Continuous Differential Equations Research Articles

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.

On a diffusive bacteriophage dynamical model for bacterial infections

Bacteriophages or phages are viruses that infect bacteria and are increasingly used to control bacterial infections. We develop a reaction–diffusion model coupling the interactive dynamic of phages and bacteria with an epidemiological bacteria-borne disease model. For the submodel without phage absorption, the basic reproduction number [Formula: see text] is computed. The disease-free equilibrium (DFE) is shown to be globally asymptotically stable whenever [Formula: see text] is less than one, while a unique globally asymptotically endemic equilibrium is proven whenever [Formula: see text] exceeds one. In the presence of phage absorption, the above stated classical condition based on [Formula: see text], as the average number of secondary human infections produced by susceptible/lysogen bacteria during their entire lifespan, is no longer sufficient to guarantee the global stability of the DFE. We thus derive an additional threshold [Formula: see text], which is the average offspring number of lysogen bacteria produced by one infected human during the phage–bacteria interactions, and prove that the DFE is globally asymptotically stable whenever both [Formula: see text] and [Formula: see text] are under unity, and infections persist uniformly whenever [Formula: see text] is greater than one. Finally, the discrete counterpart of the continuous partial differential equation model is derived by constructing a nonstandard finite difference scheme which is dynamically consistent. This consistency is shown by constructing suitable discrete Lyapunov functionals thanks to which the global stability results for the continuous model are replicated. This scheme is implemented in MatLab platform and used to assess the impact of spatial distribution of phages, on the dynamic of bacterial infections.

Closed-form continuous-time neural networks

Continuous-time neural networks are a class of machine learning systems that can tackle representation learning on spatiotemporal decision-making tasks. These models are typically represented by continuous differential equations. However, their expressive power when they are deployed on computers is bottlenecked by numerical differential equation solvers. This limitation has notably slowed down the scaling and understanding of numerous natural physical phenomena such as the dynamics of nervous systems. Ideally, we would circumvent this bottleneck by solving the given dynamical system in closed form. This is known to be intractable in general. Here, we show that it is possible to closely approximate the interaction between neurons and synapses—the building blocks of natural and artificial neural networks—constructed by liquid time-constant networks efficiently in closed form. To this end, we compute a tightly bounded approximation of the solution of an integral appearing in liquid time-constant dynamics that has had no known closed-form solution so far. This closed-form solution impacts the design of continuous-time and continuous-depth neural models. For instance, since time appears explicitly in closed form, the formulation relaxes the need for complex numerical solvers. Consequently, we obtain models that are between one and five orders of magnitude faster in training and inference compared with differential equation-based counterparts. More importantly, in contrast to ordinary differential equation-based continuous networks, closed-form networks can scale remarkably well compared with other deep learning instances. Lastly, as these models are derived from liquid networks, they show good performance in time-series modelling compared with advanced recurrent neural network models.

Validation and parameterization of a novel physics-constrained neural dynamics model applied to turbulent fluid flow

In fluid physics, data-driven models to enhance or accelerate time to solution are becoming increasingly popular for many application domains, such as alternatives to turbulence closures, system surrogates, or for new physics discovery. In the context of reduced order models of high-dimensional time-dependent fluid systems, machine learning methods grant the benefit of automated learning from data, but the burden of a model lies on its reduced-order representation of both the fluid state and physical dynamics. In this work, we build a physics-constrained, data-driven reduced order model for Navier–Stokes equations to approximate spatiotemporal fluid dynamics in the canonical case of isotropic turbulence in a triply periodic box. The model design choices mimic numerical and physical constraints by, for example, implicitly enforcing the incompressibility constraint and utilizing continuous neural ordinary differential equations for tracking the evolution of the governing differential equation. We demonstrate this technique on a three-dimensional, moderate Reynolds number turbulent fluid flow. In assessing the statistical quality and characteristics of the machine-learned model through rigorous diagnostic tests, we find that our model is capable of reconstructing the dynamics of the flow over large integral timescales, favoring accuracy at the larger length scales. More significantly, comprehensive diagnostics suggest that physically interpretable model parameters, corresponding to the representations of the fluid state and dynamics, have attributable and quantifiable impact on the quality of the model predictions and computational complexity.

A Continuation Method for Large-Scale Modeling and Control: From ODEs to PDE, a Round Trip

In this article, we present a continuation method, which transforms spatially distributed ordinary differential equation (ODE) systems into a continuous partial differential equation (PDE). We show that this continuation can be performed for both linear and nonlinear systems, including multidimensional, space-varying, and time-varying systems. When applied to a large-scale network, the continuation provides a PDE describing the evolution of a continuous-state approximation that respects the spatial structure of the original ODE. Our method is illustrated by multiple examples, including transport equations, Kuramoto equations, and heat diffusion equations. As a main example, we perform the continuation of a Newtonian system of interacting particles and obtain the Euler equations for compressible fluids, thereby providing an original solution to Hilbert’s sixth problem. Finally, we leverage our derivation of Euler equations to solve a control problem multiagent systems, by designing a nonlinear control algorithm for robot formation based on its continuous approximation.

A Robust Time-Varying Riccati-Based Control for Uncertain Nonlinear Dynamical Systems

Abstract Riccati equation-based control approaches such as linear-quadratic regulator (LQR) and time-varying LQR (TVLQR) are among the most common methods for stabilizing linear and nonlinear systems, especially in the context of optimal control. However, model inaccuracies may degrade the performance of closed-loop systems under such controllers. To mitigate this issue, this paper extends and encompasses Riccati-equation based controllers through the development of a robust stabilizing control methodology for uncertain nonlinear systems with modeling errors. We begin by linearizing the nonlinear system around a nominal trajectory to obtain a time-varying linear system with uncertainty in the system matrix. We propose a modified version of the continuous differential Riccati equation (MCDRE), whose solution is updated based upon the estimates of model uncertainty. An optimal least squares (OLS) algorithm is presented to identify this uncertainty and inform the MCDRE to update the control gains. The unification of MCDRE and OLS yields a robust time-varying Riccati-based (RTVR) controller that stabilizes uncertain nonlinear systems without the knowledge of the structure of the system's uncertainty a priori. The convergence of the system states is formally proven using a Lyapunov argument. Simulations and comparisons to the baseline backward-in-time Riccati-based controller on two real-world examples verify the benefits of our proposed control method.

НАЧАЛЬНАЯ ЗАДАЧА ДЛЯ ОБЫКНОВЕННОГО ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ ВТОРОГО ПОРЯДКА С ОПЕРАТОРОМ РАСПРЕДЕЛЕННОГО ДИФФЕРЕНЦИРОВАНИЯ

We study a linear ordinary differential equation of the second order with operator of continuously distributed differentiation with variable coefficients. Such equations belong to the class of continuous differential equations. In this paper, the continuously distributed differentiation operator is defined as an integral with summable kernel from the Riemann–Liouville fractional differentiation operator in the order of differentiation. A special case of the operator of continuously distributed differentiation is the operator of discretely distributed differentiation. For the equation under consideration, a fundamental solution is constructed in the form of a Neumann series, reducing the differential problem to the Volterra integral equation of the second kind, which is solved by the method of sequential Picard approximation. The qualitative and structural properties of the fundamental solution are proved with the help of which the solution to the Cauchy problem is found in terms of the fundamental solution using the Lagrange formula.

Continuous Limit, Rational Solutions, and Asymptotic State Analysis for the Generalized Toda Lattice Equation Associated with 3 × 3 Lax Pair

Discrete integrable nonlinear differential difference equations (NDDEs) have various mathematical structures and properties, such as Lax pair, infinitely many conservation laws, Hamiltonian structure, and different kinds of symmetries, including Lie point symmetry, generalized Lie bäcklund symmetry, and master symmetry. Symmetry is one of the very effective methods used to study the exact solutions and integrability of NDDEs. The Toda lattice equation is a famous example of NDDEs, which may be used to simulate the motions of particles in lattices. In this paper, we investigated the generalized Toda lattice equation related to 3×3 matrix linear spectral problem. This discrete equation is related to continuous linear and nonlinear partial differential equations under the continuous limit. Based on the known 3×3 Lax pair of this equation, the discrete generalized (m,3N−m)-fold Darboux transformation was constructed for the first time and extended from the 2×2 Lax pair to the 3×3 Lax pair to give its rational solutions. Furthermore, the limit states of such rational solutions are discussed via the asymptotic analysis technique. Finally, the exponential–rational mixed solutions of the generalized Toda lattice equation are obtained in the form of determinants.

New Fundamental Results on the Continuous and Discrete Integro-Differential Equations

This work studies certain perturbed and un-perturbed nonlinear systems of continuous and discrete integro-delay differential equations (IDDEs). Using the Lyapunov–Krasovskii functional (LKF) method and the Lyapunov–Razumikhin method (LRM), uniform asymptotic stability (UAS), uniform stability (US), integrability and boundedness of solutions as well as exponential stability (ES) and instability of solutions are discussed. In this paper, five new theorems and a corollary are given and three numerical applications are provided with their simulations. With this work, we aim to make new contributions to the theory of the continuous and discrete integro-differential equations.

Numerical Solutions of the Hattendorff Differential Equation for Multi-State Markov Insurance Models

We use the representation of a continuous time Hattendorff differential equation and Matlab to compute 2σt(j), the solution of a backwards in time differential equation that describes the evolution of the variance of Lt(j), the loss at time t random variable for a multi-state Markovian process, given that the state at time t is j. We demonstrate this process by solving examples of several instances of a multi-state model which a practitioner can use as a guide to solve and analyze specific multi-state models. Numerical solutions to compute the variance 2σt(j) enable practitioners and academic researchers to test and simulate various state-space scenarios, with possible transitions to and from temporary disabilities, to permanent disabilities, to and from good health, and eventually to a deceased state. The solution method presented in this paper allows researchers and practitioners to easily compute the evolution of the variance of loss without having to resort to detailed programming.

Intelligent Production Monitoring with Continuous Deep Learning Models

Summary Monitoring of production rates is essential for reservoir management, history matching, and production optimization. Traditionally, such information is provided by multiphase flowmeters or test separators. The growth of the availability of data, combined with the rapid development of computational resources, enabled the inception of digital techniques, which estimate oil, gas, and water rates indirectly. This paper discusses the application of continuous deep learning models, capable of reproducing multiphase flow dynamics for production monitoring purposes. This technique combines the time evolution properties of a dynamical system and the ability of neural networks to quantitively describe poorly understood multiphase phenomena and can be considered as a hybrid solution between data-driven and mechanistic approaches. The continuous latent ordinary differential equation (Latent ODE) approach is compared with other known machine learning methods, such as linear regression, ensemble-based model, and recurrent neural network (RNN). In addition, the framework of virtual flowmetering (VFM) is analyzed from the point of multiphase measurement technology. In this work, the application of Latent ODEs for the problem of multiphase flow rate estimation is introduced. The considered example refers to a scenario in which the topside oil, gas, and water flow rates are estimated using the data from several downhole pressure sensors. The predictive capabilities of different types of machine learning and deep learning instruments are explored using simulated production data from a multiphase flow simulator. The results demonstrate the satisfactory performance of the continuous deep learning models in comparison with other machine learning methods in terms of accuracy, where the normalized root-mean-squared error (RMSE) and mean absolute error (MAE) of prediction below 5% were achieved. While Latent ODE demonstrates the significant time required to train the model, it outperforms other methods for irregularly sampled time series, which makes it especially attractive to forecast values of multiphase rates.

Existence and Uniqueness of Mild Solution for Fractional-Order Controlled Fuzzy Evolution Equation

In this article, we investigated the existence and uniqueness of mild solutions for fractional-order controlled fuzzy evolution equations with Caputo derivatives of the controlled fuzzy nonlinear evolution equation of the form 0 c D I γ x I = α x I + P I , x I + A I W I , I ∈ 0 , T , x I 0 = x 0 , in which γ ∈ 0 , 1 , E 1 is the fuzzy metric space and I = 0 , T is a real line interval. With the help of few conditions on functions P : I × E 1 × E 1 ⟶ E 1 , W I is control and it belongs to E 1 , A ∈ F I , L E 1 , and α stands for the highly continuous fuzzy differential equation generator. Finally, a few instances of fuzzy fractional differential equations are shown.

Analysis of finite-volume discrete adjoint fields for two-dimensional compressible Euler flows

This work deals with a number of questions related to the discrete and continuous adjoint fields associated with the compressible Euler equations and classical aerodynamic functions. The consistency of the discrete adjoint equations with the corresponding continuous adjoint partial differential equations is one of them. It has been established or at least discussed only for some numerical schemes and a contribution of this article is to give the adjoint consistency conditions for the 2-D Jameson-Schmidt-Turkel scheme in cell-centered finite-volume formulation. The consistency issue is also studied here from a new heuristic point of view by discretizing the continuous adjoint equation for the discrete flow and adjoint fields. Both points of view prove to provide useful information. Besides, it has been often noted that discrete or continuous inviscid lift and drag adjoint exhibit numerical divergence close to the wall and stagnation streamline for a wide range of subsonic and transonic flow conditions. This is analyzed here using the physical source term perturbation method introduced in reference [1]. With this point of view, the fourth physical source term of [1] appears to be the only one responsible for this behavior. It is also demonstrated that the numerical divergence of the adjoint variables corresponds to the response of the flow to the convected increment of stagnation pressure and diminution of entropy created at the source and the resulting change in lift and drag.

Simulation of the Riprap Movement Using the Continuous-Time Random Walking Method

During the implementation of the riprap project, the underwater migration process of the stones is quite uncertain because of its difficulty to observe. The process of stone transportation is discrete, which makes it unsuitable to be described by a continuous differential equation. Therefore, considering the distribution of stone jumping and waiting, a continuous-time random walk (CTRW) model is established. Based on the actual engineering data, five schemes simulate the one-dimensional motion of riprap underwater and further discuss the spatial distribution and particle size of the riprap. The results show that the CTRW model can effectively predict the riverbed elevation change behavior caused by the riprap project. The suitability of the model for the prediction of riprap movement decreases first and then increases with the increase in the selected width. This indicates that the randomness of the motion of the riprap causes the width of the observation zone to have a significant effect on the overall behavior of riprap movement. When the width is large enough, the influence of the randomness of the motion can be reduced by the average movement behavior within the observation zone. While the observation time of riprap movement is from a short to long time scale, the transport behavior changes from subdiffusion to normal diffusion behavior.

Oscillation analysis of solutions of non-zero continuous linear functional equations

As a mathematical model of mechanical and electronic oscillation, the study and analysis of the oscillation characteristics of the solution of the non-zero continuous linear functional equation are of great significance in theory and practice. In view of the oscillation characteristics of the solutions of the second and third order non-zero continuous functional equations, this paper puts forward a hypothesis, studies the oscillation and asymptotics of the non-zero continuous linear functional differential equations by using the generalized Riccati transformation and the integral average technique, and establishes some new sufficient conditions for the oscillation or convergence to zero of all solutions of the equations, so as to obtain a new theorem for the solutions of the non-zero continuous linear functional equations.

Application of Stochastic Differential Equation in Insurance Portfolio Construction Involving Leverage Function and Elasticity of Debt and Equity

Leverage effect specifies the functional relationship between stock returns and volatility. As stock price declines, volatility tends to rise. Thus, the variability in market prices of a company's stock has pervasive effect when measuring the level of leverage in the capital structure. The determination of portfolio value of an investor using the continuous time second order stochastic differential equation has major consequence on leverage function. Usually, structural stochastic value of leveraged firms treats company's portfolio as equity whose underlying instrument is the company's asset. In this paper, the objectives are to theoretically: (i) measure the value of insurance company's portfolio by second order stochastic differential equation, (ii) apply Ito's rule to obtain a value on its leverage function and, (iii) obtain the analytical correspondence between equity and volatility in a leveraged company through infinitesimal calculus. The stochastic second order differential equation of portfolio value under two arguments results in equilibrium position which provides the traded price of the derivative, furthermore the linear combination of first order derivative of volatilities with respect to equity and debt is vanishingly zero based on the underlying elasticity of stock volatilities. The resultant effect is that elasticity = of debt and equity cancel out and -15E, SOSE, 51

Traveling waves of a differential-difference diffusive Kermack-McKendrick epidemic model with age-structured protection phase

We consider a general class of diffusive Kermack-McKendrick SIR epidemic models with an age-structured protection phase with limited duration, for example due to vaccination or drugs with temporary immunity. A saturated incidence rate is also considered which is more realistic than the bilinear rate. The characteristics method reduces the model to a coupled system of a reaction-diffusion equation and a continuous difference equation with a time-delay and a nonlocal spatial term caused by individuals moving during their protection phase. We study the existence and non-existence of non-trivial traveling wave solutions. We get almost complete information on the threshold and the minimal wave speed that describes the transition between the existence and non-existence of non-trivial traveling waves that indicate whether the epidemic can spread or not. We discuss how model parameters, such as protection rates, affect the minimal wave speed. The difficulty of our model is to combine a reaction-diffusion system with a continuous difference equation. We deal with our problem mainly by using Schauder's fixed point theorem. More precisely, we reduce the problem of the existence of non-trivial traveling wave solutions to the existence of an admissible pair of upper and lower solutions.

Analysis Of Modeling Methods Of Computer Engineering Digital Devices

When designing new digital instruments and devices, there are a vast variety of reasons why the finally designed devices will malfunction. To decrease the number of such failures and to increase design accuracy, various methods and systems for modeling digital devices are used. In these systems, various methods for describing signals in models of designed devices can be used. In this case, three-valued, four-valued, ..., nine-valued, thirteen-valued, as well as analog signal descriptions can be applied. Increasing signal and element models complexity in digital devices allows designing more accurate models. However, when modeling digital devices, multi-valued alphabets do not allow to increase the accuracy of modeling and research of dynamic processes in devices. This is due to the impossibility of taking into account processes and interference caused by both stray capacitances and inductances between separate components of the devices and conductors connecting them, as well as dynamic processes, caused by external electromagnetic fields affecting the device designed. Describing such processes using continuous or K-valued differential equations improves the accuracy of digital devices modeling. Nevertheless, the problems of automated testing of these devices and the automation of determining their performance remain unsolved. For automated recognition of failures in the designed digital devices, neural networks, in particular, adaptive resonant theory (ART) neural networks, can be applied, since they have an important property, the ability to retrain when additional information about failures occurs. However, neural networks also have an essential drawback: they do not allow getting more than one solution, although with K-valued differential calculus of digital devices, this can occur quite often, which makes it possible to recognize failures that can be attributed simultaneously to two or more different classes of errors, and, therefore, to recognize failures, which can be simultaneously assigned to two or more different classes, and consequently, get more accurate results. In this regard, it is necessary to develop neural networks that could recognize two or more possible solutions (or types of failures). This would expand the field of failures automated detection in the designed digital devices and determine the performance accuracy. Figs.: 3. Refs: 12 titles.

Dynamics, stability and sustainable optimal control in wolf-moose systems

A wolf-moose predator-prey system is defined as a nonlinear continuous time differential equation system. The parameters are estimated via a discretized approximation and empirical data from Isle Royale, USA, representing a 61 year period. All parameter estimates confirm the hypotheses and all P-values are below 5%. Possible dynamic equilibria are determined as explicit general functions of the parameters and as numerical values based on the empirical data. General dynamic properties of the system are determined via phase -plane analysis and nonlinear simulation. The nonlinear system is also linearized close to the single equilibrium with two strictly positive populations. The explicit equations of the time path of the linearized system are compared to the nonlinear simulation. Both methods give almost identical solutions close to the equilibrium. Far from the equilibrium, the time paths of the two methods deviate considerably. The solution is a stable converging spiral (center) (unstable diverging spiral) in case the system equilibrium prey population is located at a higher (the same) (lower) level than the population level that gives maximum sustainable net production. Based on the empirically determined expected parameter values, the system is stable but converges very slowly, as a spiral, to the two-species equilibrium. The estimated standard deviations of the parameters can be used to determine the probability that the system solution is a center or an unstable diverging spiral. The optimal management of the wolf-moose system is also determined via sustainable optimal control. Moose hunting and adjustments of the wolf population based on different prices and values of sustainable population levels are derived. Optimal hunting and stock levels are determined and reported as explicit functions of all parameters and prices.

Higher Order Analysis on the Existence of Periodic Solutions in Continuous Differential Equations via Degree Theory

Recently, the higher order averaging method for studying periodic solutions of both Lipschitz differential equations and discontinuous piecewise smooth differential equations was developed in terms of Brouwer degree theory. Between the Lipschitz and the discontinuous piecewise smooth differential equations, there is a huge class of differential equations lacking in a higher order analysis on the existence of periodic solutions, namely the class of continuous non-Lipschitz differential equations. In this paper, based on the coincidence degree theory for nonlinear operator equations, we perform a higher order analysis of continuous (non-Lipschitz) perturbed differential equations and derive sufficient conditions for the existence of periodic solutions for such systems. We apply our results to study continuous (non-Lipschitz) higher order perturbations of a harmonic oscillator.