Continuous Differential Equations Research Articles

CAG-NSPDE: Continuous adaptive graph neural stochastic partial differential equations for traffic flow forecasting

Traffic flow forecasting, which aims to predict future traffic flow given historical data, is crucial for the planning and construction of urban transportation infrastructure. Existing methods based on discrete dynamic graphs are unable to model the smooth evolution of spatial relationships and usually have high computational cost. In addition, they ignore the effect of stochastic noise on the modeling of temporal dependencies, which lead to poor predictive performance on complex intelligent transportation systems. To address these issues, we propose continuous adaptive graph neural stochastic partial differential equation (CAG-NSPDE). A continuous graph is constructed based on learnable continuous adaptive node embeddings for the modeling of evolution of spatial dependencies. To explicitly model the effect of noise, a novel architecture based on stochastic partial differential equation (SPDE) is proposed. It can simultaneously extract features from the temporal and frequency domains by reducing SPDEs to a system of ODEs in Fourier space. We perform extensive experiments on four real-world datasets and the results demonstrate the superiority of CAG-NSPDE compared with state-of-the-arts.

Quantitative analysis of social influence and digital piracy contagion with differential equations on networks

Illegal file sharing of copyrighted contents through popular file sharing networks poses an enormous threat to providers of digital contents, such as, games, softwares, music and movies. Though empirical studies of network effects on piracy is a well-studied domain, the dynamics of peer effect in the context of evolving social contagion has not been enough explored using dynamical models. In this research, we methodically study the trends of online piracy with a continuous ODE approach and differential equations on graphs to have a clear comparative view. We first formulate a compartmental model to study bifurcations and thresholds mathematically. We later move on with a network-based analysis to illustrate the proliferation of online piracy dynamics with an epidemiological approach over a social network. We figure out a solution for this online piracy problem by developing awareness among individuals and introducing media campaigns, which could be a valuable factor in eradicating and controlling online piracy. Next, using degree-block approximation, network analysis has been performed to investigate the phenomena from a heterogeneous approach and to derive the threshold condition for the persistence of piracy in the population in a steady state. Considering the dual control of positive peer influence and media-driven awareness, we examine the system through realistic parameter selection to better understand the complexity of the dynamics and suggest policy implications.

Heteroclinic solutions in singularly perturbed discontinuous differential equations

We derive Melnikov type conditions for the persistence of heteroclinic solutions in perturbed slowly varying discontinuous differential equations extending to these equations similar results for continuous differential equations.

Discrete heat equation for a periodic layered system with allowance for the interfacial thermal resistance: General formulation and dispersion analysis.

Discrete heat equations for the multilayered periodic systems with allowance for the thermal resistance between the layers and corresponding dispersion relations in ω-k space have been derived and analyzed. The discrete equations imply a finite velocity of thermal disturbances and guarantee the positiveness of the solutions. Analytical expressions for the attenuation distance, and phase and group velocities have been obtained as functions of frequency and thermal resistance between the discrete layers. These functions demonstrate unusual behavior at high frequency compared to the continuum case. Furthermore, the maximum allowed frequency and wave number for the discrete heat equation are limited, whereas there are no such limits in a continuum. The discrete equation contains an infinite hierarchy of continuous partial differential equations, which starts with the Fourier law, proceeds with the hyperbolic equation, the Guyer-Krumhansl (or Jeffreys type) equation, and then with higher-order equations. The partial differential equations with a finite number of terms are only approximations of the discrete equation, which implies that on the ultrashort space and timescales the discrete approach is preferable. This work provides a relatively simple, easy-to-adopt, conceptual tool, together with analytical expressions allowing one to study ultrafast wavelike heat conduction regimes in periodic multilayered metamaterials.

High-Frequency Space Diffusion Model for Accelerated MRI.

Diffusion models with continuous stochastic differential equations (SDEs) have shown superior performances in image generation. It can serve as a deep generative prior to solving the inverse problem in magnetic resonance (MR) reconstruction. However, low-frequency regions of k -space data are typically fully sampled in fast MR imaging, while existing diffusion models are performed throughout the entire image or k -space, inevitably introducing uncertainty in the reconstruction of low-frequency regions. Additionally, existing diffusion models often demand substantial iterations to converge, resulting in time-consuming reconstructions. To address these challenges, we propose a novel SDE tailored specifically for MR reconstruction with the diffusion process in high-frequency space (referred to as HFS-SDE). This approach ensures determinism in the fully sampled low-frequency regions and accelerates the sampling procedure of reverse diffusion. Experiments conducted on the publicly available fastMRI dataset demonstrate that the proposed HFS-SDE method outperforms traditional parallel imaging methods, supervised deep learning, and existing diffusion models in terms of reconstruction accuracy and stability. The fast convergence properties are also confirmed through theoretical and experimental validation. Our code and weights are available at https://github.com/Aboriginer/HFS-SDE.

Link prediction by continuous spatiotemporal representation via neural differential equations

With the continuous advancement of data science and machine learning, temporal link prediction has emerged as a crucial aspect of dynamic network analysis, providing significant research and application potential across various domains. While deep learning techniques have achieved remarkable results in temporal link prediction, most existing studies have focused on discrete model frameworks. These frameworks face limitations in capturing deep structural features and effectively aggregating temporal information. To address these limitations, we draw inspiration from neural differential equations to propose a Continuous Temporal Graph Neural Differential Equation (CTGNDE) network model for temporal link prediction. Specifically, we design a spatial graph Ordinary Differential Equation (ODE) to capture the spatial correlations inherent in complex spatiotemporal information. Then we employ Neural Controlled Differential Equation (Neural CDE) to learn complex evolution patterns and effectively aggregate temporal information. Finally, we characterize completely continuous and more accurate hidden state trajectories by coupling spatial and temporal messages. Experiments conducted on 10 real-world network datasets validated the superior performance of the CTGNDE model over the state-of-the-art baselines.

A traffic-fractal-element-based congestion model considering the uneven distribution of road traffic

The uneven distribution of road traffic has been proven to play a crucial role in congestion evolution within urban transportation networks. Establishing congestion models to explore congestion evolution under its impact is of great significance for predicting congestion. Current congestion models commonly describe the uneven distribution of traffic on roads through continuous differential equations or the simulation of vehicle operational rules by which to model traffic flow dynamics. However, these models fail to directly reveal the mechanism by which the uneven distribution characteristics affect congestion, which is not conducive to uncovering the fundamental factors that affect congestion evolution and general laws for predicting congestion. Therefore, this paper first constructs traffic fractal elements (TFE) based on the fractal characteristics exhibited by uneven traffic distribution. A local parameter namely the load duty ratio (LDR) is introduced to represent the uneven distribution level of TFE. By employing the fractal interpolation function (FIF), we model road traffic as the iterations of TFE. Then we define the fractal evolution rule and the load transfer rule to inscribe the influence relationship between LDR of TFE and global congestion, ultimately establishing a TFE-based congestion model. Furthermore, the proposed TFE-based model is validated using empirical data from AMAP and simulation data from a calibrated Cellular Automata (CA) model. Finally, by comparing the results with a CA-based congestion model, the LDR of TFE is confirmed to be a comprehensive and indicative parameter affecting congestion evolution. By simulating and fitting the congestion evolution data under different LDRs, our proposed model demonstrates its value in congestion prediction and analysis.

Stochastic Modeling of a Measles Outbreak in Brazil

Development of mathematical models and its numerical implementations are essential tools in epidemiological modeling. Susceptible-Infected-Recovered (SIR) compartmental model, proposed by Kermack and McKendrick, in 1927, is a widely used deterministic model which serves as a basis for more involved mathematical models. In this work, we consider two stochastic versions of the SIR model for analysing a measles outbreak in Ilha Grande, Rio de Janeiro, in 1976; Continuous Time Markov Chain and Stochastic Differential Equations. The SIR Continuous Time Markov Chain model is used to extract specific information from the measles outbreak, obtaining results in excellent agreement with the reported epidemic values. Numerical simulations are performed in Python.

Accelerated dynamics with dry friction via time scaling and averaging of doubly nonlinear evolution equations

In a Hilbert framework, for convex differentiable optimization, we analyze the long-time behavior of inertial dynamics with dry friction. In classical approaches based on asymptotically vanishing viscous damping (in accordance with Nesterov’s method), the results are expressed in terms of rapid convergence of the values of the function to its minimum value. On the other hand, dry friction induces convergence towards an approximate minimizer, typically the system stops at x when a given threshold ‖∇f(x)‖≤r is satisfied. We obtain rapid convergence results in this direction. In our approach, we start from a doubly nonlinear first-order evolution equation involving two potentials: one is the differentiable function f to be minimized, which acts on the state of the system via its gradient, and the other is the nonsmooth potential dry friction φ(x)=r‖x‖ which acts on the velocity vector via its sub-differential. To highlight the central role played by ∇f(x), we also deal with the dual formulation of these dynamics, which have a Riemannian gradient structure. We then rely on the general acceleration method recently developed by Attouch, Boţ and Nguyen, which consists in applying the method of time scaling and then averaging to a continuous differential equation of the first order in time. We thus obtain fast convergence results for second order time-evolution systems involving dry friction, asymptotically vanishing viscous damping, and Hessian-driven damping in the implicit form.

A Review of Continuous and Discontinuous Galerkin Finite Element Methodfor Differential Equations

A Review of Continuous and Discontinuous Galerkin Finite Element Methodfor Differential Equations

Fusing time-varying mosquito data and continuous mosquito population dynamics models

Climate change is arguably one of the most pressing issues affecting the world today and requires the fusion of disparate data streams to accurately model its impacts. Mosquito populations respond to temperature and precipitation in a nonlinear way, making predicting climate impacts on mosquito-borne diseases an ongoing challenge. Data-driven approaches for accurately modeling mosquito populations are needed for predicting mosquito-borne disease risk under climate change scenarios. Many current models for disease transmission are continuous and autonomous, while mosquito data is discrete and varies both within and between seasons. This study uses an optimization framework to fit a non-autonomous logistic model with periodic net growth rate and carrying capacity parameters for 15 years of daily mosquito time-series data from the Greater Toronto Area of Canada. The resulting parameters accurately capture the inter-annual and intra-seasonal variability of mosquito populations within a single geographic region, and a variance-based sensitivity analysis highlights the influence each parameter has on the peak magnitude and timing of the mosquito season. This method can easily extend to other geographic regions and be integrated into a larger disease transmission model. This method addresses the ongoing challenges of data and model fusion by serving as a link between discrete time-series data and continuous differential equations for mosquito-borne epidemiology models.

The ℒ1 controller synthesis for piecewise continuous nonlinear systems via set invariance principles

AbstractThis article provides a synthesis method of the controller for piecewise continuous nonlinear systems in terms of set invariance principles. Following the arguments of Filippov's differential equations, solutions of piecewise continuous differential equations are first characterized by set‐valued functions. Based on such characteristics of solutions equipped with set‐valued functions, the performance of piecewise continuous nonlinear systems, that is, the gain from the associated disturbance to regulated output, is formulated. To clarify the existence of a controller such that the resulting closed‐loop system satisfies the performance in the presence of piecewise continuity, the so‐called generalized controlled invariance domain is proposed. Taking this advanced argument allows us to establish a synthesis procedure for the controller by showing a lower‐semicontinuity of the set‐valued functions involved in the generalized controlled invariance domain. Finally, some numerical examples are provided to evaluate the validity of the existence condition for the controller.

Oscillation and Asymptotic Behavior of a Special Delay Third Order Nonlinear Neutral Functional Differential Equation

Abstract The oscillation theory of differential equations is an important branch of performance of differential equations, which is widely used in engineering control, vibration mechanics, mechanics, and industry. Therefore, the vibration performance of different parts has attracted people’s attention, and a lot of research work has been done. For a special class of delay differential equations - advanced piecewise continuous differential equations, the oscillation of numerical solution is discussed. The θ − method is used to discretize the equation, and the numerical method is obtained to keep the oscillation of the analytical solution of the equation, progressive conditions. At the same time, four different states of the dynamic behavior are discussed in detail for the analytical solution and the numerical solution respectively. Some numerical examples further verify the corresponding conclusions.

Stability Criteria of Delayed Memristor-Based Neural Networks via Continuous-Time Model and Interval Matrix Approach

Modeling and stability analysis of memristor-based neural networks (MNNs) are the premise of designs and applications. Different from most previous research, delayed MNNs are described by continuous differential equations with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{2n^{2}+n}$</tex-math> </inline-formula> variables where the memductances of memristors are continuously dependent on the fluxes. Both system delays and input delays are considered, and the delays and their derivatives may vary in intervals whose lower bounds are not restricted to be zero. The systems are further reduced to continuous-time neural networks (NNs) with interval matrix uncertainties, and a unified method is developed to solve the stability of delayed NNs and MNNs. Stability criteria are obtained for delayed MNNs by augmented Lyapunov functionals, Wirtinger-based integral inequality, reciprocally convex approach, and linear matrix inequalities. In the end, two numerical simulations are used to demonstrate the validity of our theorems.

Aperiodic switching event-triggered stabilization of continuous memristive neural networks with interval delays

The stabilization problem is studied for memristive neural networks with interval delays under aperiodic switching event-triggered control. Note that, most of delayed memristive neural networks models studied are discontinuous, which are not the real memristive neural networks. First, a real model of memristive neural networks is proposed by continuous differential equations, furthermore, it is simplified to neural networks with interval matrix uncertainties. Secondly, an aperiodic switching event-trigger is given, and the considered system switches between aperiodic sampled-data system and continuous event-triggered system. Thirdly, by constructing a time-dependent piecewise-defined Lyapunov functional, the stability criterion and the feedback gain design are obtained by linear matrix inequalities. Compared with the existing results, the stability criterion is with lower conservatism. Finally, two neurons are taken as examples to ensure the feasibility of the results.

Intelligent decision support tool for optimizing stochastic inventory systems under uncertainty

Pricing and production policies play a key role in ensuring the added value of supply chain systems. For perishable inventory management, the pricing and production lines must be manipulated dynamically since several uncertainties are involved in the system’s behavior. This study discusses the impact of dynamic pricing and production policies on an uncertain stochastic inventory system with perishable products. The mathematical model of the inventory management system under external disturbance is formulated using a continuous differential equation in which the price and production rates are considered as control factors to optimize total profits, which is described as an objective function. An analytical solution for the optimal pricing and production rate was obtained using the Hamilton-Jacobi-Bellman equation. The unknown disturbance was approximated using an intelligent approach called radial basis function neural network. Finally, extensive numerical simulations were presented to validate the theoretical results and optimization solutions (including the efficiency of the approximation of the unknown disturbance) for the dynamic pricing and production management strategy of an uncertain stochastic inventory system against volatile markets. The performance of the proposed method was analyzed under different stock level conditions, which highlighted the importance of keeping the inventory levels at an optimal range to ensure the profitability of business operations. This management strategy can assist a business with solutions for inventory policies while supporting decision-making processes to facilitate coping with production management disruptions.

On Mixed Steps-Collocation Schemes for Nonlinear Fractional Delay Differential Equations

This research deals with the numerical solution of fractional differential equations with delay using the method of steps and shifted Legendre (Chebyshev) collocation method. This article presents a new formula for the fractional derivatives (in the Caputo sense) of shifted Legendre polynomials. With the help of this tool and previous work of the authors, efficient numerical schemes for solving nonlinear continuous fractional delay differential equations are proposed. The proposed schemes transform the nonlinear fractional delay differential equations to a non-delay one by employing the method of steps. Then, the approximate solution is expanded in terms of Legendre (Chebyshev) basis functions. Furthermore, the convergence analysis of the proposed schemes is provided. Several practical model examples are considered to illustrate the efficiency and accuracy of the proposed schemes.

Neural Generalized Ordinary Differential Equations with Layer-Varying Parameters

Deep residual networks (ResNets) have shown state-of-the-art performance in various real-world applications. Recently, the ResNets model was reparameterized and interpreted as solutions to a continuous ordinary differential equation or Neural-ODE model. In this study, we propose a neural generalized ordinary differential equation (Neural-GODE) model with layer-varying parameters to further extend the Neural-ODE to approximate the discrete ResNets. Specifically, we use nonparametric B-spline functions to parameterize the Neural-GODE so that the trade-off between the model complexity and computational efficiency can be easily balanced. It is demonstrated that ResNets and Neural-ODE models are special cases of the proposed Neural-GODE model. Based on two benchmark datasets, MNIST and CIFAR-10, we show that the layer-varying Neural-GODE is more flexible and general than the standard Neural-ODE. Furthermore, the Neural-GODE enjoys the computational and memory benefits while performing comparably to ResNets in prediction accuracy.

Estimation of the invariant density for discretely observed diffusion processes: impact of the sampling and of the asynchronicity

We aim at estimating in a non-parametric way the density π of the stationary distribution of a d-dimensional stochastic differential equation , for , from the discrete observations of a finite sample ,…, with . We propose a kernel density estimator and we study its convergence rates for the pointwise estimation of the invariant density under anisotropic Hölder smoothness constraints. First of all, we find some conditions on the discretization step that ensures it is possible to recover the same rates as if the continuous trajectory of the process was available. As proven in the recent work [Amorino C, Gloter A. Minimax rate of estimation for invariant densities associated to continuous stochastic differential equations over anisotropic Holder classes; 2021. arXiv preprint arXiv:2110.02774], such rates are optimal and new in the context of density estimator. Then we deal with the case where such a condition on the discretization step is not satisfied, which we refer to as the intermediate regime. In this new regime we identify the convergence rate for the estimation of the invariant density over anisotropic Hölder classes, which is the same convergence rate as for the estimation of a probability density belonging to an anisotropic Hölder class, associated to n iid random variables . After that we focus on the asynchronous case, in which each component can be observed at different time points. Even if the asynchronicity of the observations complexifies the computation of the variance of the estimator, we are able to find conditions ensuring that this variance is comparable to the one of the continuous case. We also exhibit that the non-synchronicity of the data introduces additional bias terms in the study of the estimator.

CoordiNet: Constrained Dynamics Learning for State Coordination Over Graph

A neural architecture to efficiently coordinate the transitions of state variables (states) over a graph is proposed. We consider the coordination of the time evolution of the state variables associated with the nodes on a graph. The states are associated with physical attributes of agents, e.g., the speed and/or location of vehicles. Efficient coordination then corresponds to the optimization of dynamics (how fast do vehicles travel) that may be subject to constraints (travel must be collision-free). The aim of this paper is (i) the formulation of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">learnable constrained dynamics</i> which governs the transition of states under constraints over a graph and (ii) to show its industrial application. Firstly, we formulate continuous ordinary differential equations (ODEs), namely forward propagation of state transitions over a graph and backward propagation to optimize model parameters for the learnable constrained dynamics. Discretization of these continuous ODEs results in the neural architecture <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">CoordiNet</i> . Secondly, as an application of CoordiNet, we address a traffic coordination problem by learning constrained dynamics such that vehicles can travel as fast as possible without collisions. Simulation experiments of traffic coordination confirm that our method maximizes the vehicles' speed states while avoiding collisions.