

Abstract DNA and other biopolymers are being investigated as new computing substrates and alternative to silicon-based digital computers. However, the established top-down design of biomolecular interaction networks remains challenging and does not fully exploit biomolecular self-assembly capabilities. Outside the field of computation, directed evolution has been used as a tool for goal directed optimization of DNA sequences. Here, we propose integrating directed evolution with DNA-based reservoir computing to enable in-material optimization and adaptation. Simulations of colloidal bead networks connected via DNA strands demonstrate a physical reservoir capable of non-linear time-series prediction tasks, including Volterra series and Mackey–Glass chaotic dynamics. Reservoir computing performance, quantified by normalized mean squared error (NMSE), strongly depends on network topology, suggesting task-specific optimal network configurations. Implementing genetic algorithms to evolve DNA-encoded network connectivity effectively identified well-performing reservoir networks. Directed evolution improved reservoir performance across multiple tasks, outperforming random network selection. Remarkably, sequential training on distinct tasks resulted in reservoir populations maintaining performance on prior tasks. Our findings indicate that DNA-bead networks offer sufficient complexity for reservoir computing, and that directed evolution robustly optimizes performance.

The computational demands of next-generation (Next-Gen) communication systems pose major challenges for real-time signal processing, particularly in digital predistortion (DPD), which is essential for linearizing power amplifier (PA) nonlinearities. While traditional DPD methods-such as polynomial and Volterra series models-remain prevalent, neural network (NN)-based approaches offer superior modeling accuracy and adaptability. However, their deployment is hindered by high computational complexity, limited scalability, and hardware integration challenges. This review presents a comprehensive analysis of NN-based DPD techniques and hardware acceleration strategies for efficient real-time implementation. We assess the strengths of various NN architectures-deep, convolutional, recurrent, and hybrid-and evaluate their tradeoffs across graphics processing unit (GPU), field-programmable gate arrays (FPGA), and application-specific integrated circuits (ASIC) platforms. We also examine key challenges, including fragmented evaluation standards and limited real-world validation. Finally, we outline future directions emphasizing model-hardware codesign, reconfigurable computing, and on-chip learning to enable scalable, energy-efficient DPD for 5G, 6G, and beyond.

Abstract Sensor-derived time series vibrational signals are difficult to classify under variable operating conditions, particularly during RPM shifts in frequency overlapping scenarios. The current preprocessing techniques struggle to detect frequency overlapped faults, as their shifting frequency patterns fail to align with the fixed scales. The proposed study designed a frequency-aware transformer fusion, an adaptive hybrid architecture that combines geometric structures, repetitive transients, and long-range dependencies within a variable frequency overlapping context. The utilization of Continuous wavelet transforms (CWT) and Volterra series form transient energy map and provide short-term memory, which replaces the absolute frequency analysis to the pattern recognition. After that, the encoded features are processed via a dual block, which consists of two concurrent branches: a Kronecker convolutional feature pyramid (KCFP) for repetitive transients and a channel attention MLP block that highlights semantic structures within the transient energy maps. The outputs of these parallel branches are fused through a Swin Transformer module, which uses frequency-aware shifting windows to confidently diagnose frequency overlapped faults. This modular fusion strategy enables the model to learn robust representations across both local and global temporal scales. Experimental validation on multi-class sensor position datasets obtained from Shandong University, CWRU and PU demonstrates superior classification accuracy. Ablation experiments demonstrate the significance of the dual-branch and dual-attention mechanisms, enhancing diagnostic consistency under frequency overlapped conditions.

Transmission at ever higher data rates increasingly demands more advanced digital signal processing techniques, raising both power consumption and operational costs. Here, we introduce a photonic/plasmonic artificial neural network (ANN) using plasmonic modulators to directly mitigate nonlinear signal distortions carried by an optical carrier. This first-of-its-kind plasmonic ANN achieves an ultracompact footprint and high-speed operation and markedly reduces the need for electronic processing. We compare our plasmonic ANN against a traditional digital feed-forward equalizer and a Volterra series, as well as the corresponding digital ANN. The results demonstrate that an astonishingly small ANN outperforms classical equalizers by attaining higher SNR at smaller computational effort. While the digital ANN offers an ideal implementation, executing the ANN on our first plasmonic chip already shows remarkable equalization performance with minimal components. The findings reveal a path toward ultracompact, high-speed, power-efficient, low-latency alternatives to conventional signal processing.

The D band presents a possible solution for accommodating ultrahigh-capacity data transmission. Multiple-input–multiple-output (MIMO) technology can further double system transmission capacity. Our study explores the integration of photonics-aided millimeter wave with MIMO technologies. Based on phonics-aided technology, we realized the transmission of 10-Gbaud D-band polarization-division multiplexing (PDM) discrete multitone (DMT) 16QAM signals at 125 GHz over a 1.2-km wireless link in an Intensity-modulation (IM)/direct-detection (DD) radio-over-fiber (RoF) system. However, apart from severe nonlinear impairment and PDM crosstalk, the D-band DMT RoF system is subject to phase noise (PN). Traditional Volterra series and deep neural network (DNN) equalizers can only compensate for nonlinear impairments but are poor at correcting phase distortion. To realize the mitigation of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$X/Y$</tex-math> </inline-formula> polarization crosstalk, nonlinear impairments, PN, and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</i>/<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> imbalance, we propose a novel MIMO 2-D convolutional neural network (2D-CNN) by addressing complex-valued two PDM signals and their phase information simultaneously. This model uniquely combines the ability to mitigate these four critical impairments within a single processing framework, enhancing the robustness of MIMO IM/DD RoF systems. Our MIMO 2D-CNN significantly improves the BER performance compared to traditional digital-signal-processing (DSP) models, providing an almost tenfold enhancement. The net bit rate can achieve 31.49 bit/s under the 7% hard decision forward error correction (HD-FEC) threshold of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3.8\times 10^{-3}$</tex-math> </inline-formula>. Meanwhile, due to its unique ability to extract information from 2-D space–time images, the advanced MIMO 2D-CNN can effectively compensate for the PN of the signals with minimal phase rotation. It surpasses the limitations of Volterra series and MIMO 2D-DNN approaches, which struggle to handle such PN. Moreover, regarding receiver sensitivity, our MIMO 2D-CNN outperforms the previously proposed MIMO 2D-DNN by over 1 dB at the BER of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3.8\times 10^{-3}$</tex-math> </inline-formula>. In the future, we expect that the low-PN MIMO 2D-CNN, with its accurate BER decision ability, will be a valuable tool for further investigations into MIMO IM/DD RoF networks.

Structural damage detection based on coherence-statistical analysis of Volterra kernels

This paper proposes an approach to efficiently mitigate the perceptible distortion in the secondary sound field, which contributes to the degradation of the active noise control accuracy used for spatial sound field control system. Our proposed system uses a Volterra inverse filter to reduce these distortions. In this filtering for active noise control, the primary challenge lies in the high computational load associated with the processing related to the Volterra kernel. We propose a new method to enhance the sparsity of the Volterra kernel, which contributes to the reduction of computational load, by focusing on the physical mechanisms behind the generation of these distortions Simulation results show that the proposed method suppresses these distortions and improves noise reduction by approximately 10 dB in specific frequency bands.

Abstract An analytical method is presented to find the stability domain of PID controller parameters for nonlinear time-delay systems by utilizing the Volterra series and its generalized frequency response function (GFRF), and all the parameters in the derived region can stabilize the system. According to the conditions for closed-loop L 2 stabilization, the procedures are derived to determine the PID controller stabilization region. The simulation results show the accuracy of the stability domain. Meanwhile, this proposed method preserves the nonlinear characteristics and is suitable for nonlinear time-delay systems with arbitrary-order nonlinearity. It also brings convenience to the tuning of PID parameters, providing a new idea for the practical design of nonlinear time-delay systems.

This article is devoted to the study of the physical properties of deterministic, chaotic, and stochastic signals used in modern information systems with the aim of enhancing their spectral efficiency. Special attention is given to chaotic and stochastic signals, which, due to their unique structures, ensure a high level of information security. However, they require further development to achieve optimal utilization of the frequency spectrum. The study addresses a pressing issue in communication systems: the need to balance high spectral efficiency with robust data protection and interference resilience. Deterministic signals, such as harmonic signals, traditionally used in narrowband systems, are characterized by simplicity and predictability but suffer from low spectral efficiency and poor resistance to interference. In contrast, chaotic and stochastic signals exhibit significant advantages in terms of security and interference immunity but require advanced signal processing techniques to overcome challenges associated with their wide spectral bandwidth and energy consumption. The research methodology integrates mathematical modeling and signal analysis. Traditional approaches, such as Fourier transformation, wavelet transformation, and Hilbert transformation, are compared with innovative techniques, including Volterra series and Karhunen–Loève transformation. The comparative analysis is based on evaluating spectral efficiency, interference resilience, and the energy requirements of different signal types. The results demonstrate that chaotic signals outperform deterministic and stochastic signals in terms of spectral efficiency across broader frequency ranges. However, chaotic signals require more sophisticated processing methods to ensure their stability and reliability in modern communication systems. Stochastic signals, while offering superior interference resistance and information security, exhibit lower spectral efficiency due to their broad frequency spectrum and uneven energy distribution. Innovative approaches, such as Volterra series and Karhunen–Loève transformation, significantly improve the spectral efficiency of chaotic and stochastic signals by reducing redundancy and optimizing frequency utilization. These findings highlight the need for integrating advanced signal processing methods into information systems to enhance their performance and reliability. The study's results have practical implications for the development of advanced communication systems, such as cellular networks, the Internet of Things, and satellite communication systems, where high data confidentiality and efficient spectrum usage are critical.

The unsteady pressure distribution over a pitching NACA 0012 airfoil with and without a control surface is measured in a low-speed wind tunnel. Two sets of motions (step change and oscillatory) are enforced on the wing angle of attack and the control surface deflection angle to obtain the linear and nonlinear kernels of the Volterra series reduced-order model (ROM), respectively, based on the unsteady pressure field captured by pressure taps along the chord. The experimental procedure is presented, followed by a discussion on the performance of the measurement setting using a stepper motor and pressure transducers located exterior to the wing. The lift coefficient is calculated by integrating the pressure field obtained by the pressure transducers, and the performance of the Volterra series method using linear and nonlinear terms is presented and discussed for each configuration explored in this study.

Recently, the continuous advancement of sixth-generation (6G) mobile communication technology has positioned the space-air-ground-sea integrated network (SAGSIN) as a highly promising architecture for 6G networks. However, emerging applications and intelligent access devices have significantly strained network capacity, making it increasingly difficult to meet the ever-growing demand. To address these challenges, high-frequency terahertz-waveband has been proposed as an attractive solution for next-generation communication networks. Terahertz (THz) technology, enables data rates from tens to hundreds of gigabits per second, thus making it suitable for ultrahigh-speed wireless communication. Our study explores integrating photonics-aided signal generation with THz transmission technologies in a long-haul wireless communication system with a partial over-the-sea transmission. However, nonlinear distortions often occur for the THz-band system, the traditional Volterra-series nonlinear equalizers (VNEs) do not perform well in our experiment since the Volterra series cannot give an accurate polynomial approximation of the nonlinear channel response for the complicated over-the-sea channel conditions. A complex-valued multiple-symbol output (CVMSO) neural network (NN) equalizer is proposed to solve this problem. In this article, we achieved the transmission of a 2 GBaud THz-band quadrature phase shift keying (QPSK) signal over a 20.1 km wireless link at 125 GHz using photonics-assisted technology. Meanwhile, we establish a 20.1 km THz wireless communication model over the sea, revealing that elevated temperature and humidity markedly enhance air attenuation loss, adversely affecting the THz communication system. With the CVMSO equalization method, we achieved a bit error rate (BER) below 1.56 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times$</tex-math> </inline-formula> <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{-2}$</tex-math> </inline-formula> and a 1.8 dB receiver gain compared with VNE. Furthermore, the CVMSO NN equalizer demonstrated a 5.7% reduction in computational complexity and a reduction of 2000 training lengths because it can utilize the network information trained by the current input symbols for equalizing following multiple symbols, thereby minimizing unnecessary computations. Moreover, we extend our research on the complex climate environment in the over-the-sea channel, instructive for long-haul THz over-the-sea communications.

The Volterra series has been used in nonlinear system identification (NLSI) for decades; its frequency-domain counterpart allows a generalization of 'resonance curves' for nonlinear systems-so-called higher-order frequency-response functions (HFRFs). Estimating the terms in the series has often proved to be a challenge; however, the (comparatively) recent uptake of machine-learning technology into engineering dynamics has led to advances in the identification of the series-both for the Volterra kernels themselves and for the HFRFs. The current paper provides an overview of a number of approaches based on neural networks, Gaussian processes (GPs) and reproducing kernel Hilbert spaces (RKHSs), and presents new results for multi-input multi-output (MIMO) systems based on neural networks.This article is part of the theme issue 'Frontiers of applied inverse problems in science and engineering'.

The problem of input signals recovery is one of the key problems in many branches of science and technology: in measurement technology for dynamic measurements; in control tasks, where the control is based on the input signal (stabilization task); in tasks of image restoration and filtering, etc., which determines the relevance of the development of methods of input signal recovery. The relevance of development of input signal recovery methods is increasing every year and becomes the most pronounced at operation of measuring and control systems in extreme and harsh operating conditions. Non-stationarity and nonlinearity in these conditions are the most pronounced, accounting of which is a prerequisite for the creation of new and improvement of existing information-measuring and control systems. The paper proposes methods for determining the input signal of nonlinear dynamic systems described by the Volterra functional series. The methods are based on the solving of a nonlinear integral equation, which is defined by a finite segment of the Volterra series. The input signal recovery is carried out under the assumption that the integral transforms of Volterra kernels possess factorization based on Borel's theorem, which leads to a nonlinear algebraic equation. Recovery methods of continuous nonlinear dynamic systems described by a finite Volterra series and their discrete analog are considered. When recovering the input signal, for continuous systems the integral Laplace transform is applied, for discrete systems the Z-transform is applied. An example of a mathematical solution to the problem of restoring a discrete one-dimensional signal is given, illustrating the efficiency and effectiveness of the developed methods. The results of studies on restoring signals of nonlinear dynamic systems will be useful to specialists engaged in theoretical research and mathematical modeling in the field of digital signal processing and vector analysis of electrical circuits.