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We consider the problem of modeling heterogeneous materials where microscale dynamics and interactions affect global behavior. In the presence of heterogeneities in material microstructure it is often impractical, if not impossible, to provide quantitative characterization of material response. The goal of this work is to develop a Bayesian framework for uncertainty quantification (UQ) in material response prediction when using nonlocal models. Our approach combines the nonlocal operator regression (NOR) technique and Bayesian inference. Specifically, additive independent identically distributed Gaussian noise is employed to model the discrepancy between the nonlocal model and the data. Then, we use a Markov chain Monte Carlo (MCMC) method to sample the posterior probability distribution on parameters involved in the nonlocal constitutive law and associated modeling discrepancies relative to higher-fidelity computations. As an application, we consider the propagation of stress waves through a one-dimensional heterogeneous bar with randomly generated microstructure. Several numerical tests illustrate the construction, enabling UQ in nonlocal model predictions. Although nonlocal models have become popular means for homogenization, their statistical calibration with respect to high-fidelity models has not been presented before. This work is a first step in this direction, focused on Bayesian parameter calibration. [ABSTRACT FROM AUTHOR]

We introduce a mathematically rigorous formulation for a nonlocal interface problem with jumps and propose an asymptotically compatible finite element discretization for the weak form of the interface problem. After proving the well-posedness of the weak form, we demonstrate that solutions to the nonlocal interface problem converge to the corresponding local counterpart when the nonlocal data are appropriately prescribed. Several numerical tests in one and two dimensions show the applicability of our technique, its numerical convergence to exact nonlocal solutions, its convergence to the local limit when the horizons vanish, and its robustness with respect to the patch test. [ABSTRACT FROM AUTHOR]

Nonlocal vector calculus, which is based on the nonlocal forms of gradient, divergence, and Laplace operators in multiple dimensions, has shown promising applications in fields such as hydrology, mechanics, and image processing. In this work, we study the analytical underpinnings of these operators. We rigorously treat compositions of nonlocal operators, prove nonlocal vector calculus identities, and connect weighted and unweighted variational frameworks. We combine these results to obtain a weighted fractional Helmholtz decomposition which is valid for sufficiently smooth vector fields. Our approach identifies the function spaces in which the stated identities and decompositions hold, providing a rigorous foundation to the nonlocal vector calculus identities that can serve as tools for nonlocal modeling in higher dimensions. [ABSTRACT FROM AUTHOR]

Fractional equations have become the model of choice in several applications where heterogeneities at the microstructure result in anomalous diffusive behavior at the macroscale. In this work we introduce a new fractional operator characterized by a doubly‐variable fractional order and possibly truncated interactions. Under certain conditions on the model parameters and on the regularity of the fractional order we show that the corresponding Poisson problem is well‐posed. We also introduce a finite element discretization and describe an efficient implementation of the finite‐element matrix assembly in the case of piecewise constant fractional order. Through several numerical tests, we illustrate the improved descriptive power of this new operator across media interfaces. Furthermore, we present one‐dimensional and two‐dimensional h‐convergence results that show that the variable‐order model has the same convergence behavior as the constant‐order model. [ABSTRACT FROM AUTHOR]

A mathematical framework is provided for a substructuring‐based domain decomposition (DD) approach for nonlocal problems that features interactions between points separated by a finite distance. Here, by substructuring it is meant that a traditional geometric configuration for local partial differential equation (PDE) problems is used in which a computational domain is subdivided into non‐overlapping subdomains. In the nonlocal setting, this approach is substructuring‐based in the sense that those subdomains interact with neighboring domains over interface regions having finite volume, in contrast to the local PDE setting in which interfaces are lower dimensional manifolds separating abutting subdomains. Key results include the equivalence between the global, single‐domain nonlocal problem and its multi‐domain reformulation, both at the continuous and discrete levels. These results provide the rigorous foundation necessary for the development of efficient solution strategies for nonlocal DD methods. [ABSTRACT FROM AUTHOR]

We present a novel definition of variable-order fractional Laplacian on R n \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document} based on a natural generalization of the standard Riesz potential. Our definition holds for values of the fractional parameter spanning the entire open set (0, n/2). We then discuss some properties of the fractional Poisson's equation involving this operator and we compute the corresponding Green's function, for which we provide some instructive examples for specific problems. [ABSTRACT FROM AUTHOR]

Nonlocal and fractional-order models capture effects that classical partial differential equations cannot describe; for this reason, they are suitable for a broad class of engineering and scientific applications that feature multiscale or anomalous behavior. This has driven a desire for a vector calculus that includes nonlocal and fractional gradient, divergence and Laplacian type operators, as well as tools such as Green's identities, to model subsurface transport, turbulence, and conservation laws. In the literature, several independent definitions and theories of nonlocal and fractional vector calculus have been put forward. Some have been studied rigorously and in depth, while others have been introduced ad-hoc for specific applications. The goal of this work is to provide foundations for a unified vector calculus by (1) consolidating fractional vector calculus as a special case of nonlocal vector calculus, (2) relating unweighted and weighted Laplacian operators by introducing an equivalence kernel, and (3) proving a form of Green's identity to unify the corresponding variational frameworks for the resulting nonlocal volume-constrained problems. The proposed framework goes beyond the analysis of nonlocal equations by supporting new model discovery, establishing theory and interpretation for a broad class of operators, and providing useful analogues of standard tools from the classical vector calculus. [ABSTRACT FROM AUTHOR]

The implementation of finite element methods (FEMs) for nonlocal models with a finite range of interaction poses challenges not faced in the partial differential equations (PDEs) setting. For example, one has to deal with weak forms involving double integrals which lead to discrete systems having higher assembly and solving costs due to possibly much lower sparsity compared to that of FEMs for PDEs. In addition, one may encounter nonsmooth integrands. In many nonlocal models, nonlocal interactions are limited to bounded neighborhoods that are ubiquitously chosen to be Euclidean balls, resulting in the challenge of dealing with intersections of such balls with the finite elements. We focus on developing recipes for the efficient assembly of FEM stiffness matrices and on the choice of quadrature rules for the double integrals that contribute to the assembly efficiency and also posses sufficient accuracy. A major feature of our recipes is the use of approximate balls, e.g. several polygonal approximations of Euclidean balls, that, among other advantages, mitigate the challenge of dealing with ball-element intersections. We provide numerical illustrations of the relative accuracy and efficiency of the several approaches we develop. [ABSTRACT FROM AUTHOR]

Nonlocal models provide exceptional simulation fidelity for a broad spectrum of scientific and engineering applications. However, wider deployment of nonlocal models is hindered by several modeling and numerical challenges. Among those, we focus on the nontrivial prescription of nonlocal boundary conditions, or volume constraints, that must be provided on a layer surrounding the domain where the nonlocal equations are posed. The challenge arises from the fact that, in general, data are provided on surfaces (as opposed to volumes) in the form of force or pressure data. In this paper we introduce an efficient, flexible, and physically consistent technique for an automatic conversion of surface (local) data into volumetric data that does not have any constraints on the geometry of the domain or on the regularity of the nonlocal solution and that is not tied to any discretization. We show that our formulation is well-posed and that the limit of the nonlocal solution, as the nonlocality vanishes, is the local solution corresponding to the available surface data. Quadratic convergence rates are proved for the strong energy and $L^2$ convergence. We illustrate the theory with one-dimensional numerical tests whose results provide the groundwork for realistic simulations. [ABSTRACT FROM AUTHOR]

We design and analyze solution techniques for a linear-quadratic optimal control problem involving the integral fractional Laplacian. We derive existence and uniqueness results, first-order optimality conditions, and regularity estimates for the optimal variables. We propose two strategies to discretize the fractional optimal control problem: a semidiscrete approach, where the control is not discretized---the so-called variational discretization approach; and a fully discrete approach, where the control variable is discretized with piecewise constant functions. Both schemes rely on the discretization of the state equation with the finite element space of continuous piecewise polynomials of degree one. We derive a priori error estimates for both solution techniques. We illustrate the theory with two-dimensional numerical tests. [ABSTRACT FROM AUTHOR]

Brood and social parasites exploit the parental care of their host and often integrate into host colonies using different strategies at different moments during their life cycle. In social wasps, the obligate social parasite Polistes atrimandibularis , using chemical mimicry and chemical insignificance, avoids detection by its host, Polistes biglumis , which has an efficient nestmate recognition system. Enslaved host foundresses accept and care for parasite brood as if it were their own. This observation raises the following question: how exactly do parasite brood escape detection and increase host acceptance? Here, we compared the cuticular chemical signatures of parasite brood and host brood. Their signatures differed in composition: parasite larvae and pupae had particularly high levels of alkenes (notably 9-nonacosene), which were absent in host larvae and pupae. Given this result, we tested the effect of 9-nonacosene on host foundresses, and documented that they reduced their aggression. Our findings address a novel integration strategy for social parasite brood as they suggest that the hydrocarbon serves as an appeasement substance and promotes host tolerance of parasite larvae and pupae, which are chemically disparate from host larvae and pupae. Highlights • We compared the cuticular signatures of parasite brood and host brood in social wasps. • Their signatures differed and parasite larvae and pupae had alkenes, absent in hosts. • Behavioural tests showed that one alkene diminishes the host aggressive response. • A hydrocarbon may promote host acceptance of parasite brood (appeasement). [ABSTRACT FROM AUTHOR]

Social insects recognize their nestmates by means of a cuticular hydrocarbon signature shared by colony members, but how nest signature changes across time has been rarely tested in longitudinal studies and in the field. In social wasps, the chemical signature is also deposited on the nest surface, where it is used by newly emerged wasps as a reference to learn their colony odor. Here, we investigate the temporal variations of the chemical signature that wasps have deposited on their nests. We followed the fate of the colonies of the social paper wasp Polistes biglumis in their natural environment from colony foundation to decline. Because some colonies were invaded by the social parasite Polistes atrimandibularis, we also tested the effects of social parasites on the nest signature. We observed that, as the season progresses, the nest signature changed; the overall abundance of hydrocarbons as well as the proportion of longer-chain and branched hydrocarbons increased. Where present, social parasites altered the host-nest signature qualitatively (adding parasite-specific alkenes) and quantitatively (by interfering with the increase in overall hydrocarbon abundance). Our results show that 1) colony odor is highly dynamic both in colonies controlled by legitimate foundresses and in those controlled by social parasites; 2) emerged offspring contribute little to colony signature, if at all, in comparison to foundresses; and 3) social parasites, that later mimic host signature, initially mark host nests with species-specific hydrocarbons. This study implies that important updating of the neural template used in nestmate recognition should occur in social insects. [ABSTRACT FROM AUTHOR]

Nonlocal models allow for the description of phenomena which cannot be captured by classical partial differential equations. The availability of efficient solvers is one of the main concerns for the use of nonlocal models in real world engineering applications. We present a domain decomposition solver that is inspired by substructuring methods for classical local equations. In numerical experiments involving finite element discretizations of scalar and vectorial nonlocal equations of integrable and fractional type, we observe improvements in solution time of up to 14.6x compared to commonly used solver strategies. [ABSTRACT FROM AUTHOR]

The presence and evolution of defects that appear in the manufacturing process play a vital role in the failure mechanisms of engineering materials. In particular, the collective behavior of dislocation dynamics at the mesoscale leads to avalanche, strain bursts, intermittent energy spikes, and nonlocal interactions producing anomalous features across different time- and length-scales, directly affecting plasticity, void and crack nucleation. Discrete Dislocation Dynamics (DDD) simulations are often used at the meso-level, but the cost and complexity increase dramatically with simulation time. To further understand how the anomalous features propagate to the continuum, we develop a probabilistic model for dislocation motion constructed from the position statistics obtained from DDD simulations. We obtain the continuous limit of discrete dislocation dynamics through a Probability Density Function for the dislocation motion, and propose a nonlocal transport model for the PDF. We develop a machine-learning framework to learn the parameters of the nonlocal operator with a power-law kernel, connecting the anomalous nature of DDD to the origin of its corresponding nonlocal operator at the continuum, facilitating the integration of dislocation dynamics into multi-scale, long-time material failure simulations. [ABSTRACT FROM AUTHOR]

Sound scattering layers (SSLs) are important components of oceanic ecosystems with ubiquitous distribution throughout the world's oceans. This vertical movement is an important mechanism for exchanging organic matter from the surface to the deep ocean, as many of the organisms comprising SSLs serve as prey resources for linking the lower trophic levels to larger predators. Variations in abundance and taxonomic composition of mesopelagic organisms were quantified using repeated discrete net sampling and acoustics over a 30-h survey, performed during 26–27 June 2011 at single site (27°28’51”N and 88°27’54”W) in the northern Gulf of Mexico. We acoustically classified the mesopelagic SSL into four broad taxonomic categories, crustacean and small non-swimbladdered fish (CSNSBF), large non-swimbladdered fish (LNSBF), swimbladdered fish (SBF) and unclassified and we quantified the abundance of mesopelagic organisms over three discrete depth intervals; epipelagic (0–200 m); upper mesopelagic (200–600 m) and lower mesopelagic (600–1000 m). Irrespective of the acoustic categories at dusk part of the acoustic energy redistributed from the mesopelagic into the upper epipelagic (shallower than 100 m) remaining however below the thermocline depth. At night higher variability in species composition was observed between 100 and 200 m suggested that a redistribution of organisms may also occur within the upper portion of the water column. Along the upper mesopelagic backscatter spectra from CSNSBF migrated between 400 and 460 m while spectra from the other categories moved to shallower depths (300 and 350 m), resulting in habitat separation from CSNSBF. Relatively small vertical changes in both acoustic backscatter and center of mass metrics of the deep mesopelagic were observed for CNSBF and LNSBF suggesting that these animals may be tightly connected to deeper (below 1000 m) mesopelagic habitats, and do not routinely migrate into the epipelagic. [ABSTRACT FROM AUTHOR]

We develop and analyze an optimization-based method for the coupling of nonlocal and local diffusion problems with mixed volume constraints and boundary conditions. The approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the nonlocal and local domains, and the controls are virtual volume constraints and boundary conditions. When some assumptions on the kernel functions hold, we prove that the resulting optimization problem is well-posed and discuss its implementation using Sandia’s agile software components toolkit. The latter provides the groundwork for the development of engineering analysis tools, while numerical results for nonlocal diffusion in three-dimensions illustrate key properties of the optimization-based coupling method. [ABSTRACT FROM AUTHOR]

We consider the problem of finding a mesh such that 1) it is the closest, with respect to a suitable metric, to a given source mesh having the same connectivity, and 2) the volumes of its cells match a set of prescribed positive values that are not necessarily equal to the cell volumes in the source mesh. This volume correction problem arises in important simulation contexts, such as satisfying a discrete geometric conservation law and solving transport equations by incremental remapping or similar semi-Lagrangian transport schemes. In this paper we formulate volume correction as a constrained optimization problem in which the distance to the source mesh defines an optimization objective, while the prescribed cell volumes, mesh validity and/or cell convexity specify the constraints. We solve this problem numerically using a sequential quadratic programming (SQP) method whose performance scales with the mesh size. To achieve scalable performance we develop a specialized multigrid-based preconditioner for optimality systems that arise in the application of the SQP method to the volume correction problem. Numerical examples illustrate the importance of volume correction, and showcase the accuracy, robustness and scalability of our approach. [ABSTRACT FROM AUTHOR]

A control problem constrained by a nonlocal steady diffusion equation that arises in several applications is studied. The control is the right-hand side forcing function and the objective of control is a standard matching functional. A recently developed nonlocal vector calculus is exploited to define a weak formulation of the state system. When sufficient conditions on certain kernel functions and the volume constraints hold, the existence and uniqueness of the optimal state and control is demonstrated and an optimality system is derived. We demonstrate the convergence, as the nonlocal interactions vanish, of the optimal nonlocal state to the optimal state of a local PDEconstrained control problem. We also define continuous and discontinuous Galerkin finite element discretizations of the optimality system for which we derive a priori error estimates. Numerical examples are provided illustrating these convergence results and also illustrating the differences between optimal controls and states obtained for the nonlocal diffusion equations and for PDEs. In particular, we observe that using nonlocal models result in a better match to nonsmooth target functions and a reduced cost of control. We also provide a one-dimensional numerical example of nonlocal heat conduction in a bar for which we determine the optimal heat source that results in an optimal match to a reference temperature distribution. [ABSTRACT FROM AUTHOR]

Abstract: We analyze a nonlocal diffusion operator having as special cases the fractional Laplacian and fractional differential operators that arise in several applications. In our analysis, a nonlocal vector calculus is exploited to define a weak formulation of the nonlocal problem. We demonstrate that, when sufficient conditions on certain kernel functions hold, the solution of the nonlocal equation converges to the solution of the fractional Laplacian equation on bounded domains as the nonlocal interactions become infinite. We also introduce a continuous Galerkin finite element discretization of the nonlocal weak formulation and we derive a priori error estimates. Through several numerical examples we illustrate the theoretical results and we show that by solving the nonlocal problem it is possible to obtain accurate approximations of the solutions of fractional differential equations circumventing the problem of treating infinite-volume constraints. [Copyright &y& Elsevier]

Hydroacoustic data collected during two echosurveys carried out in the Sicily Channel in 1998 and 2002 were analysed to investigate the distribution and spatial structure of small pelagic fish species in relation to the sedimentological nature of the sea bottom. The study was carried out on two contiguous areas (labelled ZONE 1 and ZONE 2) of the continental shelf off the southern coast of Sicily, characterised by different dominant texture, ‘sand’ for ZONE 1 and ‘clayey-silt’ for ZONE 2. Simultaneous information on small pelagic fish schools and the seabed was obtained using a quantitative echo-sounder (SIMRAD EK500) that measures echoes due to the scattering from both fish schools and the bottom surface. Acoustically determined fish school and seabed data were integrated, respectively, with information on species composition obtained by experimental fishing hauls, and with granulometric information obtained from the analysis of in situ sediment samples. The results indicate a general preference of small pelagic fish schools for seabeds of finer granulometry. First, the occurrence of fish schools was higher over the acoustically classified ‘soft’ seabeds of ZONE 2. Secondly, although ZONE 2 represents <30% of the total length of daytime acoustic tracks analysed in this study, in both surveys the bulk of fish biomass (>60%) was concentrated over ‘soft’ seabed substrates of ZONE 2. Different species composition and/or behaviour of fish schools in the two areas investigated were postulated in relation to seabed conditions. Specifically, over the hard and soft bottoms of ZONE 2, fish schools were found at lower depths and at shallower bottom depths compared to ZONE 1. Furthermore, over the softer bottoms of ZONE 2, fish schools exhibiting a more ‘pelagic’ behaviour ( i.e. detected at a greater distance from the bottom) showed a preference for softer (and finer) seabed substrate conditions. Conversely, fish schools exhibiting a more ‘demersal’ behaviour ( i.e. at a smaller distance from the bottom) were mostly found on relatively harder substrates. [ABSTRACT FROM AUTHOR]

As a nonlocal extension of continuum mechanics, peridynamics has been widely and effectively applied in different fields where discontinuities in the field variables arise from an initially continuous body. An important component of the constitutive model in peridynamics is the influence function which weights the contribution of all the interactions over a nonlocal region surrounding a point of interest. Recent work has shown that in solid mechanics the influence function has a strong relationship with the heterogeneity of a material's micro-structure. However, determining an accurate influence function analytically from a given micro-structure typically requires lengthy derivations and complex mathematical models. To avoid these complexities, the goal of this paper is to develop a data-driven regression algorithm to find the optimal bond-based peridynamic model to describe the macro-scale deformation of linear elastic medium with periodic heterogeneity. We generate macro-scale deformation training data by averaging over periodic micro-structure unit cells and add a physical energy constraint representing the homogenized elastic modulus of the micro-structure to the regression algorithm. We demonstrate this scheme for examples of one- and two-dimensional linear elastodynamics and show that the energy constraint improves the accuracy of the resulting peridynamic model. • Machine learning is used to find the peridynamic influence function from synthetic high-fidelity data. • Macro-scale deformation of a material with micro-scale heterogeneity can be accurately predicted by the learned model. • The use of a physical constraint imposed on the objective function reduces the amount of training data needed. [ABSTRACT FROM AUTHOR]

Nonlocal operators of fractional type are a popular modeling choice for applications that do not adhere to classical diffusive behavior; however, one major challenge in nonlocal simulations is the selection of model parameters. In this work we propose an optimization-based approach to parameter identification for fractional models with an optional truncation radius. We formulate the inference problem as an optimal control problem where the objective is to minimize the discrepancy between observed data and an approximate solution of the model, and the control variables are the fractional order and the truncation length. For the numerical solution of the minimization problem we propose a gradient-based approach, where we enhance the numerical performance by an approximation of the bilinear form of the state equation and its derivative with respect to the fractional order. Several numerical tests in one and two dimensions illustrate the theoretical results and show the robustness and applicability of our method. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study a Helmholtz problem with a spectral fractional Laplacian, instead of the standard Laplacian. Recently, it has been established that such a fractional Helmholtz problem better captures the underlying behavior in geophysical electromagnetics. We establish the well-posedness and regularity of this problem. We introduce a hybrid spectral-finite element approach to discretize it and show well-posedness of the discrete system. In addition, we derive a priori discretization error estimates. Finally, we introduce an efficient solver that scales as well as the best possible solver for the classical integer-order Helmholtz equation. We conclude with several illustrative examples that confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

Characterizing and quantifying microstructure evolution is critical to forming quantitative relationships between material processing conditions, resulting microstructure, and observed properties. Machine-learning methods are increasingly accelerating the development of these relationships by treating microstructure evolution as a pattern recognition problem, discovering relationships explicitly or implicitly. These methods often rely on identifying low-dimensional microstructural fingerprints as latent variables. However, using inappropriate latent variables can lead to challenges in learning meaningful relationships. In this work, we survey and discuss the ability of various linear and nonlinear dimensionality reduction methods including principal component analysis, autoencoders, and diffusion maps to quantify and characterize the learned latent space microstructural representations and their time evolution. We characterize latent spaces by their ability to represent high-dimensional microstructural data in terms of compression achieved as a function of the number of latent dimensions required to represent the data accurately, their accuracy based on their reconstruction performance, and the smoothness of the microstructural trajectories in latent dimension. We quantify these metrics for common microstructure evolution problems in material science including spinodal decomposition of a binary metallic alloy, thin film deposition of a binary metallic alloy, dendritic growth, and grain growth in a polycrystal. This study provides considerations and guidelines for choosing dimensionality reduction methods when considering materials problems that involve high dimensional data and a variety of features over a range of lengths and time scales. [Display omitted] [ABSTRACT FROM AUTHOR]

Physics-based deep learning frameworks have shown to be effective in accurately modeling the dynamics of complex physical systems with generalization capability across problem inputs. However, time-independent problems pose the challenge of requiring long-range exchange of information across the computational domain for obtaining accurate predictions. In the context of graph neural networks (GNNs), this calls for deeper networks, which, in turn, may compromise or slow down the training process. In this work, we present two GNN architectures to overcome this challenge—the edge augmented GNN and the multi-GNN. We show that both these networks perform significantly better than baseline methods, such as MeshGraphNets, when applied to time-independent solid mechanics problems. Furthermore, the proposed architectures generalize well to unseen domains, boundary conditions, and materials. Here, the treatment of variable domains is facilitated by a novel coordinate transformation that enables rotation and translation invariance. By broadening the range of problems that neural operators based on graph neural networks can tackle, this paper provides the groundwork for their application to complex scientific and industrial settings. [ABSTRACT FROM AUTHOR]

Numerical simulations are used to study the dynamics of a developing suspension Poiseuille flow with monodispersed and bidispersed neutrally buoyant particles in a planar channel, and machine learning is applied to learn the evolving stresses of the developing suspension. The particle stresses and pressure develop on a slower time scale than the volume fraction, indicating that once the particles reach a steady volume fraction profile, they rearrange to minimize the contact pressure on each particle. We consider the timescale for stress development and how the stress development connects to particle migration. For developing monodisperse suspensions, we present a new physics-informed Galerkin neural network that allows for learning the particle stresses when direct measurements are not possible. We show that when a training set of stress measurements is available, the MOR-physics operator learning method can also capture the particle stresses accurately. [ABSTRACT FROM AUTHOR]

Neural operators [1–5] have recently become popular tools for designing solution maps between function spaces in the form of neural networks. Differently from classical scientific machine learning approaches that learn parameters of a known partial differential equation (PDE) for a single instance of the input parameters at a fixed resolution, neural operators approximate the solution map of a family of PDEs [6,7]. Despite their success, the uses of neural operators are so far restricted to relatively shallow neural networks and confined to learning hidden governing laws. In this work, we propose a novel nonlocal neural operator, which we refer to as nonlocal kernel network (NKN), that is resolution independent, characterized by deep neural networks, and capable of handling a variety of tasks such as learning governing equations and classifying images. Our NKN stems from the interpretation of the neural network as a discrete nonlocal diffusion reaction equation that, in the limit of infinite layers, is equivalent to a parabolic nonlocal equation, whose stability is analyzed via nonlocal vector calculus. The resemblance with integral forms of neural operators allows NKNs to capture long-range dependencies in the feature space, while the continuous treatment of node-to-node interactions makes NKNs resolution independent. The resemblance with neural ODEs, reinterpreted in a nonlocal sense, and the stable network dynamics between layers allow for generalization of NKN's optimal parameters from shallow to deep networks. This fact enables the use of shallow-to-deep initialization techniques [8]. Our tests show that NKNs outperform baseline methods in both learning governing equations and image classification tasks and generalize well to different resolutions and depths. • We propose a novel nonlocal neural operator, which we refer to as nonlocal kernel network (NKN). • NKN is resolution independent and characterized by deep neural networks. • In the limit of infinite layers, NKN is equivalent to a parabolic nonlocal equation with its stability theoretically proved. • Tests show that NKNs outperform baseline methods in various tasks and generalize well to different resolutions and depths. [ABSTRACT FROM AUTHOR]

In this paper, we design efficient quadrature rules for finite element (FE) discretizations of nonlocal diffusion problems with compactly supported kernel functions. Two of the main challenges in nonlocal modeling and simulations are the prohibitive computational cost and the nontrivial implementation of discretization schemes, especially in three‐dimensional settings. In this work, we circumvent both challenges by introducing a parametrized mollifying function that improves the regularity of the integrand, utilizing an adaptive integration technique, and exploiting parallelization. We first show that the "mollified" solution converges to the exact one as the mollifying parameter vanishes, then we illustrate the consistency and accuracy of the proposed method on several two‐ and three‐dimensional test cases. Furthermore, we demonstrate the good scaling properties of the parallel implementation of the adaptive algorithm and we compare the proposed method with recently developed techniques for efficient FE assembly. [ABSTRACT FROM AUTHOR]

Casting nonlocal problems in variational form and discretizing them with the finite element (FE) method facilitates the use of nonlocal vector calculus to prove well-posedness, convergence, and stability of such schemes. Employing an FE method also facilitates meshing of complicated domain geometries and coupling with FE methods for local problems. However, nonlocal weak problems involve the computation of a double-integral, which is computationally expensive and presents several challenges. In particular, the inner integral of the variational form associated with the stiffness matrix is defined over the intersections of FE mesh elements with a ball of radius δ , where δ is the range of nonlocal interaction. Identifying and parameterizing these intersections is a nontrivial computational geometry problem. In this work, we propose a quadrature technique where the inner integration is performed using quadrature points distributed over the full ball, without regard for how it intersects elements, and weights are computed based on the generalized moving least squares method. Thus, as opposed to all previously employed methods, our technique does not require element-by-element integration and fully circumvents the computation of element–ball intersections. This paper considers one- and two-dimensional implementations of piecewise linear continuous FE approximations, focusing on the case where the element size h and the nonlocal radius δ are proportional, as is typical of practical computations. When boundary conditions are treated carefully and the outer integral of the variational form is computed accurately, the proposed method is asymptotically compatible in the limit of h ∼ δ → 0 , featuring at least first-order convergence in L 2 for all dimensions, using both uniform and nonuniform grids. Moreover, in the case of uniform grids, the proposed method passes a patch test and, according to numerical evidence, exhibits an optimal, second-order convergence rate. Our numerical tests also indicate that, even for nonuniform grids, second-order convergence can be observed over a substantial pre-asymptotic regime. • A GMLS-based quadrature is proposed for finite element discrete nonlocal problems. • Our procedure does not require the computation of element-ball intersections. • The quadrature approach is numerically shown to be asymptotically compatible. • A preliminary numerical analysis of the convergence of the method is derived. [ABSTRACT FROM AUTHOR]

The identification and classification of macroscopic maturity stages plays a key role in the assessment of small pelagic fishery resources. The main scientific international commissions strongly recommend standardizing methodologies across countries and scientists. Unfortunately, there is still a great deal of uncertainty concerning macroscopic identification, which remains to be validated. The current paper analyses reproductive data of European anchovy ( Engraulis encrasicolus L. 1758), collected during three summer surveys (2001, 2005 and 2006) in the Strait of Sicily, to evaluate the uncertainty in the macroscopic maturity stage identification and the reliability of the macroscopic adopted scale. On board the survey vessels, the maturity stage of each fish was determined macroscopically by means of an adopted maturity scale subdivided in six stages. Later, at the laboratory, the gonads were prepared for histological examination. The histological slides were analysed, finally assigning the six maturity stages for macroscopic examinations. A correspondence table was obtained with the proportion and number of matches between the two methods. The results highlight critical aspects in the ascription of macroscopic maturity stages, particularly for the present research aim. Different recommendations were evaluated depending on the scope of the study conducted on maturity ( e.g. daily egg production, fecundity and maturity ogive computation). The most interesting results concern the misclassification of stage IV and stages III and V (the most abundant), which confirms their macroscopic similarity. Although the results are based on a small number of samples, the advantages and disadvantages of macroscopic and histological methods are discussed with the aim to increase the accuracy of correct identification and to standardize macroscopic maturity ascription criteria. [ABSTRACT FROM AUTHOR]

We propose a domain decomposition method for the efficient simulation of nonlocal problems. Our approach is based on a multi-domain formulation of a nonlocal diffusion problem where the subdomains share "nonlocal" interfaces of the size of the nonlocal horizon. This system of nonlocal equations is first rewritten in terms of minimization of a nonlocal energy, then discretized with a meshfree approximation and finally solved via a Lagrange multiplier approach in a way that resembles the finite element tearing and interconnect method. Specifically, we propose a distributed projected gradient algorithm for the solution of the Lagrange multiplier system, whose unknowns determine the nonlocal interface conditions between subdomains. Several two-dimensional numerical tests on problems as large as 191 million unknowns illustrate the strong and the weak scalability of our algorithm, which outperforms the standard approach to the distributed numerical solution of the problem. This work is the first rigorous numerical study in a two-dimensional multi-domain setting for nonlocal operators with finite horizon and, as such, it is a fundamental step towards increasing the use of nonlocal models in large scale simulations. • First rigorous numerical study in a multi-domain setting for nonlocal operators with finite horizon. • Numerical tests show that the proposed approach outperforms commonly used parallel solvers. • The parallel iterative solver exhibits excellent weak and strong scalability properties. [ABSTRACT FROM AUTHOR]

We develop a thermodynamically consistent, fractional visco-elasto-plastic model coupled with damage for anomalous materials. The model utilizes Scott-Blair rheological elements for both visco- elastic/plastic parts. The constitutive equations are obtained through Helmholtz free-energy potentials for Scott-Blair elements, together with a memory-dependent fractional yield function and dissipation inequalities. A memory-dependent Lemaitre-type damage is introduced through fractional damage energy release rates. For time-fractional integration of the resulting nonlinear system of equations, we develop a first-order semi-implicit fractional return-mapping algorithm. We also develop a finite-difference discretization for the fractional damage energy release rate, which results into Hankel-type matrix–vector operations for each time-step, allowing us to reduce the computational complexity from O (N 3) to O (N 2) through the use of Fast Fourier Transforms. Our numerical results demonstrate that the fractional orders for visco-elasto-plasticity play a crucial role in damage evolution, due to the competition between the anomalous plastic slip and bulk damage energy release rates. • Accurate and predictive modeling of material damage and failure for a wide range of materials poses multi-disciplinary challenges on experimental detection, consistent physics-informed models and efficient algorithms. We develop a thermodynamically consistent time-fractional visco-elasto-plastic model, coupled with damage for anomalous materials. • This model plays a crucial role in damage evolution, due to the competition between the 'anomalous plastic slip' and the 'bulk damage energy release rates'. In addition, the constitutive equations are obtained through Helmholtz free-energy potentials for ScottBlair elements, together with a memory-dependent fractional yield function and dissipation inequalities. • Our novel model can address 'softening', 'hysteresis' and 'low-cycle fatigue'. Moreover, the memory-dependent damage energy release rates induce anomalous damage evolutions with competing visco- elastic/plastic effects, without changing the form of Lemaitre's damage potential. • On the numerical scheme development side, we develop a full first-order semi-implicit fractional return-mapping algorithm (for the corresponding nonlinear system), along with a finite-difference discretization for the fractional damage energy release rate. The proposed discretization results into Hankel-type matrix–vector operations for each timestep, allowing us to reduce the computational complexity from O (N 3) to O (N 2) through the use of Fast Fourier Transforms. [ABSTRACT FROM AUTHOR]

This paper continues our efforts to exploit optimization and control ideas as a common foundation for the development of property-preserving numerical methods. Here we focus on a class of scalar advection equations whose solutions have fixed mass in a given Eulerian region and constant bounds in any Lagrangian volume. Our approach separates discretization of the equations from the preservation of their solution properties by treating the latter as optimization constraints. This relieves the discretization process from having to comply with additional restrictions and makes stability and accuracy the sole considerations in its design. A property-preserving solution is then sought as a state that minimizes the distance to an optimally accurate but not property-preserving target solution computed by the scheme, subject to constraints enforcing discrete proxies of the desired properties. We consider two such formulations in which the optimization variables are given by the nodal solution values and suitably defined nodal fluxes, respectively. A key result of the paper reveals that a standard Algebraic Flux Correction (AFC) scheme is a modified version of the second formulation obtained by shrinking its feasible set to a hypercube. We conclude with numerical studies illustrating the optimization-based formulations and comparing them with AFC. • We develop 2 classes of optimization-based property-preserving finite element methods. • We prove existence of optimal solutions for specific instances of these methods. • We show that Algebraic Flux Correction (AFC) is related to one of these classes. • AFC results from shrinking the feasible set of the optimization problem to hypercube. • This is the first result proving equivalence between AFC and global optimization. [ABSTRACT FROM AUTHOR]

Nonlocal models, including peridynamics, often use integral operators that embed lengthscales in their definition. However, the integrands in these operators are difficult to define from the data that are typically available for a given physical system, such as laboratory mechanical property tests. In contrast, molecular dynamics (MD) does not require these integrands, but it suffers from computational limitations in the length and time scales it can address. To combine the strengths of both methods and to obtain a coarse-grained, homogenized continuum model that efficiently and accurately captures materials' behavior, we propose a learning framework to extract, from MD data, an optimal Linear Peridynamic Solid (LPS) model as a surrogate for MD displacements. To maximize the accuracy of the learnt model we allow the peridynamic influence function to be partially negative, while preserving the well-posedness of the resulting model. To achieve this, we provide sufficient well-posedness conditions for discretized LPS models with sign-changing influence functions and develop a constrained optimization algorithm that minimizes the equation residual while enforcing such solvability conditions. This framework guarantees that the resulting model is mathematically well-posed, physically consistent, and that it generalizes well to settings that are different from the ones used during training. We illustrate the efficacy of the proposed approach with several numerical tests for single layer graphene. Our two-dimensional tests show the robustness of the proposed algorithm on validation data sets that include thermal noise, different domain shapes and external loadings, and discretizations substantially different from the ones used for training. • New well-posedness conditions for the discretized peridynamic model are provided. • Conditions are embedded in an optimization algorithm guaranteeing model solvability. • Experiments on single-layered graphene illustrate the efficacy of the method. • Robustness with respect to noise are demonstrated. • Generalization to different domain shapes and external loadings are illustrated. [ABSTRACT FROM AUTHOR]

A key challenge to nonlocal models is the analytical complexity of deriving them from first principles, and frequently their use is justified a posteriori. In this work we extract nonlocal models from data, circumventing these challenges and providing data-driven justification for the resulting model form. Extracting data-driven surrogates is a major challenge for machine learning (ML) approaches, due to nonlinearities and lack of convexity — it is particularly challenging to extract surrogates which are provably well-posed and numerically stable. Our scheme not only yields a convex optimization problem, but also allows extraction of nonlocal models whose kernels may be partially negative while maintaining well-posedness even in small-data regimes. To achieve this, based on established nonlocal theory, we embed in our algorithm sufficient conditions on the non-positive part of the kernel that guarantee well-posedness of the learnt operator. These conditions are imposed as inequality constraints to meet the requisite conditions of the nonlocal theory. We demonstrate this workflow for a range of applications, including reproduction of manufactured nonlocal kernels; numerical homogenization of Darcy flow associated with a heterogeneous periodic microstructure; nonlocal approximation to high-order local transport phenomena; and approximation of globally supported fractional diffusion operators by truncated kernels. • Derivation of well-posed sign-changing nonlocal kernels is challenging • Novel optimization framework guarantees extraction of well-posed kernels from data • Main contribution generalizes Mengesha and Du's analysis guaranteeing well-posedness • Applications to: homogenization, high-order effects, and fractional diffusion [ABSTRACT FROM AUTHOR]