Symbolic regression with polynomial neural networks and polynomial neural ordinary differential equations (ODEs) are two recent and powerful approaches for equation recovery of many science and engineering problems. However, these methods provide point estimates for the model parameters and are currently unable to accommodate noisy data. We address this challenge by developing and validating the following Bayesian inference methods: the Laplace approximation, Markov Chain Monte Carlo (MCMC) sampling methods, and variational inference. We have found the Laplace approximation to be the best method for this class of problems. Our work can be easily extended to the broader class of symbolic neural networks to which the polynomial neural network belongs.

Some applications of deep learning require not only to provide accurate results but also to quantify the amount of confidence in their prediction. The management of an electric power grid is one of these cases: to avoid risky scenarios, decision-makers need both precise and reliable forecasts of, for example, power loads. For this reason, point forecasts are not enough hence it is necessary to adopt methods that provide an uncertainty quantification. This work focuses on reservoir computing as the core time series forecasting method, due to its computational efficiency and effectiveness in predicting time series. While the RC literature mostly focused on point forecasting, this work explores the compatibility of some popular uncertainty quantification methods with the reservoir setting. Both Bayesian and deterministic approaches to uncertainty assessment are evaluated and compared in terms of their prediction accuracy, computational resource efficiency and reliability of the estimated uncertainty, based on a set of carefully chosen performance metrics.

Deep neural networks (NNs) are known for their high-prediction performances. However, NNs are prone to yield unreliable predictions when encountering completely new situations without indicating their uncertainty. Bayesian variants of NNs (BNNs), such as Monte Carlo (MC) dropout BNNs, do provide uncertainty measures and simultaneously increase the prediction performance. The only disadvantage of BNNs is their higher computation time during test time because they rely on a sampling approach. Here we present a single-shot MC dropout approximation that preserves the advantages of BNNs while being as fast as NNs. Our approach is based on moment propagation (MP) and allows to analytically approximate the expected value and the variance of the MC dropout signal for commonly used layers in NNs, i.e. convolution, max pooling, dense, softmax, and dropout layers. The MP approach can convert an NN into a BNN without re-training given the NN has been trained with standard dropout. We evaluate our approach on different benchmark datasets and a simulated toy example in a classification and regression setting. We demonstrate that our single-shot MC dropout approximation resembles the point estimate and the uncertainty estimate of the predictive distribution that is achieved with an MC approach, while being fast enough for real-time deployments of BNNs. We show that using part of the saved time to combine our MP approach with deep ensemble techniques does further improve the uncertainty measures.

Offshore wind structures are subject to deterioration mechanisms throughout their operational lifetime. Even if the deterioration evolution of structural elements can be estimated through physics-based deterioration models, the uncertainties involved in the process hurdle the selection of lifecycle management decisions. In this scenario, the collection of relevant information through an efficient monitoring system enables the reduction of uncertainties, ultimately driving more optimal lifecycle decisions. However, a full monitoring instrumentation implemented on all wind turbines in a farm might become unfeasible due to practical and economical constraints. Besides, certain load monitoring systems often become defective after a few years of marine environment exposure. Addressing the aforementioned concerns, a farm-wide virtual load monitoring scheme directed by a fleet-leader wind turbine offers an attractive solution. Fetched with data retrieved from a fully-instrumented wind turbine, a model can be trained and then deployed, thus yielding load predictions of non-fully monitored wind turbines, from which only standard data remains available. In this paper, we propose a virtual load monitoring framework formulated via Bayesian neural networks (BNNs) and we provide relevant implementation details needed for the construction, training, and deployment of BNN data-based virtual monitoring models. As opposed to their deterministic counterparts, BNNs intrinsically announce the uncertainties associated with generated load predictions and allow to detect inaccurate load estimations generated for non-fully monitored wind turbines. The proposed virtual load monitoring is thoroughly tested through an experimental campaign in an operational offshore wind farm and the results demonstrate the effectiveness of BNN models for fleet-leader-based farm-wide virtual monitoring.

We propose an optimization algorithm for Variational Inference (VI) in complex models. Our approach relies on natural gradient updates where the variational space is a Riemann manifold. We develop an efficient algorithm for Gaussian Variational Inference that implicitly satisfies the positive definite constraint on the variational covariance matrix. Our Exact manifold Gaussian Variational Bayes (EMGVB) provides exact but simple update rules and is straightforward to implement. Due to its black-box nature, EMGVB stands as a ready-to-use solution for VI in complex models. Over five datasets, we empirically validate our feasible approach on different statistical, econometric, and deep learning models, discussing its performance with respect to baseline methods.

Machine Learning (ML) algorithms that perform classification may predict the wrong class, experiencing misclassifications. It is well-known that misclassifications may have cascading effects on the encompassing system, possibly resulting in critical failures. This paper proposes SPROUT, a Safety wraPper thROugh ensembles of UncertainTy measures, which suspects misclassifications by computing uncertainty measures on the inputs and outputs of a black-box classifier. If a misclassification is detected, SPROUT blocks the propagation of the output of the classifier to the encompassing system. The resulting impact on safety is that SPROUT transforms erratic outputs (misclassifications) into data omission failures, which can be easily managed at the system level. SPROUT has a broad range of applications as it fits binary and multi-class classification, comprising image and tabular datasets. We experimentally show that SPROUT always identifies a huge fraction of the misclassifications of supervised classifiers, and it is able to detect all misclassifications in specific cases. SPROUT implementation contains pre-trained wrappers, it is publicly available and ready to be deployed with minimal effort.

Spectrum cartography (SC), also known as radio map estimation (RME), aims at crafting multi-domain (e.g., frequency and space) radio power propagation maps from limited sensor measurements. While early methods often lacked theoretical support, recent works have demonstrated that radio maps can be provably recovered using low-dimensional models -- such as the block-term tensor decomposition (BTD) model and certain deep generative models (DGMs) -- of the high-dimensional multi-domain radio signals. However, these existing provable SC approaches assume that sensors send real-valued (full-resolution) measurements to the fusion center, which is unrealistic. This work puts forth a quantized SC framework that generalizes the BTD and DGM-based SC to scenarios where heavily quantized sensor measurements are used. A maximum likelihood estimation (MLE)-based SC framework under a Gaussian quantizer is proposed. Recoverability of the radio map using the MLE criterion are characterized under realistic conditions, e.g., imperfect radio map modeling and noisy measurements. Simulations and real-data experiments are used to showcase the effectiveness of the proposed approach.

The ability to replicate predictions by machine learning (ML) or artificial intelligence (AI) models and results in scientific workflows that incorporate such ML/AI predictions is driven by numerous factors. An uncertainty-aware metric that can quantitatively assess the reproducibility of quantities of interest (QoI) would contribute to the trustworthiness of results obtained from scientific workflows involving ML/AI models. In this article, we discuss how uncertainty quantification (UQ) in a Bayesian paradigm can provide a general and rigorous framework for quantifying reproducibility for complex scientific workflows. Such as framework has the potential to fill a critical gap that currently exists in ML/AI for scientific workflows, as it will enable researchers to determine the impact of ML/AI model prediction variability on the predictive outcomes of ML/AI-powered workflows. We expect that the envisioned framework will contribute to the design of more reproducible and trustworthy workflows for diverse scientific applications, and ultimately, accelerate scientific discoveries.

Regression analysis is a key area of interest in the field of data analysis and machine learning which is devoted to exploring the dependencies between variables, often using vectors. The emergence of high dimensional data in technologies such as neuroimaging, computer vision, climatology and social networks, has brought challenges to traditional data representation methods. Tensors, as high dimensional extensions of vectors, are considered as natural representations of high dimensional data. In this book, the authors provide a systematic study and analysis of tensor-based regression models and their applications in recent years. It groups and illustrates the existing tensor-based regression methods and covers the basics, core ideas, and theoretical characteristics of most tensor-based regression methods. In addition, readers can learn how to use existing tensor-based regression methods to solve specific regression tasks with multiway data, what datasets can be selected, and what software packages are available to start related work as soon as possible. Tensor Regression is the first thorough overview of the fundamentals, motivations, popular algorithms, strategies for efficient implementation, related applications, available datasets, and software resources for tensor-based regression analysis. It is essential reading for all students, researchers and practitioners of working on high dimensional data.

Extended target tracking estimates the centroid and shape of the target in space and time. In various situations where extended target tracking is applicable, the presence of multiple targets can lead to interference, particularly when they maneuver behind one another in a sensor like a camera. Nonetheless, when dealing with multiple extended targets, there's a tendency for them to share similar shapes within a group, which can enhance their detectability. For instance, the coordinated movement of a cluster of aerial vehicles might cause radar misdetections during their convergence or divergence. Similarly, in the context of a self-driving car, lane markings might split or converge, resulting in inaccurate lane tracking detections. A well-known joint probabilistic data association coupled (JPDAC) filter can address this problem in only a single-point target tracking. A variation of JPDACF was developed by introducing a nonparametric Spatio-Temporal Joint Probabilistic Data Association Coupled Filter (ST-JPDACF) to address the problem for extended targets. Using different kernel functions, we manage the dependency of measurements in space (inside a frame) and time (between frames). Kernel functions are able to be learned using a limited number of training data. This extension can be used for tracking the shape and dynamics of nonparametric dependent extended targets in clutter when targets share measurements. The proposed algorithm was compared with other well-known supervised methods in the interfering case and achieved promising results.