The vulnerability of machine learning models to adversarial attacks remains a critical security challenge. Traditional defenses, such as adversarial training, typically robustify models by minimizing a worst-case loss. However, these deterministic approaches do not account for uncertainty in the adversary's attack. While stochastic defenses placing a probability distribution on the adversary exist, they often lack statistical rigor and fail to make explicit their underlying assumptions. To resolve these issues, we introduce a formal Bayesian framework that models adversarial uncertainty through a stochastic channel, articulating all probabilistic assumptions. This yields two robustification strategies: a proactive defense enacted during training, aligned with adversarial training, and a reactive defense enacted during operations, aligned with adversarial purification. Several previous defenses can be recovered as limiting cases of our model. We empirically validate our methodology, showcasing the benefits of explicitly modeling adversarial uncertainty.

Haag duality is a strong notion of locality for two-dimensional lattice quantum spin systems, requiring that the commutant of the algebra of observables supported in a cone-like region coincides with the algebra of observables in its complement. Originally introduced within algebraic quantum field theory, Haag duality has recently become pivotal in the operator-algebraic analysis of quantum many-body systems. In particular, it plays a key role in the description of anyonic excitations, which are widely believed to classify two-dimensional non-chiral gapped quantum phases of matter. Prior to this work, Haag duality had only been rigorously established for Kitaev's quantum double models with abelian input groups. In this paper, we establish that two-dimensional tensor network states based on biconnected $C^*$-weak Hopf algebras satisfy Haag duality. These states include as particular cases Kitaev quantum double and Levin-Wen string-net models, and we expect them to encompass representatives of all non-chiral phases. Our proof relies on deriving an operator-algebraic sufficient condition for Haag duality in finite systems, which we verify using tensor-network methods. As part of our analysis, we extend the tensor-network bulk-boundary correspondence, construct explicit commuting parent Hamiltonians for these models, and prove that in the biconnected case they satisfy the local topological quantum order condition.

Federated Learning is a machine learning approach that enables the training of a deep learning model among several participants with sensitive data that wish to share their own knowledge without compromising the privacy of their data. In this research, the authors employ a secured Federated Learning method with an additional layer of privacy and proposes a method for addressing the non-IID challenge. Moreover, differential privacy is compared with chaotic-based encryption as layer of privacy. The experimental approach assesses the performance of the federated deep learning model with differential privacy using both IID and non-IID data. In each experiment, the Federated Learning process improves the average performance metrics of the deep neural network, even in the case of non-IID data.

Whilst the Universal Approximation Theorem guarantees the existence of approximations to Sobolev functions -- the natural function spaces for PDEs -- by Neural Networks (NNs) of sufficient size, low-regularity solutions may lead to poor approximations in practice. For example, classical fully-connected feed-forward NNs fail to approximate continuous functions whose gradient is discontinuous when employing strong formulations like in Physics Informed Neural Networks (PINNs). In this article, we propose the use of regularity-conforming neural networks, where a priori information on the regularity of solutions to PDEs can be employed to construct proper architectures. We illustrate the potential of such architectures via a two-dimensional (2D) transmission problem, where the solution may admit discontinuities in the gradient across interfaces, as well as power-like singularities at certain points. In particular, we formulate the weak transmission problem in a PINNs-like strong formulation with interface and continuity conditions. Such architectures are partially explainable; discontinuities are explicitly described, allowing the introduction of novel terms into the loss function. We demonstrate via several model problems in one and two dimensions the advantages of using regularity-conforming architectures in contrast to classical architectures. The ideas presented in this article easily extend to problems in higher dimensions.

Machine learning (ML) methods have been successfully employed in identifying variables that can predict the equity premium of individual stocks. In this paper, we investigate if ML can also be helpful in selecting variables relevant for optimal portfolio choice. To address this question, we parameterize minimum-variance portfolio weights as a function of a large pool of firm-level characteristics as well as their second-order and cross-product transformations, yielding a total of 4,610 predictors. We find that the gains from employing ML to select relevant predictors are substantial: minimum-variance portfolios achieve lower risk relative to sparse specifications commonly considered in the literature, especially when non-linear terms are added to the predictor space. Moreover, some of the selected predictors that help decreasing portfolio risk also increase returns, leading to minimum-variance portfolios with good performance in terms of Shape ratios in some situations. Our evidence suggests that ad-hoc sparsity can be detrimental to the performance of minimum-variance characteristics-based portfolios.

Quantum Information Theory, the standard formalism used to represent information contained in quantum systems, is based on complex Hilbert spaces (CQT). It was recently shown that it predicts correlations in quantum networks which cannot be explained by Real Quantum Theory (RQT), a quantum theory with real Hilbert spaces instead of complex ones, when three parties are involved in a quantum network with non-trivial locality constraints. In this work, we study a scenario with $N+1$ parties sharing quantum systems in a star network. Here, we construct a "conditional" multipartite Bell inequality that exhibits a gap between RQT and CQT, which linearly increases with $N$ and is thus arbitrarily large in the asymptotic limit. This implies, that, as the number of parties grows, Hilbert space formalism based on real numbers becomes exceedingly worse at describing complex networks of quantum systems. Furthermore, we also compute the tolerance of this gap to experimental errors.

Let A be a square matrix with real entries. The spread of A is defined as the maximum of the distances among the eigenvalues of A. Let $S_m[a,b]$ denote the set of all $m\times m$ symmetric matrices with entries in the real interval $[a,b]$ and let $S_m\{a,b\}$ be the subset of $S_m[a,b]$ of Bohemian matrices with population from only the extremal elements $\{a,b\}$. S. M. Fallat and J. J. Xing in 2012 proposed the following conjecture: the maximum spread in $S_m[a,b]$ is attained by a rank $2$ matrix in $S_m\{a,b\}$. X. Zhan had proved previously that the conjecture was true for $S_m[-a,a]$ with $a>0$. We will show how to interpret this problem geometrically, via polynomial resultants, in order to be able to treat this conjecture from a computational point of view. This will allow us to prove that this conjecture is true for several formerly open cases.

We study the linear convergence of Frank-Wolfe algorithms over product polytopes. We analyze two condition numbers for the product polytope, namely the \emph{pyramidal width} and the \emph{vertex-facet distance}, based on the condition numbers of individual polytope components. As a result, for convex objectives that are $\mu$-Polyak-{\L}ojasiewicz, we show linear convergence rates quantified in terms of the resulting condition numbers. We apply our results to the problem of approximately finding a feasible point in a polytope intersection in high-dimensions, and demonstrate the practical efficiency of our algorithms through empirical results.

Pricing decisions stand out as one of the most critical tasks a company faces, particularly in today's digital economy. As with other business decision-making problems, pricing unfolds in a highly competitive and uncertain environment. Traditional analyses in this area have heavily relied on game theory and its variants. However, an important drawback of these approaches is their reliance on common knowledge assumptions, which are hardly tenable in competitive business domains. This paper introduces an innovative personalized pricing framework designed to assist decision-makers in undertaking pricing decisions amidst competition, considering both buyer's and competitors' preferences. Our approach (i) establishes a coherent framework for modeling competition mitigating common knowledge assumptions; (ii) proposes a principled method to forecast competitors' pricing and customers' purchasing decisions, acknowledging major business uncertainties; and, (iii) encourages structured thinking about the competitors' problems, thus enriching the solution process. To illustrate these properties, in addition to a general pricing template, we outline two specifications - one from the retail domain and a more intricate one from the pension fund domain.

We revisit the nonlinear second-order differential equations $$ \ddot{x}(t)=a (x )\dot{x}(t)^2+b(t)\dot{x}(t) $$ where $a(x)$ and $b(t)$ are arbitrary functions on their argument from the perspective of Lie-Hamilton systems. For the particular choice $a(x)=3/x$ and $b(t)=1/t$, these equations reduce to the Buchdahl equation considered in the context of General Relativity. It is shown that these equations are associated to the 'book' Lie algebra $\mathfrak{b}_2$, determining a Lie-Hamilton system for which the corresponding $t$-dependent Hamiltonian and the general solution of the equations are given. The procedure is illustrated considering several particular cases. We also make use of the quantum deformation of $\mathfrak{b}_2$ with quantum deformation parameter $z$ (where $q={\rm e}^z$), leading to a deformed generalized Buchdahl equation. Applying the formalism of Poisson-Hopf deformations of Lie-Hamilton systems, we derive the corresponding deformed $t$-dependent Hamiltonian, as well as its general solution. The generalized Buchdahl equation is further extended to the oscillator Lie-Hamilton algebra $\mathfrak{h}_4\supset \mathfrak{b}_2$, together with its quantum deformation, and the corresponding (deformed) equations are also analyzed for their exact solutions. The presence of the quantum deformation parameter $z$ is interpreted as the introduction of an integrable perturbation of the (initial) generalized Buchdahl equation, which is described in detail in its linear approximation. Finally, it is also shown that, under quantum deformations, the higher-dimensional deformed generalized Buchdahl equations from either the book or the oscillator algebras do not reduce to a sum of copies of the initial system but to intrinsic coupled systems governed by $z$.

Variational Physics-Informed Neural Networks often suffer from poor convergence when using stochastic gradient-descent-based optimizers. By introducing a Least Squares solver for the weights of the last layer of the neural network, we improve the convergence of the loss during training in most practical scenarios. This work analyzes the computational cost of the resulting hybrid Least-Squares/Gradient-Descent optimizer and explains how to implement it efficiently. In particular, we show that a traditional implementation based on backward-mode automatic differentiation leads to a prohibitively expensive algorithm. To remedy this, we propose using either forward-mode automatic differentiation or an ultraweak-type scheme that avoids the differentiation of trial functions in the discrete weak formulation. The proposed alternatives are up to one hundred times faster than the traditional one, recovering a computational cost-per-iteration similar to that of a conventional gradient-descent-based optimizer alone. To support our analysis, we derive computational estimates and conduct numerical experiments in one- and two-dimensional problems.

The computation of the topology of a real algebraic plane curve is greatly simplified if there are no more than one critical point in each vertical line: the general position condition. When this condition is not satisfied, then a finite number of changes of coordinates will move the initial curve to one in general position. We will show many cases where the topology of the considered curve around a critical point is very easy to compute even if the curve is not in general position. This will be achieved by introducing a new family of formulae describing, in many cases and through subresultants, the multiple roots of a univariate polynomial as rational functions of the considered polynomial involving at most one square root. This new approach will be used to show that the topology of cubics, quartics and quintics can be computed easily even if the curve is not in general position and to characterise those higher degree curves where this approach can be used. We will apply also this technique to determine the intersection curve of two quadrics and to study how to characterise the type of the curve arising when intersecting two ellipsoids.

This paper provides an efficient computational scheme to handle general security games from an adversarial risk analysis perspective. Two cases in relation to single-stage and multi-stage simultaneous defend-attack games motivate our approach to general setups which uses bi-agent influence diagrams as underlying problem structure and augmented probability simulation as core computational methodology. Theoretical convergence and numerical, modeling, and implementation issues are thoroughly discussed. A disinformation war case study illustrates the relevance of the proposed approach.

We examine the challenges associated with numerical integration when applying Neural Networks to solve Partial Differential Equations (PDEs). We specifically investigate the Deep Ritz Method (DRM), chosen for its practical applicability and known sensitivity to integration inaccuracies. Our research demonstrates that both standard deterministic integration techniques and biased stochastic quadrature methods can lead to incorrect solutions. In contrast, employing high-order, unbiased stochastic quadrature rules defined on integration meshes in low dimensions is shown to significantly enhance convergence rates at a comparable computational expense with respect to low-order methods like Monte Carlo. Additionally, we introduce novel stochastic quadrature approaches designed for triangular and tetrahedral mesh elements, offering increased adaptability for handling complex geometric domains. We highlight that the variance inherent in the stochastic gradient acts as a bottleneck for convergence. Furthermore, we observe that for gradient-based optimisation, the crucial factor is the accurate integration of the gradient, rather than just minimizing the quadrature error of the loss function itself.

Bayesian Optimization (BO) is a powerful method for optimizing black-box functions by combining prior knowledge with ongoing function evaluations. BO constructs a probabilistic surrogate model of the objective function given the covariates, which is in turn used to inform the selection of future evaluation points through an acquisition function. For smooth continuous search spaces, Gaussian Processes (GPs) are commonly used as the surrogate model as they offer analytical access to posterior predictive distributions, thus facilitating the computation and optimization of acquisition functions. However, in complex scenarios involving optimization over categorical or mixed covariate spaces, GPs may not be ideal. This paper introduces Simulation Based Bayesian Optimization (SBBO) as a novel approach to optimizing acquisition functions that only requires sampling-based access to posterior predictive distributions. SBBO allows the use of surrogate probabilistic models tailored for combinatorial spaces with discrete variables. Any Bayesian model in which posterior inference is carried out through Markov chain Monte Carlo can be selected as the surrogate model in SBBO. We demonstrate empirically the effectiveness of SBBO using various choices of surrogate models in applications involving combinatorial optimization. choices of surrogate models.

We present a new method to analytically prove global stability in ghost-ridden dynamical systems. Our proposal encompasses all prior results and consequentially extends them. In particular, we show that stability can follow from a conserved quantity that is unbounded from below, contrary to expectation. Novel examples illustrate all our results. Our findings take root on a careful examination of the literature, here comprehensively reviewed for the first time. This work lays the mathematical basis for ulterior extensions to field theory and quantization, and it constitutes a gateway for inter-disciplinary research in dynamics and integrability.

In this note we establish simple and verifiable analytical conditions for a power series f in the class K, and its associated Khinchin family, to be Gaussian. We give several moment criteria for general power series with non-negative coefficients in K and also provide several applications of these criteria.

Data privacy is a major concern in industries such as healthcare or finance. The requirement to safeguard privacy is essential to prevent data breaches and misuse, which can have severe consequences for individuals and organisations. Federated learning is a distributed machine learning approach where multiple participants collaboratively train a model without compromising the privacy of their data. However, a significant challenge arises from the differences in feature spaces among participants, known as non-IID data. This research introduces a novel federated learning framework employing fuzzy cognitive maps, designed to comprehensively address the challenges posed by diverse data distributions and non-identically distributed features in federated settings. The proposal is tested through several experiments using four distinct federation strategies: constant-based, accuracy-based, AUC-based, and precision-based weights. The results demonstrate the effectiveness of the approach in achieving the desired learning outcomes while maintaining privacy and confidentiality standards.

After over a century of internal combustion engines ruling the transport sector, electric vehicles appear to be on the verge of gaining traction due to a slew of advantages, including lower operating costs and lower CO2 emissions. By using the Vehicle-to-Grid (or Grid-to-Vehicle if Electric vehicles (EVs) are utilized as load) approach, EVs can operate as both a load and a source. Primary frequency regulation and congestion management are two essential characteristics of this technology that are added to an industrial microgrid. Industrial Microgrids are made up of different energy sources such as wind farms and PV farms, storage systems, and loads. EVs have gained a lot of interest as a technique for frequency management because of their ability to regulate quickly. Grid reliability depends on this quick reaction. Different contingency, state of charge of the electric vehicles, and a varying number of EVs in an EV fleet are considered in this work, and a proposed control scheme for frequency management is presented. This control scheme enables bidirectional power flow, allowing for primary frequency regulation during the various scenarios that an industrial microgrid may encounter over the course of a 24-h period. The presented controller will provide dependable frequency regulation support to the industrial microgrid during contingencies, as will be demonstrated by simulation results, achieving a more reliable system. However, simulation results will show that by increasing a number of the EVs in a fleet for the Vehicle-to-Grid approach, an industrial microgrid\'s frequency can be enhanced even further.

Currently, wind energy is one of the most important sources of renewable energy. Offshore locations for wind turbines are increasingly exploited because of their numerous advantages. However, offshore wind farms require high investment in maintenance service. Due to its complexity and special requirements, maintenance service is usually outsourced by wind farm owners. In this paper, we propose a novel approach to determine, quantify, and reduce the possible conflicts of interest between owners and maintenance suppliers. We created a complete techno-economic model to address this problem from an impartial point of view. An iterative process was developed to obtain statistical results that can help stakeholders negotiate the terms of the contract, in which the availability of the wind farm is the reference parameter by which to determine penalisations and incentives. Moreover, a multi-objective programming problem was addressed that maximises the profits of both parties without losing the alignment of their interests. The main scientific contribution of this paper is the maintenance analysis of offshore wind farms from two perspectives: that of the owner and the maintenance supplier. This analysis evaluates the conflicts of interest of both parties. In addition, we demonstrate that proper adjustment of some parameters, such as penalisation, incentives, and resources, and adequate control of availability can help reduce this conflict of interests.