We generalize the framework of Fredholm Neural Networks, to learn non-expansive integral operators arising in Fredholm Integral Equations (FIEs) of the second kind in arbitrary dimensions. We first present the proposed Fredholm Integral Neural Operators (FREDINOs), for FIEs and prove that they are universal approximators of linear and non-linear integral operators and corresponding solution operators. We furthermore prove that the learned operators are guaranteed to be contractive, thereby strictly satisfying the mathematical property required for the convergence of the fixed point scheme. Finally, we also demonstrate how FREDINOs can be used to learn the solution operator of non-linear elliptic PDEs, via a Boundary Integral Equation (BIE) formulation. We assess the proposed methodology numerically, via several benchmark problems: linear and non-linear FIEs in arbitrary dimensions, as well as a non-linear elliptic PDE in 2D. Built on tailored mathematical/numerical analysis theory, FREDINOs offer high-accuracy approximations and interpretable schemes, making them well suited for scientific machine learning/numerical analysis computations.

We deal with the solution of the forward problem for high-dimensional parabolic PDEs with random feature (projection) neural networks (RFNNs). We first prove that there exists a single-hidden layer neural network with randomized heat-kernels arising from the fundamental solution (Green's functions) of the heat operator, that we call HEATNET, that provides an unbiased universal approximator to the solution of parabolic PDEs in arbitrary (high) dimensions, with the rate of convergence being analogous to the ${O}(N^{-1/2})$, where $N$ is the size of HEATNET. Thus, HEATNETs are explainable schemes, based on the analytical framework of parabolic PDEs, exploiting insights from physics-informed neural networks aided by numerical and functional analysis, and the structure of the corresponding solution operators. Importantly, we show how HEATNETs can be scaled up for the efficient numerical solution of arbitrary high-dimensional parabolic PDEs using suitable transformations and importance Monte Carlo sampling of the integral representation of the solution, in order to deal with the singularities of the heat kernel around the collocation points. We evaluate the performance of HEATNETs through benchmark linear parabolic problems up to 2,000 dimensions. We show that HEATNETs result in remarkable accuracy with the order of the approximation error ranging from $1.0E-05$ to $1.0E-07$ for problems up to 500 dimensions, and of the order of $1.0E-04$ to $1.0E-03$ for 1,000 to 2,000 dimensions, with a relatively low number (up to 15,000) of features.

We propose a statistical duality among the preferences and endowments of the agents. Under this duality, the logarithmic prices of random trades among agents in a decentralized economy converge in expectation to the logarithm of the Walrasian equilibrium price in a centralized economy.

We study the problem of model aggregation within the Wasserstein space for probability measures on the real line. Given a fixed finite collection of candidate probability models, we consider the associated class of Wasserstein barycenters and develop a data-driven calibration framework in which the aggregation weights are statistically learned from empirical information associated with a target distribution. From a variational perspective based on Γ-convergence, we establish consistency of the resulting aggregation scheme, showing that empirical minimizers converge to the minimizers of the actual problem, along with the associated barycentric estimators, under mild conditions. The performance of the proposed method is evaluated through synthetic experiments and illustrated on a real dataset from a temperature monitoring network of sensors.

Building on our previous work on Fredholm Neural Networks (Fredholm NNs/ FNNs) for solving integral equations, we extend the framework to inverse problems for linear and nonlinear elliptic partial differential equations. The proposed scheme consists of a custom-designed deep neural network (DNN) in which the number of layers, weights, biases and hyperparameters are computed in an explainable manner based on a fixed-point scheme, and we therefore refer to this as the Potential Fredholm Neural Network (PFNN). We first build the PFNN as a method for solving the forward problem, showing that this approach ensures both a high accuracy and explainability, achieving small errors in the interior of the domain, and near machine-precision on the boundary. We then use this approach to solve inverse problems for elliptic PDEs, and provide a rigorous proof for the consistency of the scheme and error bounds for both the interior and boundary of the domain, tied directly to the architecture of the PFNN. In particular, we show that these error bounds depend on the approximation of the boundary function and the integral discretization scheme, both of which directly correspond to components of the Fredholm NN architecture. In this way, we construct an explainable scheme that provides accurate solutions to the inverse problems, whilst still explicitly respecting the boundary conditions, due to the architecture of the PFNN. We assess the performance of the proposed scheme for linear and semi-linear elliptic PDEs in two and three dimensions.

Applications of Fredholm integral equations (FIEs) include the solution of ordinary and partial differential equations (ODEs and PDEs) as well as the solution of inverse problems. Within the family of explainable machine learning techniques, we present Fredholm neural networks schemes for the solution of linear and nonlinear FIEs of the second kind. We first prove that Fredholm NNs provide accurate solutions. We then provide a rigorous approach for determining the values of the hyperparameters and trainable/explainable weights and biases of the DNN, by directly connecting their values to the underlying mathematical theory. We illustrate the applicability and performance of the proposed Fredholm NNs to the solution of forward and inverse linear and nonlinear problems, including elliptic PDEs and boundary value problems. We show that the proposed scheme achieves a significant numerical approximation accuracy across both the domain and the boundary. The proposed methodology provides insight into the connection between neural networks and classical numerical methods, and we posit that it can have applications in fields such as uncertainty quantification and explainable artificial intelligence. Thus, we believe that it will trigger further advances in the intersection and fruitful cross pollination between scientific machine learning and numerical analysis.

Deep neural networks have been extensively used for the solution of both the forward and the inverse problem for dynamical systems. However, their implementation necessitates optimizing a high-dimensional space of parameters and hyperparameters. This fact, along with the requirement of substantial computational resources, pose a barrier to achieving high numerical accuracy, but also interpretability. Here, to address the above challenges, we present Random Projection- based Operator Networks (RandONets): shallow networks with random projections and tailormade numerical analysis methods that learn accurately and fast linear and nonlinear operators. Building on previous works, we prove that RandOnets are universal approximators of linear and nonlinear operators. Due to their simplicity, RandONets provide a one-step transformation of the input space, facilitating interpretability. For the evaluation of their performance, we focus on operators of PDEs. We show, that RandONets outperform by several orders of magnitude, both in terms of numerical approximation accuracy and computational cost, the "vanilla" DeepONets. Hence, we believe that our method will trigger further developments in the field of scientific machine learning, for the development of new ''light"schemes that will provide high accuracy while reducing dramatically the computational cost. A MATLAB toolbox for RandONets, including demos, is available on GitHub at https://github.com/GianlucaFabiani/RandONets.

We introduce a spatial economic growth model where space is described as a network of interconnected geographic locations and we study a corresponding finite-dimensional optimal control problem on a graph with state constraints. Economic growth models on networks are motivated by the nature of spatial economic data, which naturally possess a graph-like structure: this fact makes these models well-suited for numerical implementation and calibration. The network setting is different from the one adopted in the related literature, where space is modeled as a subset of a Euclidean space, which gives rise to infinite dimensional optimal control problems. After introducing the model and the related control problem, we prove existence and uniqueness of an optimal control and a regularity result for the value function, which sets up the basis for a deeper study of the optimal strategies. Then, we focus on specific cases where it is possible to find, under suitable assumptions, an explicit solution of the control problem. Finally, we discuss the cases of networks of two and three geographic locations.

Increasing the generation of electric power from renewable energy sources (RESs) creates important challenges to transmission system operators (TSOs) for balancing the power system. To address these challenges, adequate system flexibility is required. In this context, TSOs carry out flexibility assessment studies to evaluate the flexibility level of the power system and ensure that a stable operation of the transmission system under high RESs integration can be achieved. These studies take into consideration numerous scenarios incorporating different assumptions for temperature, RESs penetration, load growth, and hydraulic conditions. Until now, flexibility studies usually solve the standard unit commitment problem and evaluate if the flexibility level is adequate. Although this approach provides quite accurate results, the computational requirements are significant, resulting in limiting the scenarios chosen for examination. In this paper, deep learning approaches are examined, and more precisely, an integrated system of two recurrent neural networks with long short-term memory cells is designed to carry out the flexibility assessment task, aiming at the reduction in the computational time required by the optimization process. The output of this neural network system is then used to calculate the probability of flexibility shortages. The proposed method is evaluated based on data from the Hellenic transmission system, providing quite promising results in (a) accurately calculating the probability of insufficient flexibility and (b) achieving a significant decrease in computational time. This novel approach could notably facilitate TSOs since more scenarios can be included, exploiting the computational efficiency of the method. In this way, a more complete evaluation of the flexibility level of the power system can be achieved and thus help to ensure the stable and reliable operation of the transmission system.

Forward-looking economic agents operating in a finite continuous geographic area choose how much to innovate at each point in time and space. Based on this assumption, the present study incorporates spatial interactions in endogenous growth models, addressing the criticism that such models are inconsistent with empirical evidence. More specifically, we introduce spatial production spillovers, knowledge diffusion across space, and the capability for spatial heterogeneity into a standard expanding variety growth model based on R&D. We study the properties of equilibrium and optimal allocations and argue that the characteristics are different from those of the non-spatial model, which alter the appropriate policy measures. Finally, we provide numerical examples demonstrating the importance of spatial dependent policy measures in achieving a balanced regional development.

In this paper we propose a consensus group decision making scheme under model uncertainty consisting of an iterative two-stage procedure based on the concept of Fréchet barycenter. Each stage consists of two steps: the agents first update their position in the opinion metric space adopting a local barycenter characterized by the agents’ immediate interactions and then a moderator makes a proposal in terms of a global barycenter, checking for consensus at each stage. In cases of large heterogeneous groups, the procedure can be complemented by an auxiliary initial homogenization stage, consisting of a clustering procedure in opinion space, leading to large homogeneous groups for which the aforementioned procedure will be applied. The scheme is illustrated in examples motivated from environmental economics.

Catastrophe (CAT) bonds necessitate a robust construction with regard to the estimated probability measure of their triggering parameter. This article concentrates on earthquakes as the primary natural catastrophe of concern. By leveraging the geometry of active faults for estimating default probability, we utilize seismic event information spanning up to 15,000 years in the past—thereby surpassing the restricted time range of available historical catalogs commonly used in other analyses, which typically cover only a few hundred years. This article introduces the design and pricing methodology of CAT bonds employing autoregressive neural networks, extending the standard VAR Nelson-Siegel model for yield curves. It presents a case study focused on the region of Greece, estimating that an additional spread of approximately 500 basis points over LIBOR constitutes the minimum premium required to attract an investor to undertake the associated risk. This premium could be absorbed by insured parties as an alternative to the conventional insurance process.

We present a linear stability analysis of physics-informed random projection neural networks (PI-RPNNs), for the numerical solution of the initial value problem (IVP) of (stiff) ODEs. We begin by proving that PI-RPNNs are uniform approximators of the solution to ODEs. We then provide a constructive proof demonstrating that PI-RPNNs offer consistent and asymptotically stable numerical schemes, thus convergent schemes. In particular, we prove that multi-collocation PI-RPNNs guarantee asymptotic stability. Our theoretical results are illustrated via numerical solutions of benchmark examples including indicative comparisons with the backward Euler method, the midpoint method, the trapezoidal rule, the 2-stage Gauss scheme, and the 2- and 3-stage Radau schemes.

This study presents a novel approach to assessing food security risks at the national level, employing a probabilistic scenario-based framework that integrates both Shared Socioeconomic Pathways (SSP) and Representative Concentration Pathways (RCP). This innovative method allows each scenario, encompassing socio-economic and climate factors, to be treated as a model capable of generating diverse trajectories. This approach offers a more dynamic understanding of food security risks under varying future conditions. The paper details the methodologies employed, showcasing their applicability through a focused analysis of food security challenges in Egypt and Ethiopia, and underscores the importance of considering a spectrum of socio-economic and climatic factors in national food security assessments.

This authoritative Encyclopedia provides an innovative approach to theory, reviews, applications and examples relevant to the basic concepts of water science and water management issues in order to facilitate better interdisciplinary cooperation.

In this paper, we study a stochastic parabolic problem involving a nonlocal diffusion operator associated with nonlocal Robin-type boundary conditions. The stochastic dynamics under consideration are driven by a mixture of a classical Brownian and a fractional Brownian motions with Hurst index $H\in(\frac{1}{2}, 1).$ We first establish local in existence result of the considered model and then explore conditions under which the resulting SPDE exhibits finite-time quenching. Using the probability distribution of perpetual integral functional of Brownian motion as well as tail estimates of fractional Brownian motion we provide analytic estimates for certain statistics of interest, such as quenching times and the corresponding quenching probabilities. The existence of global in time solutions is also investigated and as a consequence a lower estimate of the quenching time is also derived. Our analytical results demonstrate the non-trivial impact of the considered noise on the dynamics of the system. Next, a connection of a special case of the examined model is drawn in the context of MEMS technology. Finally, a numerical investigation of the considered model for a fractional Laplacian diffusion and Dirichlet-type boundary conditions is delivered.