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In this paper, we propose a deep learning framework for solving high-dimensional partial integro-differential equations (PIDEs) based on the temporal difference learning. We introduce a set of Lévy processes and construct a corresponding reinforcement learning model. To simulate the entire process, we use deep neural networks to represent the solutions and nonlocal terms of the equations. Subsequently, we train the networks using the temporal difference error, the termination condition, and properties of the nonlocal terms as the loss function. The relative error of the method reaches O (10- 3) in 100-dimensional experiments and O (10- 4) in one-dimensional pure jump problems. Additionally, our method demonstrates the advantages of low computational cost and robustness, making it well-suited for addressing problems with different forms and intensities of jumps. [ABSTRACT FROM AUTHOR]

The Gray--Scott model is a set of reaction-diffusion equations that describes chemical systems far from equilibrium. Interest in this model stems from its ability to generate spatio-temporal structures, including pulses, spots, stripes, and self-replicating patterns. We consider an extension of this model in which the spread of the different chemicals is assumed to be nonlocal and can thus be represented by an integral operator. In particular, we focus on the case of strictly positive, symmetric, L¹ convolution kernels that have a finite second moment. Modeling the equations on a finite interval, we prove the existence of small-time weak solutions in the case of nonlocal Dirichlet and Neumann boundary constraints. We then use this result to develop a finite element numerical scheme that helps us explore the effects of nonlocal diffusion on the formation of pulse solutions. [ABSTRACT FROM AUTHOR]

Spatial systems with heterogeneities are ubiquitous in nature, from precipitation, temperature, and soil gradients controlling vegetation growth to morphogen gradients controlling gene expression in embryos. Such systems, generally described by nonlinear dynamical systems, often display complex parameter dependence and exhibit bifurcations. The dynamics of heterogeneous spatially extended systems passing through bifurcations are still relatively poorly understood, yet recent theoretical studies and experimental data highlight the resulting complex behaviors and their relevance to real-world applications. We explore the consequences of spatial heterogeneities passing through bifurcations via two examples strongly motivated by applications. These model systems illustrate that studying heterogeneity-induced behaviors in spatial systems is crucial for a better understanding of ecological transitions and functional organization in brain development. [ABSTRACT FROM AUTHOR]

Steady gravity-capillary periodic waves on the surface of a thin viscous liquid film supported by an air stream on an inclined wall are investigated. Based on lubrication approximation and thin air-foil theory, this problem is reduced to an integro-differential equation. The small-amplitude analysis is carried out to obtain two analytical solutions up to the second order. Numerical computation shows there exist two distinct primary bifurcation branches starting from infinitesimal waves, which approach solitary wave configuration in the long-wave limit when the values of physical parameters are above certain thresholds. New families of solutions manifest themselves either as secondary bifurcation occurring on primary branches or as isolated solution branches. The limiting configurations of the primary solution branches with the increase of two parameters are studied in two different cases, where one and two limiting configurations are obtained, respectively. For the latter case, the approximation of the configurations is given. [ABSTRACT FROM AUTHOR]

In this manuscript, an original numerical procedure for the nonlinear peridynamics on arbitrarily shaped two-dimensional (2D) closed manifolds is proposed. When dealing with non-parameterized 2D manifolds at the discrete scale, the problem of computing geodesic distances between two non-adjacent points arise. Here, a routing procedure is implemented for computing geodesic distances by reinterpreting the triangular computational mesh as a non-oriented graph, thus returning a suitable and general method. Moreover, the time integration of the peridynamics equation is demanded to a P-(EC)k formulation of the implicit\beta-Newmark scheme. The convergence of the overall proposed procedure is questioned and rigorously proved. Its abilities and limitations are analyzed by simulating the evolution of a 2D sphere. The performed numerical investigations are mainly motivated by the issues related to the insurgence of singularities in the evolution problem. The obtained results return an interesting picture of the role played by the nonlocal character of the integrodifferential equation in the intricate processes leading to the spontaneous formation of singularities in real materials. [ABSTRACT FROM AUTHOR]

A unified discrete gradient structure of the second order nonuniform integral averaged approximations for the Caputo fractional derivative and the Riemann--Liouville fractional integral is established in this paper. The required constraint of the step-size ratio is weaker than that found in the literature. With the proposed discrete gradient structure, the energy stability of the variable step Crank--Nicolson type numerical schemes is derived immediately, which is essential to the longtime simulations of the time fractional gradient flows and the nonlinear integro-differential models. The discrete energy dissipation laws fit seamlessly into their classical counterparts as the fractional indexes tend to one. In particular, we provide a framework for the stability analysis of variable step numerical schemes based on the scalar auxiliary variable type approaches. The time fractional Swift--Hohenberg model and the time fractional sine-Gordon model are taken as two examples to elucidate the theoretical results at great length. Extensive numerical experiments using the adaptive time-stepping strategy are provided to verify the theoretical results in the time multiscale simulations. [ABSTRACT FROM AUTHOR]

Plasticity of metals is the emergent phenomenon of many crystal defects (dislocations) which interact and move on microscopic time and length scales. Two of the commonly used models to describe such dislocation dynamics are the Peierls-Nabarro model and the so-called discrete dislocation dynamics model. However, the consistency between these two models is known only for a few number of dislocations or up to the first time at which two dislocations collide. In this paper we resolve these restrictions, and establish the consistency for any number of dislocations and without any restriction on their initial position or orientation. In more detail, the evolutive Peierls-Nabarro model which we consider describes the evolution of a phase-field function vε(t, x) which represents the atom deformation in a crystal. The model is a reaction-diffusion equation of Allen-Cahn type with the half Laplacian. The small parameter ε is the ratio between the atomic distance and the typical distance between phase transitions in vε. The position of a phase transition determines the position of a dislocation, and the sign of the transition (up or down) determines the orientation. The goal of this paper is to derive the asymptotic behavior of the function vε as ε → 0 up to arbitrary end time T; in particular beyond collisions. We prove that vε converges to a piecewise constant function v, whose jump points in the spatial variable satisfy the ODE system which represents discrete dislocation dynamics with annihilation. Our proof method is to explicitly construct and patch together several sub- and supersolutions of vε, and to show that they converge to the same limit v. [ABSTRACT FROM AUTHOR]

This work presents a probabilistic scheme for solving semilinear nonlocal diffusion equations with volume constraints and integrable kernels. The nonlocal model of interest is defined by a time-dependent semilinear partial integro-differential equation (PIDE), in which the integrodifferential operator consists of both local convection-diffusion and nonlocal diffusion operators. Our numerical scheme is based on the direct approximation of the nonlinear Feynman-Kac formula that establishes a link between nonlinear PIDEs and stochastic differential equations. The exploitation of the Feynman-Kac representation avoids solving dense linear systems arising from nonlocal operators. Compared with existing stochastic approaches, our method can achieve first-order convergence after balancing the temporal and spatial discretization errors, which is a significant improvement of existing probabilistic/stochastic methods for nonlocal diffusion problems. Error analysis of our numerical scheme is established. The effectiveness of our approach is shown in two numerical examples. The first example considers a three-dimensional nonlocal diffusion equation to numerically verify the error analysis results. The second example presents a physics problem motivated by the study of heat transport in magnetically confined fusion plasmas. [ABSTRACT FROM AUTHOR]

In this article we consider a class of nonlinear integro-differential equations of the form infτ Є T { Є Rd (u(x + y) + u(x-y)-2u(x)) kτ (x,y)/| y| d+2s dy + bτ (x). u(x) + gτ (x)}-λ* = 0 in Rd, where 0 < λ (2-2s)≤ kτ (2-2s), s Є (1/2, 1). The above equation appears in the study of ergodic control problems in Rd when the controlled dynamics is governed by pure-jump Lévy processes characterized by the kernels kτ |y|-d-2s and the drift bτ. Under a Foster-Lyapunov condition, we establish the existence of a unique solution pair (u,λ*) satisfying the above equation, provided we set u(0) =0. Results are then extended to cover the HJB equations of mixed local-nonlocal type and this significantly improves the results in [A. Arapostathis et al., SIAM J. Control Optim., 57 (2019), pp. 1516-1540]. [ABSTRACT FROM AUTHOR]

Since the introduction of the perfectly matched layer (PML) technique by B\'erenger [J. P. B\'erenger, J. Comput. Phys., 114 (1994), pp. 185--200] to solve the time-dependent Maxwell's equations in unbounded domains, many different PML models have been developed and adopted for various wave propagation problems in unbounded domains. In this paper, we are interested in a physically inspired PML model proposed by Ziolkowski [R. W. Ziolkowski, Comput. Methods Appl. Mech. Engrg., 169 (1999), pp. 237--262]. To reduce the unknowns and make more efficient numerical methods, we reformulate the original model in integro-differential form. We propose and analyze two novel finite element methods for solving this equivalent PML model. Stability and convergence analysis are established for both schemes. Numerical results are presented to support our analysis and demonstrate the effectiveness of wave absorption of this equivalent PML. [ABSTRACT FROM AUTHOR]

The discrete gradient structure and the positive definiteness of discrete fractional integrals or derivatives are fundamental to the numerical stability in long-time simulation of nonlinear integro-differential models. We build up a discrete gradient structure for a class of second-order variable-step approximations of the fractional Riemann--Liouville integral and the fractional Caputo derivative. Then certain variational energy dissipation laws at discrete levels of the corresponding variable-step Crank--Nicolson-type methods are established for time-fractional Allen--Cahn and timefractional Klein--Gordon-type models. They are shown to be asymptotically compatible with the associated energy laws of the classical Allen--Cahn and Klein--Gordon equations in the associated fractional order limits. Numerical examples together with an adaptive time-stepping procedure are provided to demonstrate the effectiveness of our second-order methods. [ABSTRACT FROM AUTHOR]

In this article we consider a class of nonlinear integro-differential equations of the form infτ Є T { Є Rd (u(x + y) + u(x-y)-2u(x)) kτ (x,y)/| y| d+2s dy + bτ (x). u(x) + gτ (x)}-λ* = 0 in Rd, where 0 < λ (2-2s)≤ kτ (2-2s), s Є (1/2, 1). The above equation appears in the study of ergodic control problems in Rd when the controlled dynamics is governed by pure-jump Lévy processes characterized by the kernels kτ |y|-d-2s and the drift bτ. Under a Foster-Lyapunov condition, we establish the existence of a unique solution pair (u,λ*) satisfying the above equation, provided we set u(0) =0. Results are then extended to cover the HJB equations of mixed local-nonlocal type and this significantly improves the results in [A. Arapostathis et al., SIAM J. Control Optim., 57 (2019), pp. 1516-1540]. [ABSTRACT FROM AUTHOR]

This paper is addressed to a study of time optimal control problems for a family of systems governed by parabolic integro-differential equations. The target set is a closed ball centered at the origin of the state space and the control functions are distributed. We prove the optimal time and the associated optimal controls for the parabolic integro-differential equations, where the integral terms have a small perturbation and are close to those of the original problem for a parabolic equation. The method of the main result relies on the bang-bang property of optimal controls, the convergence of the associated minimization problems, and the connections between time optimal control problems and norm optimal control problems. [ABSTRACT FROM AUTHOR]

We present two diagrammatic Monte Carlo methods for quantum systems coupled with harmonic baths, whose dynamics are described by integro-differential equations. The first approach can be considered as a reformulation of the Dyson series, and the second one, called "bold-thin-bold diagrammatic Monte Carlo," is based on resummation of the diagrams in the Dyson series to accelerate its convergence. The underlying mechanism of the governing equations associated with the two methods lies in the recurrence relation of the path integrals, which is the most costly part in the numerical methods. The proposed algorithms give an extension to the work of Cai, Lu, and Yang [Comput. Phys. Commun. 278 (2022), 108417], where the algorithms are designed based on reusing the previous calculations of bath influence functionals. Compared with the algorithms therein, our methods further include the reuse of system-associated functionals and show better performance in terms of computational efficiency and memory cost. We demonstrate the two methods in the framework of spin-boson model, and numerical experiments are carried out to verify the validity of the methods. [ABSTRACT FROM AUTHOR]

The aim of this paper is to construct and analyze explicit exponential Runge--Kutta methods for the temporal discretization of linear and semilinear integro-differential equations. By expanding the errors of the numerical method in terms of the solution, we derive order conditions that form the basis of our error bounds for integro-differential equations. The order conditions are further used for constructing numerical methods. The convergence analysis is performed in a Hilbert space setting, where the smoothing effect of the resolvent family is heavily used. For the linear case, we derive the order conditions for general order p and prove convergence of order p, whenever these conditions are satisfied. In the semilinear case, we consider in addition spatial discretization by a spectral Galerkin method, and we require locally Lipschitz continuous nonlinearities. We derive the order conditions for orders one and two, construct methods satisfying these conditions, and prove their convergence. Finally, some numerical experiments illustrating our theoretical results are given. [ABSTRACT FROM AUTHOR]

Neural field models are nonlinear integro-differential equations for the evolution of neuronal activity, and they are a prototypical large-scale, coarse-grained neuronal model in continuum cortices. Neural fields are often simulated heuristically and, in spite of their popularity in mathematical neuroscience, their numerical analysis is not yet fully established. We introduce generic projection methods for neural fields and derive a priori error bounds for these schemes. We extend an existing framework for stationary integral equations to the time-dependent case, which is relevant for neuroscience applications. We find that the convergence rate of a projection scheme for a neural field is determined to a great extent by the convergence rate of the projection operator. This abstract analysis, which unifies the treatment of collocation and Galerkin schemes, is carried out in operator form, without resorting to quadrature rules for the integral term, which are introduced only at a later stage, and whose choice depends on the choice of the projector. Using an elementary time stepper as an example, we demonstrate that the error in a time stepper has two separate contributions: one from the projector and one from the time discretization. We give examples of concrete projection methods: two collocation schemes (piecewise linear and spectral collocation) and two Galerkin schemes (finite element and spectral Galerkin); for each of them we derive error bounds from the general theory, introduce several discrete variants, provide implementation details, and present reproducible convergence tests. [ABSTRACT FROM AUTHOR]

We present a mean-field formalism for modeling firing-rate statistics of brain regions whose neurons exhibit atypical firing patterns and heterogeneous electrophysiological properties. We apply the formalism to the suprachiasmatic nucleus (SCN)--the human circadian pacemaker--whose neurons can intrinsically exhibit depolarized low-amplitude membrane oscillations (DLAMOs), depolarization block (DB), and standard action potential firing at different times of day. Further, gammaaminobutyric acid reversal potentials and molecular circadian phases of SCN neurons, among other properties, vary across the network and/or slowly over time. Our formalism consists of a system of integro-differential equations describing the time evolution of the mean and standard deviation of synaptic conductances across the network. Electrophysiological properties of SCN neurons are incorporated by computing responses to synaptic conductance inputs of a Hodgkin--Huxley-type SCN neuron model that exhibits DLAMOs and DB. Such responses are then averaged over distributions of relevant quantities and included in the differential equations. Results suggest mechanisms by which physiologically relevant changes to firing activities may arise, highlighting means by which the amplitude of firing rates may shrink, the standard deviation of firing rates may grow, and by which a mid-day dip in firing rates may appear. For instance, results show that a large spread in circadian phases across SCN neurons reduces the size of oscillations in SCN network firing activity across the 24-hour day, identifying a mechanism by which heterogeneities in neuron electrophysiology could influence circadian rhythms. [ABSTRACT FROM AUTHOR]

We consider a jump-diffusion process on a bounded domain with reflection at the boundary and establish long-term results for a general additive process of its path. This includes the long-term behavior of its occupation time in the interior and on the boundaries. We derive a characterization of the large deviation rate function which quantifies the rate of exponential decay of probabilities of rare events for paths of the process. The characterization relies on a solution of a partial integro-differential equation with boundary constraints. We develop a practical implementation of our results in terms of a numerical solution for the PIDE. We illustrate the method on a few standard examples (Brownian motion, birth-death processes) and on a particular jump-diffusion model arising from applications to biochemical reactions. [ABSTRACT FROM AUTHOR]

A class of evolution equations with nonlocal diffusion is considered in this work. These are integro-differential equations arising as models of propagation phenomena in continuum media with nonlocal interactions including neural tissue, porous media, peridynamics, and models with fractional diffusion, as well as continuum limits of interacting dynamical systems. The principal challenge of numerical integration of nonlocal systems stems from the lack of spatial regularity of the data and solutions intrinsic to nonlocal models. To overcome this problem we propose a semidiscrete numerical scheme based on the combination of sparse Monte Carlo and discontinuous Galerkin methods. Our method requires minimal assumptions on the regularity of the data. In particular, the kernel of the nonlocal diffusivity is assumed to be a square integrable function and may be singular or discontinuous. An important feature of our method is the use of sparsity. Sparse sampling of points in the Monte Carlo approximation of the nonlocal term allows us to use fewer discretization points without compromising the accuracy. For kernels with singularities, more points are selected automatically in the regions near the singularities. We prove convergence of the numerical method and estimate the rate of convergence. There are two principal ingredients in the error of the numerical method related to the use of Monte Calro and Galerkin approximations, respectively. We analyze both errors. Two representative examples of discontinuous kernels are presented. The first example features a kernel with a singularity, while the kernel in the second example experiences jump discontinuity. We show how the information about the singularity in the former case and the geometry of the discontinuity set in the latter translate into the rate of convergence of the numerical procedure. In addition, we illustrate the rate of convergence estimate with a numerical example of an initial value problem, for which an explicit analytic solution is available. Numerical results are consistent with analytical estimates. [ABSTRACT FROM AUTHOR]

In this article, a class of backward stochastic Volterra integro-differential equations (BSVIDEs) is introduced and studied. It is worthy mentioning that the proposed BSVIDEs cannot be covered by the existing backward stochastic Volterra integral equations (BSVIEs), and they also have the nice flow property such that Itô's formula becomes quite applicable. It is found that BSVIDEs can provide a neat sufficient condition for the solvability of BSVIEs with generator depending on the diagonal value of the solutions. As applications, the optimal control problems in terms of maximum principles and linear quadratic control problems of optimal control for forward stochastic Volterra integro-differential equations are investigated. In contrast with the BSVIEs in current literature, some interesting phenomena and advantages of BSVIDEs are revealed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss the mathematical modeling of Brownian active particle systems, a recently popular paradigmatic system for self-propelled particles. We present four microscopic models with different types of repulsive interactions between particles and their associated macroscopic models, which are formally obtained using different coarse-graining methods. The macroscopic limits are integro-differential equations for the density in phase space (positions and orientations) of the particles and may include nonlinearities in both the diffusive and advective components. In contrast to passive particles, systems of active particles can undergo phase separation without any attractive interactions, a mechanism known as motility-induced phase separation (MIPS). We explore the onset of such a transition for each model in the parameter space of occupied volume fraction and P'eclet number via a linear stability analysis and numerical simulations at both the microscopic and macroscopic levels. We establish that one of the models, namely, the mean-field model which assumes long-range repulsive interactions, cannot explain the emergence of MIPS. In contrast, MIPS is observed for the remaining three models that assume short-range interactions that localize the interaction terms in space. [ABSTRACT FROM AUTHOR]

We model an artificial root which grows in the soil for underground prospecting. Its evolution is described by a controlled system of two integro-partial differential equations: one for the growth of the body and the other for the elongation of the tip. At any given time, the angular velocity of the root is obtained by solving a minimization problem with state constraints. We prove the existence of solutions to the evolution problem, up to the first time where a "breakdown configuration" is reached. Some numerical simulations are performed to test the effectiveness of our feedback control algorithm. [ABSTRACT FROM AUTHOR]

We study the Becker-Döring bubblelator, a variant of the Becker-Döring coagulationfragmentation system that models the growth of clusters by gain or loss of monomers. Motivated by models of gas evolution oscillators from physical chemistry, we incorporate the injection of monomers and depletion of large clusters. For a wide range of physical rates, the Becker-Döring system itself exhibits a dynamic phase transition as mass density increases past a critical value. We connect the Becker-Döring bubblelator to a transport equation coupled with an integrodifferential equation for the excess monomer density by formal asymptotics in the near-critical regime. For suitable injection/depletion rates, we argue that time-periodic solutions appear via a Hopf bifurcation. Numerics confirm that the generation and removal of large clusters can become desynchronized, leading to temporal oscillations associated with bursts of large-cluster nucleation. [ABSTRACT FROM AUTHOR]

Predictive modeling is key for saving time and resources in manufacturing processes such as fermentation arising in food and chemical manufacturing. To make reliable predictions, realistic models representing the most important process features are required. Several models describing the white wine fermentation process already exist. However, all of these models lack a combination of features, such as the importance of oxygen at the beginning of the process, the consumption of sugar due to yeast activity, and the toxicity of alcohol on the yeast cells combined with the single-cell yeast dynamics. This work introduces a new population balance model representing all these features in one model. It is based on a system of highly nonlinear weakly hyperbolic partial/ordinary integro-differential equations which poses a number of theoretical and numerical challenges. This paper increases the understanding of the latter and of the process itself by combining theoretical with numerical investigations. Existence and uniqueness of solutions to a simplified problem are studied based on semigroup theory. For the numerical solution of the problem, a numerical methodology based on a finite volume scheme combined with a time implicit scheme is derived. The impact of the initial cell distribution on the dynamics is studied. The detailed model is compared to a simpler model based on ordinary differential equations. The observed differences for different initial cell distributions and distinct models turn out to be smaller than expected. The outcomes of this paper are specifically relevant for applied mathematicians, winemakers, and process engineers. [ABSTRACT FROM AUTHOR]

The classical state-space approach to optimal estimation of stochastic processes is efficient when the driving noises are generated by martingales. In particular, the weight function of the optimal linear filter, which solves a complicated operator equation in general, simplifies to the Riccati ordinary differential equation in the martingale case. This reduction lies in the foundations of the Kalman--Bucy approach to linear optimal filtering. In this paper we consider a basic Kalman--Bucy model with noises, generated by independent fractional Brownian motions, and develop a new method of asymptotic analysis of the integro-differential filtering equation arising in this case. We establish existence of the steady-state error limit and find its asymptotic scaling in the high signal-to-noise regime. Closed form expressions are derived in a number of important cases. [ABSTRACT FROM AUTHOR]

Particle-based stochastic reaction-diffusion (PBSRD) models are a popular approach for studying biological systems involving both noise in the reaction process and diffusive transport. In this work we derive coarse-grained deterministic partial integro-differential equation (PIDE) models that provide a mean field approximation to the volume reactivity PBSRD model, a model commonly used for studying cellular processes. We formulate a weak measure-valued stochastic process (MVSP) representation for the volume reactivity PBSRD model, demonstrating for a simplified but representative system that it is consistent with the commonly used Doi Fock space representation of the corresponding forward equation. We then prove the convergence of the general volume reactivity model MVSP to the mean field PIDEs in the large-population (i.e., thermodynamic) limit. [ABSTRACT FROM AUTHOR]

We consider the sparse grid approximation of the Riccati operator P arising from closed loop parabolic control problems. In particular, we concentrate on the linear quadratic regulator (LQR) problems, i.e., we are looking for an optimal control uopt in the linear state feedback form uopt(t, ·) = Px(t, ·), where x(t, ·) is the solution of the controlled partial differential equation (PDE) for a time point t. Under sufficient regularity assumptions, the Riccati operator P fulfills the algebraic Riccati equation (ARE) AP + PA PBB P + Q = 0, where A, B, and Q are linear operators associated to the LQR problem. By expressing P in terms of an integral kernel p, the weak form of the ARE leads to a nonlinear partial integro-differential equation (IDE) for the kernel p-the Riccati-IDE. We represent the kernel function as an element of a sparse grid space, which considerably reduces the cost to solve the Riccati IDE. Numerical results are given to validate the approach. [ABSTRACT FROM AUTHOR]

In this paper we continue investigating the optimal dividend and investment problems under the Sparre Andersen model. More precisely, we try to give a more complete description of the optimal strategy when the claim frequency is a renewal process and therefore semi-Markovian, for which it is well-known that the barrier strategy is no longer optimal (cf. [H. Albrecher and J. Hartinger, Hermis J. Comp. Math. Appl., 7 (2006), pp. 109-122]). Building on our previous work [L. Bai, J. Ma, and X. Xing, Ann. Appl. Probab., 27 (2017), pp. 3588-3632], where we established the dynamic programming principle via a backward Markovization procedure and proved that the value function is the unique constrained viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation, which is a nonlocal, nonlinear, and degenerate parabolic partial integro-differential equation on an unbounded domain, in this paper we show that the optimal strategy is still of a band type but in a more complicated dynamic fashion. The main technical obstacles in constructing and validating the optimal strategy include the regularity of the value function, due to the fundamental degeneracy of the HJB equation caused by the Markovization procedure, and the well-posedness of the closed-loop stochastic system, given the "band" nature of the optimal strategy. Some of the technical results in this paper are purely analytical and therefore interesting in their own right. [ABSTRACT FROM AUTHOR]

Reaction-diffusion PDEs and particle-based stochastic reaction-diffusion (PBSRD) models are commonly used approaches for modeling the spatial dynamics of chemical and biological systems. Standard reaction-diffusion PDE models ignore the underlying stochasticity of spatial transport and reactions and are often described as appropriate in regimes where there are large numbers of particles in a system. Recent studies have proven the rigorous large-population limit of PBSRD models, showing the resulting mean-field models (MFMs) correspond to nonlocal systems of partial-integro differential equations. In this work we explore the rigorous relationship between standard reaction-diffusion PDE models and the derived MFM. We prove that the former can be interpreted as an asymptotic approximation to the later in the limit that bimolecular reaction kernels are short-range and averaging. As the reactive interaction length scale approaches zero, we prove the MFMs converge at second order to standard reaction-diffusion PDE models. In proving this result we also establish local well-posedness of the MFM model in time for general systems and global well-posedness for specific reaction systems and kernels. Finally, we illustrate the agreement and disagreement between the MFM, SM, and underlying particle model for several numerical examples. [ABSTRACT FROM AUTHOR]

We study a uniform-in-time continuum limit of the lattice Winfree model (LWM) on an inifinite cylinder under restricted initial datum and suitable assumptions on system functions such as natural frequency functions and the coupling strength function. Roughly speaking, the continuum Winfree model (CWM) is an integro-differential equation for the temporal evolution of a Winfree phase field over an infinite cylinder. For bounded measurable initial phase field and natural frequency functions, we can establish a unique global existence of classical solutions to the CWM in a large coupling regime. At the same time, we can also see that a classical solution to the CWM can be obtained as an L¹-limit of a sequence of lattice solutions to the LWM under a suitable framework. This provides a good approximation of solution to the CWM as a sequence of solutions to the LWM. Moreover, we verify that the continuum limit of a sequence of equilibria to the LWM tends to a unique stationary profile of the CWM. [ABSTRACT FROM AUTHOR]

Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation. The PINN algorithm is simple, and it can be applied to different types of PDEs, including integro-differential equations, fractional PDEs, and stochastic PDEs. Moreover, from an implementation point of view, PINNs solve inverse problems as easily as forward problems. We propose a new residual- based adaptive refinement (RAR) method to improve the training efficiency of PINNs. For pedagogical reasons, we compare the PINN algorithm to a standard finite element method. We also present a Python library for PINNs, DeepXDE, which is designed to serve both as an educational tool to be used in the classroom as well as a research tool for solving problems in computational science and engineering. Specifically, DeepXDE can solve forward problems given initial and boundary conditions, as well as inverse problems given some extra measurements. DeepXDE supports complex-geometry domains based on the technique of constructive solid geometry and enables the user code to be compact, resembling closely the mathematical formulation. We introduce the usage of DeepXDE and its customizability, and we also demonstrate the capability of PINNs and the user- friendliness of DeepXDE for five different examples. More broadly, DeepXDE contributes to the more rapid development of the emerging scientific machine learning field. [ABSTRACT FROM AUTHOR]

In this work, we develop a mathematical framework for predicting and quantifying virus diversity evolution during infection of a host organism. It is specified as a virus density distribution with respect to genotype and time governed by a reaction-diffusion integro-differential equation taking virus mutations, replication, and elimination by immune cells and medical treatment into account. Conditions for the existence of virus strains that correspond to localized density distributions in the space of genotypes are determined. It is shown that common viral evolutionary traits like diversification and extinction are driven by nonlocal interactions via immune responses, target-cell competition, and therapy. This provides us with a mechanistic explanation for clinically relevant properties like immune escape and drug resistance selection, and allows us to link virus genotypes to phenotypes. [ABSTRACT FROM AUTHOR]

A mathematical model for simulating biological field effect transistor (Bio-FET) experiments is introduced. It takes the form of a nonlinear equation that describes evolution of reacting species concentration at the boundary coupled to a diffusion equation. Using analytic techniques, this coupled system of equations is reduced to a singular integrodifferential equation (IDE). A numerical approximation of this equation is developed that achieves greater than firstorder accuracy in time and greater than second-order accuracy in space, despite the presence of a singular temporal convolution kernel and a discontinuous boundary condition. The mathematical model was validated using Bio-FET data, and stochastic regression was employed to separate signal from noise. Results show that our IDE provides a robust way of estimating important parameters such as diffusion coefficients and kinetic rate constants. [ABSTRACT FROM AUTHOR]

Deterministic models of vegetation often summarize, at a macroscopic scale, a multitude of intrinsically random events occurring at a microscopic scale. We bridge the gap between these scales by demonstrating convergence to a mean-field limit for a general class of stochastic models representing each individual ecological event in the limit of large system size. The proof relies on classical stochastic coupling techniques that we generalize to cover spatially extended interactions. The mean-field limit is a spatially extended non-Markovian process characterized by nonlocal integro-differential equations describing the evolution of the probability for a patch of land to be in a given state (the generalized Kolmogorov equations (GKEs) of the process). We thus provide an accessible general framework for spatially extending many classical finite-state models from ecology and population dynamics. We demonstrate the practical effectiveness of our approach through a detailed comparison of our limiting spatial model and the finite-size version of a specific savanna-forest model, the so-called Staver--Levin model. There is remarkable dynamic consistency between the GKEs and the finite-size system in spite of almost sure forest extinction in the finite-size system. To resolve this apparent paradox, we show that the extinction rate drops sharply when nontrivial equilibria emerge in the GKEs, and that the finite-size system's quasi-stationary distribution (stationary distribution conditional on nonextinction) closely matches the bifurcation diagram of the GKEs. Furthermore, the limit process can support periodic oscillations of the probability distribution and thus provides an elementary example of a jump process that does not converge to a stationary distribution. In spatially extended settings, environmental heterogeneity can lead to waves of invasion and front-pinning phenomena. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behavior of solutions to systems of strongly coupled integral equations with oscillatory coefficients. The system of equations is motivated by a peridynamic model of the deformation of heterogeneous media that additionally accounts for shortrange forces. We consider the vanishing nonlocality limit on the same length scale as the heterogeneity and show that the system's effective behavior is characterized by a coupled system of local equations that are elliptic in the sense of Legendre--Hadamard. This effective system is characterized by a fourth-order tensor that shares properties with Cauchy elasticity tensors that appear in the classical equilibrium equations for linearized elasticity. [ABSTRACT FROM AUTHOR]

This paper considers the problem of interaction of multiple straight plane cracks in the presence of the Steigmann--Ogden surface energy on the crack boundaries. The general boundary conditions with the Steigmann--Ogden surface elasticity are derived in the arc length parametrization of the crack boundary for arbitrary smooth crack geometry. The plane stress problem of linear elasticity for a plate with multiple straight cracks is solved by using integral representations of complex potentials. The resulting system of singular integro-differential equations is reduced to a system of Fredholm equations of the second kind. The existence and uniqueness of the solution are discussed. The numerical results are presented for two straight cracks located on the same straight line. The results are compared to the results of the classical problem for a plate with multiple cracks in the absence of the Steigmann--Ogden energy and to the results for a single straight crack in the plate with the Steigmann--Ogden surface energy on the boundary. [ABSTRACT FROM AUTHOR]

We rigorously prove the existence of traveling fronts in neural field models with lateral inhibition coupling types and smooth sigmoidal firing rates. With Heaviside firing rates as our base point (where unique traveling fronts exist), we repeatedly apply the implicit function theorem in Banach spaces to provide a nonmonotone version of the homotopy approach originally proposed by Ermentrout and McLeod [Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), pp. 461-478] in their seminal study of monotone fronts in purely excitatory models. By comparing smooth and Heaviside firing rates, we develop global wave speed and profile comparisons that guide our analysis, leading to uniqueness (modulo translation) in the perturbative case. Moreover, we establish a meaningful a priori existence result; we prove existence holds for a range of firing rates, independent of continuation path. [ABSTRACT FROM AUTHOR]

We study a class of nonlinear integrodifferential equations on a subspace of all probability measures on the real line related to the optimal control of McKean-Vlasov jump-diffusions. We develop an intrinsic notion of viscosity solutions that does not rely on the lifting to a Hilbert space and prove a comparison theorem for these solutions. We also show that the value function is the unique viscosity solution. [ABSTRACT FROM AUTHOR]

In 2006 Armstrong, Painter, and Sherratt formulated a nonlocal differential equation model for cell-cell adhesion. For the one-dimensional case on a bounded domain we derive various types of biological boundary conditions, describing adhesive, repulsive, and neutral boundaries. We prove local and global existence and uniqueness for the resulting integrodifferential equations. In numerical simulations we consider adhesive, repulsive, and neutral boundary conditions, and we show that the solutions mimic known behavior of uid adhesion to boundaries. In addition, we observe interior pattern formation due to cell-cell adhesion. [ABSTRACT FROM AUTHOR]

Motivated by fractional derivative models in viscoelasticity, a class of semilinear stochastic Volterra integro-differential equations, and their deterministic counterparts, are considered. A generalized exponential Euler method, named here the Mittag{Leer Euler integrator, is used for the temporal discretization, while the spatial discretization is performed by the spectral Galerkin method. The temporal rate of strong convergence is found to be (almost) twice compared to when the backward Euler method is used together with a convolution quadrature for time discretization. Numerical experiments that validate the theory are presented. [ABSTRACT FROM AUTHOR]

In this work, Holder continuity is obtained for solutions to the nonlocal kinetic Fokker-Planck equation and to a family of related equations with general integro-differential operators. These equations can be seen as a generalization of the Fokker-Planck equation, or as a linearization of non-cutoff Boltzmann. Difficulties arise because our equations are hypoelliptic, so we utilize the theory of averaging lemmas. Regularity is obtained using De Giorgi's method, so it does not depend on the regularity of initial conditions or coefficients. This work assumes stronger constraints on the nonlocal operator than in the work of Imbert and Silvestre [The Weak Harnack Inequality for the Boltzmann Equation Without Cut-Off, arXiv:1608.07571, 2016] but allows unbounded source terms. [ABSTRACT FROM AUTHOR]

The paper deals with the existence of standing wave solutions for the Schrödinger-- Poisson system with prescribed mass in dimension N = 2. This leads to investigating the existence of normalized solutions for an integrodifferential equation involving a logarithmic convolution potential, namely, -Δu + λu + γ(log| · | * |u|2)u = a|u|p-2u in R2, fR2 |u|2dx = c, where c > 0 is a given real number. Under different assumptions on γ ∈ R, a ∈ R, p > 2, we prove several existence and multiplicity results. Here λ ∈ R appears as a Lagrange parameter and is part of the unknowns. With respect to the related higher-dimensional cases, the presence of the logarithmic kernel, which is unbounded from above and below, makes the structure of the solution set much richer, forcing the implementation of new ideas to catch the normalized solutions. [ABSTRACT FROM AUTHOR]

We propose an impulsive integro-differential model to describe the dynamics of an invading species with a pulsive reproduction stage and a nonlocal dispersal stage. We first establish a threshold-type result on the global dynamics of the system in a bounded domain, and present an application to insect pests outbreak in terms of the critical domain size. For the spatial spread in an unbounded domain, we then prove the existence of the invasion speed and its coincidence with the minimal speed for monotone traveling waves. Numerical simulations are also carried out to illustrate our analytical results. [ABSTRACT FROM AUTHOR]

This paper studies the propagation of plasmons on graphene when the Drude weight is varied in time. The phenomenon of plasmon propagation is modeled by considering the graphene as a conductive sheet. Under the assumption that the field is oscillatory in the direction parallel to the sheet, it can be shown that the coupled electromagnetic field can be reduced to a single timedependent equation describing the current density on the sheet. The current density depends on the wave number\xi and is shown to satisfy an integro-differential equation in time. Well-posedness for this equation is established. A numerical scheme to solve the current equations based on convolution quadrature is developed. An approximate equation, based on large\xi with the physical interpretation of a quasi-static approximation, is derived and its accuracy assessed. The phenomena of wave reversal and parametric amplification are studied. Numerical calculations are conducted to address several theoretical issues as well as to demonstrate the main ideas. [ABSTRACT FROM AUTHOR]

We first present a new delay-dependent fractional generalization of Halanay-like inequality to characterize the asymptotic behavior of fractional functional differential equations (F-FDEs). Then we study the dissipativity of F-FDEs with a bounded absorbing set and the asymptotic stability and the contractivity of F-FDEs with algebraically contractive rate. Two numerical schemes are further constructed for F-FDEs based on Grünwald--Letnikov formula and L1 method for Caputo fractional derivative, together with linear interpolation for the functional terms. These two schemes are proved to be dissipative and contractive and can preserve the exact decay rate as the continuous equations. These results can be directly applied to some special cases of F-FDEs, such as the fractional delay differential equations, fractional integro-differential equations, and fractional delay integro-differential equations. Finally, several numerical examples are given to illustrate the advantages of the structure-preserving numerical methods. In particular, we shall compare the numerical performance of our schemes and the popular predictor-corrector algorithms for F-FDEs and demonstrate that our schemes are more efficient and robust, especially for some stiff systems. [ABSTRACT FROM AUTHOR]

In this paper the implicit-explicit (IMEX) two-step backward differentiation formula (BDF2) method with variable step-size, due to the nonsmoothness of the initial data, is developed for solving parabolic partial integro-differential equations (PIDEs), which describe the jump-diffusion option pricing model in finance. It is shown that the variable step-size IMEX BDF2 method is stable for abstract PIDEs under suitable time step restrictions. Based on the time regularity analysis of abstract PIDEs, the consistency error and the global error bounds for the variable step-size IMEX BDF2 method are provided. After time semidiscretization, spatial differential operators are treated by using finite difference methods, and the jump integral is computed using the composite trapezoidal rule. A local mesh refinement strategy is also considered near the strike price because of the nonsmoothness of the payoff function. Numerical results illustrate the effectiveness of the proposed method for European and American options under jump-diffusion models. [ABSTRACT FROM AUTHOR]

We rigorously derive an integro-differential equation as a model for the possible onset of regenerative chatter during a turning process using a lathe. The cut is made parallel to the axis of rotation of the spindle. The model allows the spindle speed to continuously vary with time, which results in the presence of a variable time delay determined from a threshold condition. We present a number of conditions sufficient for the elimination of chatter; these emphasize sufficiently low feed rate or sufficiently high spindle speed. Numerical simulations cast light on the effect of a sinusoidally varying spindle speed, a feature of some modern lathes. Spindle speed variation can cure chatter but does not necessarily do so and can fail at higher values of the tool feed rate. [ABSTRACT FROM AUTHOR]

In ecology a number of spatiotemporal datasets on cyclic populations reveal periodic traveling waves of abundance. This calls for studies of periodic traveling wave solutions of ecologically realistic mathematical models. For many species, such models must include long-range dispersal. However, mathematical theory on periodic traveling waves is almost entirely restricted to reactiondi ffusion equations, which assume purely local dispersal. I study integrodifferential equation models in which dispersal is represented via a convolution. The dispersal kernel is assumed to be of either Gaussian or Laplace form; in either case it contains a parameter scaling the width of the kernel. I show that as this parameter tends to zero, the integrodifferential equation asymptotically approaches a reaction-diffusion model. I exploit this limit to determine the effect of a small degree of nonlocality in dispersal on periodic traveling wave properties and on the selection of a periodic traveling wave solution by localized perturbation of an unstable steady state. My analysis concerns equations of λ-ω" type, which are the normal form of a large class of oscillatory systems close to a Hopf bifurcation point. I finish the paper by showing how my results can be used to determine the effect of nonlocal dispersal on spatiotemporal dynamics in a predator-prey system. [ABSTRACT FROM AUTHOR]

Integrodifference equations are discrete-time cousins of reaction-diffusion equations. Like their continuous-time counterparts, they are used to model spreading phenomena in ecology and other sciences. Unlike their continuous-time counterparts, even scalar integrodifference equations can exhibit nonmonotone dynamics. Few authors studied the existence of spreading speeds and traveling waves in the nonmonotone case; previous numerical simulations indicated the existence of traveling two-cycles. Our numerical observations indicate the presence of several spreading speeds and multiple traveling wave profiles in these equations. We generalize the concept of a spreading speed to encompass this situation and prove the existence of such generalized spreading speeds and associated traveling waves in the corresponding second-iterate operator. Our numerical simulations let us conjecture that these spreading speeds could be linearly determined. We prove the existence of bistable traveling waves in a related second-iterate operator. We relate our results to the existence of stacked waves and to dynamical stabilization. [ABSTRACT FROM AUTHOR]

We consider an infinite horizon optimal control problem for a continuous-time Markov chain X in a finite set I with noise-free partial observation. The observation process is defined as Yt = h(Xt), t ≥ 0, where h is a given map defined on I. The observation is noise-free in the sense that the only source of randomness is the process X itself. The aim is to minimize a discounted cost functional and study the associated value function V . After transforming the control problem with partial observation into one with complete observation (the separated problem) using filtering equations, we provide a link between the value function v associated with the latter control problem and the original value function V . Then, we present two different characterizations of v (and indirectly of V ): on one hand as the unique fixed point of a suitably defined contraction mapping and on the other hand as the unique constrained viscosity solution (in the sense of Soner) of a HJB integro-differential equation. Under suitable assumptions, we finally prove the existence of an optimal control. [ABSTRACT FROM AUTHOR]