In many practical fields like engineering, finance and physics, we need to explore the effects of uncertainties borne by input or parameters, which is the task of the uncertainty quantification (UQ). In a specific UQ problem, if the system is represented as stochastic partial differential equations, then stochastic Galerkin methods like generalised polynomial chaos (gPC) [29] can be used to obtain statistics of the distribution of the unknown solution. As long as a numerical or analytical solver for the system is available, the renowned Monte Carlo (MC) method [8] can be applied to obtain a distribution of the quantities of interest. However, that is usually prohibitively expensive due to the slow convergence of the Monte Carlo method, despite the fact that the running of the computer code for solving some complicated systems one single time might take days. Thus, to some extent we may be willing to compromise accuracy to reduce running time through formulating a cheap surrogate [2].

Motivated by the goal of endowing robots with a means for focusing attention in order to operate reliably in complex, uncertain, and time-varying environments, we consider how a robot can (i) determine which portions of its environment to pay attention to at any given point in time, (ii) infer changes in context (e.g., task or environment dynamics), and (iii) switch its attention accordingly. In this work, we tackle these questions by modeling context switches in a time-varying Markov decision process (MDP) framework. We utilize the theory of bisimulation-based state abstractions in order to synthesize mechanisms for paying attention to context-relevant information. We then present an algorithm based on Bayesian inference for detecting changes in the robot's context (task or environment dynamics) as it operates online, and use this to trigger switches between different abstraction-based attention mechanisms. Our approach is demonstrated on two examples: (i) an illustrative discrete-state tracking problem, and (ii) a continuous-state tracking problem implemented on a quadrupedal hardware platform. These examples demonstrate the ability of our approach to detect context switches online and robustly ignore task-irrelevant distractors by paying attention to context-relevant information.

The present paper comes across the main steps that laid from Zadeh's fuzziness ana Atanassov's intuitionistic fuzzy sets to Smarandache's indeterminacy and to Molodstov's soft sets. Two hybrid methods for assessment and decision making respectively under fuzzy conditions are also presented through suitable examples that use soft sets and real intervals as tools. The decision making method improves an earlier method of Maji et al. Further, it is described how the concept of topological space, the most general category of mathematical spaces, can be extended to fuzzy structures and how to generalize the fundamental mathematical concepts of limit, continuity compactness and Hausdorff space within such kind of structures. In particular, fuzzy and soft topological spaces are defined and examples are given to illustrate these generalizations.

A key goal of unsupervised learning is to go beyond density estimation and sample generation to reveal the structure inherent within observed data. Such structure can be expressed in the pattern of interactions between explanatory latent variables captured through a probabilistic graphical model. Although the learning of structured graphical models has a long history, much recent work in unsupervised modelling has instead emphasised flexible deep-network-based generation, either transforming independent latent generators to model complex data or assuming that distinct observed variables are derived from different latent nodes. Here, we extend amortised variational inference to incorporate structured factors over multiple variables, able to capture the observation-induced posterior dependence between latents that results from ``explaining away'' and thus allow complex observations to depend on multiple nodes of a structured graph. We show that appropriately parametrised factors can be combined efficiently with variational message passing in rich graphical structures. We instantiate the framework in nonlinear Gaussian Process Factor Analysis, evaluating the structured recognition framework using synthetic data from known generative processes. We fit the GPFA model to high-dimensional neural spike data from the hippocampus of freely moving rodents, where the model successfully identifies latent signals that correlate with behavioural covariates.

Simulation-based inference (SBI) solves statistical inverse problems by repeatedly running a stochastic simulator and inferring posterior distributions from model-simulations. To improve simulation efficiency, several inference methods take a sequential approach and iteratively adapt the proposal distributions from which model simulations are generated. However, many of these sequential methods are difficult to use in practice, both because the resulting optimisation problems can be challenging and efficient diagnostic tools are lacking. To overcome these issues, we present Truncated Sequential Neural Posterior Estimation (TSNPE). TSNPE performs sequential inference with truncated proposals, sidestepping the optimisation issues of alternative approaches. In addition, TSNPE allows to efficiently perform coverage tests that can scale to complex models with many parameters. We demonstrate that TSNPE performs on par with previous methods on established benchmark tasks. We then apply TSNPE to two challenging problems from neuroscience and show that TSNPE can successfully obtain the posterior distributions, whereas previous methods fail. Overall, our results demonstrate that TSNPE is an efficient, accurate, and robust inference method that can scale to challenging scientific models.

The best way to model, understand, and quantify the information contained in complex systems is an open question in physics, mathematics, and computer science. The uncertain relationship between entropy and complexity further complicates this question. With ideas drawn from the object-relations theory of psychology, this paper develops an object-relations model of complex systems which generalizes to systems of all types, including mathematical operations, machines, biological organisms, and social structures. The resulting Complex Information Entropy (CIE) equation is a robust method to quantify complexity across various contexts. The paper also describes algorithms to iteratively update and improve approximate solutions to the CIE equation, to recursively infer the composition of complex systems, and to discover the connections among objects across different lengthscales and timescales. Applications are discussed in the fields of engineering design, atomic and molecular physics, chemistry, materials science, neuroscience, psychology, sociology, ecology, economics, and medicine.

Forecasting and decision-making are generally modeled as two sequential steps with no feedback, following an open-loop approach. In this paper, we present application-driven learning, a new closed-loop framework in which the processes of forecasting and decision-making are merged and co-optimized through a bilevel optimization problem. We present our methodology in a general format and prove that the solution converges to the best estimator in terms of the expected cost of the selected application. Then, we propose two solution methods: an exact method based on the KKT conditions of the second-level problem and a scalable heuristic approach suitable for decomposition methods. The proposed methodology is applied to the relevant problem of defining dynamic reserve requirements and conditional load forecasts, offering an alternative approach to current \emph{ad hoc} procedures implemented in industry practices. We benchmark our methodology with the standard sequential least-squares forecast and dispatch planning process. We apply the proposed methodology to an illustrative system and to a wide range of instances, from dozens of buses to large-scale realistic systems with thousands of buses. Our results show that the proposed methodology is scalable and yields consistently better performance than the standard open-loop approach.

Choosing the values of hyper-parameters in sparse Bayesian learning (SBL) can significantly impact performance. However, the hyper-parameters are normally tuned manually, which is often a difficult task. Most recently, effective automatic hyper-parameter tuning was achieved by using an empirical auto-tuner. In this work, we address the issue of hyper-parameter auto-tuning using neural network (NN)-based learning. Inspired by the empirical auto-tuner, we design and learn a NN-based auto-tuner, and show that considerable improvement in convergence rate and recovery performance can be achieved.

Agent-based modeling (ABM) is a well-established paradigm for simulating complex systems via interactions between constituent entities. Machine learning (ML) refers to approaches whereby statistical algorithms 'learn' from data on their own, without imposing a priori theories of system behavior. Biological systems -- from molecules, to cells, to entire organisms -- consist of vast numbers of entities, governed by complex webs of interactions that span many spatiotemporal scales and exhibit nonlinearity, stochasticity and intricate coupling between entities. The macroscopic properties and collective dynamics of such systems are difficult to capture via continuum modelling and mean-field formalisms. ABM takes a 'bottom-up' approach that obviates these difficulties by enabling one to easily propose and test a set of well-defined 'rules' to be applied to the individual entities (agents) in a system. Evaluating a system and propagating its state over discrete time-steps effectively simulates the system, allowing observables to be computed and system properties to be analyzed. Because the rules that govern an ABM can be difficult to abstract and formulate from experimental data, there is an opportunity to use ML to help infer optimal, system-specific ABM rules. Once such rule-sets are devised, ABM calculations can generate a wealth of data, and ML can be applied there too -- e.g., to probe statistical measures that meaningfully describe a system's stochastic properties. As an example of synergy in the other direction (from ABM to ML), ABM simulations can generate realistic datasets for training ML algorithms (e.g., for regularization, to mitigate overfitting). In these ways, one can envision various synergistic ABM$\rightleftharpoons$ML loops. This review summarizes how ABM and ML have been integrated in contexts that span spatiotemporal scales, from cellular to population-level epidemiology.

Where am I? This is one of the most critical questions that any intelligent system should answer to decide whether it navigates to a previously visited area. This problem has long been acknowledged for its challenging nature in simultaneous localization and mapping (SLAM), wherein the robot needs to correctly associate the incoming sensory data to the database allowing consistent map generation. The significant advances in computer vision achieved over the last 20 years, the increased computational power, and the growing demand for long-term exploration contributed to efficiently performing such a complex task with inexpensive perception sensors. In this article, visual loop closure detection, which formulates a solution based solely on appearance input data, is surveyed. We start by briefly introducing place recognition and SLAM concepts in robotics. Then, we describe a loop closure detection system's structure, covering an extensive collection of topics, including the feature extraction, the environment representation, the decision-making step, and the evaluation process. We conclude by discussing open and new research challenges, particularly concerning the robustness in dynamic environments, the computational complexity, and scalability in long-term operations. The article aims to serve as a tutorial and a position paper for newcomers to visual loop closure detection.