This paper introduces a method for the nonparametric Bayesian learning of nonlinear operators, through the use of the Volterra series with kernels represented using Gaussian processes (GPs), which we term the nonparametric Volterra kernels model (NVKM). When the input function to the operator is unobserved and has a GP prior, the NVKM constitutes a powerful method for both single and multiple output regression, and can be viewed as a nonlinear and nonparametric latent force model. When the input function is observed, the NVKM can be used to perform Bayesian system identification. We use recent advances in efficient sampling of explicit functions from GPs to map process realisations through the Volterra series without resorting to numerical integration, allowing scalability through doubly stochastic variational inference, and avoiding the need for Gaussian approximations of the output processes. We demonstrate the performance of the model for both multiple output regression and system identification using standard benchmarks.

Modeling nonlinear systems with Volterra series is challenging because the number of kernel coefficients grows exponentially with the model order. This work introduces Bayesian Tensor Network Volterra kernel machines (BTN-V), extending the Bayesian Tensor Network framework to Volterra system identification. BTN-V represents Volterra kernels using canonical polyadic decomposition, reducing model complexity from O(I^D) to O(DIR). By treating all tensor components and hyperparameters as random variables, BTN-V provides predictive uncertainty estimation at no additional computational cost. Sparsity-inducing hierarchical priors enable automatic rank determination and the learning of fading-memory behavior directly from data, improving interpretability and preventing overfitting. Empirical results demonstrate competitive accuracy, enhanced uncertainty quantification, and reduced computational cost.

The identification of high-dimensional nonlinear dynamical systems via the Volterra series has significant potential, but has been severely hindered by the curse of dimensionality. Tensor Network (TN) methods such as the Modified Alternating Linear Scheme (MVMALS) have been a breakthrough for the field, offering a tractable approach by exploiting the low-rank structure in Volterra kernels. However, these techniques still encounter prohibitive computational and memory bottlenecks due to high-order polynomial scaling with respect to input dimension. To overcome this barrier, we introduce the Tensor Head Averaging (THA) algorithm, which significantly reduces complexity by constructing an ensemble of localized MVMALS models trained on small subsets of the input space. In this paper, we present a theoretical foundation for the THA algorithm. We establish observable, finite-sample bounds on the error between the THA ensemble and a full MVMALS model, and we derive an exact decomposition of the squared error. This decomposition is used to analyze the manner in which subset models implicitly compensate for omitted dynamics. We quantify this effect, and prove that correlation between the included and omitted dynamics creates an optimization incentive which drives THA's performance toward accuracy superior to a simple truncation of a full MVMALS model. THA thus offers a scalable and theoretically grounded approach for identifying previously intractable high-dimensional systems.

This paper introduces a method for the nonparametric Bayesian learning of nonlinear operators, through the use of the Volterra series with kernels represented using Gaussian processes (GPs), which we term the nonparametric Volterra kernels model (NVKM). When the input function to the operator is unobserved and has a GP prior, the NVKM constitutes a powerful method for both single and multiple output regression, and can be viewed as a nonlinear and nonparametric latent force model. When the input function is observed, the NVKM can be used to perform Bayesian system identification. We use recent advances in efficient sampling of explicit functions from GPs to map process realisations through the Volterra series without resorting to numerical integration, allowing scalability through doubly stochastic variational inference, and avoiding the need for Gaussian approximations of the output processes. We demonstrate the performance of the model for both multiple output regression and system identification using standard benchmarks.

In aerospace and mechanical engineering, the design process for new products relies on hierarchies of mathematical models, the physical complexity of which may be imposed by computational costs or dictated by regulations. These models typically incorporate parameters to account for various operating conditions and configurations. In the framework of optimization, for example, hundreds of parameters (design variables) may be required to define the configuration of a system. Similarly, uncertainty propagation may necessitate defining a complex parameter space to account for variations in geometrical imperfections, material properties, or flow conditions. The design process, especially during optimization and uncertainty quantification, often involves numerous evaluations of the system's mathematical models across a wide range of points in the parameter space. Computational costs vary with the level of model fidelity: lower fidelity models are traditionally used for computationally intensive evaluations, while higher fidelity models - often involving nonlinear partial differential equations discretized on fine grids - are typically reserved for later stages of the design process.

The damage detection problem becomes a more difficult task when the intrinsically nonlinear behavior of the structures and the natural data variation are considered in the analysis because both phenomena can be confused with damage if linear and deterministic approaches are implemented. Therefore, this work aims the experimental application of a stochastic version of the Volterra series combined with a novelty detection approach to detect damage in an initially nonlinear system taking into account the measured data variation, caused by the presence of uncertainties. The experimental setup is composed by a cantilever beam operating in a nonlinear regime of motion, even in the healthy condition, induced by the presence of a magnet near to the free extremity. The damage associated with mass changes in a bolted connection (nuts loosed) is detected based on the comparison between linear and nonlinear contributions of the stochastic Volterra kernels in the total response, estimated in the reference and damaged conditions. The experimental measurements were performed on different days to add natural variation to the data measured. The results obtained through the stochastic proposed approach are compared with those obtained by the deterministic version of the Volterra series, showing the advantage of the stochastic model use when we consider the experimental data variation with the capability to detect the presence of the damage with statistical confidence. Besides, the nonlinear metric used presented a higher sensitivity to the occurrence of the damage compared with the linear one, justifying the application of a nonlinear metric when the system exhibits intrinsically nonlinear behavior.

The damage detection problem in mechanical systems, using vibration measurements, is commonly called Structural Health Monitoring (SHM). Many tools are able to detect damages by changes in the vibration pattern, mainly, when damages induce nonlinear behavior. However, a more difficult problem is to detect structural variation associated with damage, when the mechanical system has nonlinear behavior even in the reference condition. In these cases, more sophisticated methods are required to detect if the changes in the response are based on some structural variation or changes in the vibration regime, because both can generate nonlinearities. Among the many ways to solve this problem, the use of the Volterra series has several favorable points, because they are a generalization of the linear convolution, allowing the separation of linear and nonlinear contributions by input filtering through the Volterra kernels. On the other hand, the presence of uncertainties in mechanical systems, due to noise, geometric imperfections, manufacturing irregularities, environmental conditions, and others, can also change the responses, becoming more difficult the damage detection procedure. An approach based on a stochastic version of Volterra series is proposed to be used in the detection of a breathing crack in a beam vibrating in a nonlinear regime of motion, even in reference condition (without crack). The system uncertainties are simulated by the variation imposed in the linear stiffness and damping coefficient. The results show, that the nonlinear analysis done, considering the high order Volterra kernels, allows the approach to detect the crack with a small propagation and probability confidence, even in the presence of uncertainties.

Gradient boosted decision trees have achieved remarkable success in several domains, particularly those that work with static tabular data. However, the application of gradient boosted models to signal processing is underexplored. In this work, we introduce gradient boosted filters for dynamic data, by employing Hammerstein systems in place of decision trees. We discuss the relationship of our approach to the Volterra series, providing the theoretical underpinning for its application. We demonstrate the effective generalizability of our approach with examples.

The computation of classical higher-order statistics such as higher-order moments or spectra is difficult for images due to the huge number of terms to be estimated and interpreted. We propose an alternative ap- proach in which multiplicative pixel interactions are described by a se- ries of Wiener functionals. Since the functionals are estimated implicitly via polynomial kernels, the combinatorial explosion associated with the classical higher-order statistics is avoided. First results show that image structures such as lines or corners can be predicted correctly, and that pixel interactions up to the order of five play an important role in natural images. Most of the interesting structure in a natural image is characterized by its higher-order statistics.