Ordinary Differential Equation Approach Research Articles

Applying Neural ODEs to Derive a Mechanism-Based Model for Characterizing Maturation-Related Serum Creatinine Dynamics in Preterm Newborns.

Serum creatinine in neonates follows complex dynamics due to maturation processes, most pronounced in the first few weeks of life. The development of a mechanism-based model describing complex dynamics requires high expertise in pharmacometric (PMX) modeling and substantial model development time. A recently published machine learning (ML) approach of low-dimensional neural ordinary differential equations (NODEs) is capable of modeling such data from newborns automatically. However, this efficient data-driven approach in itself does not result in a clinically interpretable model. In this work, an approach to deriving an interpretable model with reasonable PMX-type functions is presented. This "translation" was applied to derive a PMX model for serum creatinine in neonates considering maturation processes and covariates. The developed model was compared to a previously published mechanism-based PMX model whereas both models had similar mechanistic structures. The developed model was then utilized to simulate serum creatinine concentrations in the first few weeks of life considering different covariate values for gestational age and birth weight. The reference serum creatinine values derived from these simulations are consistent with observed serum creatinine values and previously published reference values. Thus, the presented NODE-based ML approach to model complex serum creatinine dynamics in newborns and derive interpretable, mathematical-statistical components similar to those in a conventional PMX model demonstrates a novel, viable approach to facilitate the modeling of complex dynamics in clinical settings and pediatric drug development.

Effectiveness of a One-fifth Hybrid Block Approach for Second Order Ordinary Differential Equations

Effectiveness of a One-fifth Hybrid Block Approach for Second Order Ordinary Differential Equations

Jost function analysis of elastic and inelastic nucleon–nucleon scattering with a new model potential

The analysis of nucleon–nucleon elastic and inelastic scattering data is presented by considering the combined interaction of nuclear and electromagnetic potentials of equal range. The exact analytical off-shell solutions of the nonrelativistic Schrödinger equation for the effective potential with the centrifugal term are investigated using ordinary differential equation approach. Numerical results of scattering phase shifts, differential cross-sections and Transition matrices are in sensible conformity with previous works.

A Modified Approach for Forecasting Relative Humidity in Indoor Premises

In order to forecast certain microclimatic parameters in a room, such as relative humidity, it is necessary to collect a large amount of data. For this reason, an electronic system for providing comfortable conditions in a room has been developed, which is based on the becoming largely popular Internet of Things (IoT) technology. On the basis of the obtained data, the future change of the relative humidity in a room can be forecasted, which aims to ensure the health and well-being of its inhabitants. It should be noted that working with large datasets requires more sophisticated methods for prediction, and for this reason, this paper presents results obtained from a modified ordinary differential equation approach for forecasting relative humidity. For the purposes of the study, a program code was developed in Matlab. The resulting predicted relative humidity values are comparable to actual measured values, which would help to improve indoor air quality and provide thermal comfort to occupants.

Consistent and fast inference in compartmental models of epidemics using Poisson Approximate Likelihoods

AbstractAddressing the challenge of scaling-up epidemiological inference to complex and heterogeneous models, we introduce Poisson approximate likelihood (PAL) methods. In contrast to the popular ordinary differential equation (ODE) approach to compartmental modelling, in which a large population limit is used to motivate a deterministic model, PALs are derived from approximate filtering equations for finite-population, stochastic compartmental models, and the large population limit drives consistency of maximum PAL estimators. Our theoretical results appear to be the first likelihood-based parameter estimation consistency results which apply to a broad class of partially observed stochastic compartmental models and address the large population limit. PALs are simple to implement, involving only elementary arithmetic operations and no tuning parameters, and fast to evaluate, requiring no simulation from the model and having computational cost independent of population size. Through examples we demonstrate how PALs can be used to: fit an age-structured model of influenza, taking advantage of automatic differentiation in Stan; compare over-dispersion mechanisms in a model of rotavirus by embedding PALs within sequential Monte Carlo; and evaluate the role of unit-specific parameters in a meta-population model of measles.

Neural ordinary differential equations with irregular and noisy data.

Measurement noise is an integral part of collecting data of a physical process. Thus, noise removal is necessary to draw conclusions from these data, and it often becomes essential to construct dynamical models using these data. We discuss a methodology to learn differential equation(s) using noisy and irregularly sampled measurements. In our methodology, the main innovation can be seen in the integration of deep neural networks with the neural ordinary differential equations (ODEs) approach. Precisely, we aim at learning a neural network that provides (approximately) an implicit representation of the data and an additional neural network that models the vector fields of the dependent variables. We combine these two networks by constraints using neural ODEs. The proposed framework to learn a model describing the vector field is highly effective under noisy measurements. The approach can handle scenarios where dependent variables are unavailable at the same temporal grid. Moreover, a particular structure, e.g. second order with respect to time, can easily be incorporated. We demonstrate the effectiveness of the proposed method for learning models using data obtained from various differential equations and present a comparison with the neural ODE method that does not make any special treatment to noise. Additionally, we discuss an ensemble approach to improve the performance of the proposed approach further.

Vegetation responses to variations in climate: A combined ordinary differential equation and sequential Monte Carlo estimation approach

Vegetation responses to variation in climate are a current research priority in the context of accelerated shifts generated by climate change. However, the interactions between environmental and biological factors still represent one of the largest uncertainties in projections of future scenarios, since the relationship between drivers and ecosystem responses has a complex and nonlinear nature. We aimed to develop a model to study the vegetation's primary productivity dynamic response to temporal variations in climatic conditions as measured by rainfall, temperature and radiation. Thus, we propose a new way to estimate the vegetation response to climate via a non-autonomous version of a classical growth curve, with a time-varying growth rate and carrying capacity parameters according to climate variables. With a Sequential Monte Carlo Estimation to account for complexities in the climate-vegetation relationship to minimize the number of parameters. The model was applied to six key sites identified in a previous study, consisting of different arid and semiarid rangelands from North Patagonia, Argentina. For each site, we selected the time series of MODIS NDVI, and climate data from ERA5 Copernicus hourly reanalysis from 2000 to 2021. After calculating the time series of the a posteriori distribution of parameters, we analyzed the explained capacity of the model in terms of the linear coefficient of determination and the parameters distribution variation. Results showed that most rangelands recorded changes in their sensitivity over time to climatic factors, but vegetation responses were heterogeneous and influenced by different drivers. Differences in this climate-vegetation relationship were recorded among different cases: (1) a marginal and decreasing sensitivity to temperature and radiation, respectively, but a high sensitivity to water availability; (2) high and increasing sensitivity to temperature and water availability, respectively; and (3) a case with an abrupt shift in vegetation dynamics driven by a progressively decreasing sensitivity to water availability, without any changes in the sensitivity either to temperature or radiation. Finally, we also found that the time scale, in which the ecosystem integrated the rainfall phenomenon in terms of the width of the window function used to convolve the rainfall series into a water availability variable, was also variable in time. This approach allows us to estimate the connection degree between ecosystem productivity and climatic variables. The capacity of the model to identify changes over time in the vegetation-climate relationship might inform decision-makers about ecological transitions and the differential impact of climatic drivers on ecosystems.

Stochasticity of disease spreading derived from the microscopic simulation approach for various physical contact networks

COVID-19 has emphasized that a precise prediction of a disease spreading is one of the most pressing and crucial issues from a social standpoint. Although an ordinary differential equation (ODE) approach has been well established, stochastic spreading features might be hard to capture accurately. Perhaps, the most important factors adding such stochasticity are the effect of the underlying networks indicating physical contacts among individuals. The multi-agent simulation (MAS) approach works effectively to quantify the stochasticity. We systematically investigate the stochastic features of epidemic spreading on homogeneous and heterogeneous networks. The study quantitatively elucidates that a strong microscopic locality observed in one- and two-dimensional regular graphs, such as ring and lattice, leads to wide stochastic deviations in the final epidemic size (FES). The ensemble average of FES observed in this case shows substantial discrepancies with the results of ODE based mean-field approach. Unlike the regular graphs, results on heterogeneous networks, such as Erdős–Rényi random or scale-free, show less stochastic variations in FES. Also, the ensemble average of FES in heterogeneous networks seems closer to that of the mean-field result. Although the use of spatial structure is common in epidemic modeling, such fundamental results have not been well-recognized in literature. The stochastic outcomes brought by our MAS approach may lead to some implications when the authority designs social provisions to mitigate a pandemic of un-experienced infectious disease like COVID-19.

Analysis of a mathematical model of the population dynamics of drug addiction among youths

Drug addiction amongst youths has been identified as one of the global menace that has affected negatively the development and productivity of young people which has come to stay from time pass to the present day and has resulted to various behavioural disorder such as criminality, psychiatric issues and death. In this paper, a 5 sub-class model is formulated to analyze the population dynamics of drug addiction among youths and the effect of addiction on their development and productivity towards contributing their quota to the human population. The model was developed using ordinary differential equations approach. It was determined that the basic reproduction number which is a vital tool for measuring the impact of drug addiction in human was obtained using the next generation operator method. Stability analysis was carried out and it was discovered that drug-free equilibrium exists and it was shown to be locally and globally asymptotically stable whenever effective basic reproduction number is less than one; and unstable whenever effective basic reproduction number is greater than one. It was discovered that drug abuse persisted among youths and are both locally and globally asymptotically stable whenever effective basic reproduction number is greater than one and unstable whenever effective basic reproduction number is less than one.

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.

Qualitative properties of different numerical methods for the inhomogeneous geometric Brownian motion

We provide a comparative analysis of qualitative features of different numerical methods for the inhomogeneous geometric Brownian motion (IGBM). The limit distribution of the IGBM exists, its conditional and asymptotic mean and variance are known and the process can be characterised according to Feller’s boundary classification. We compare the frequently used Euler–Maruyama and Milstein methods, two Lie–Trotter and two Strang splitting schemes and two methods based on the ordinary differential equation (ODE) approach, namely the classical Wong–Zakai approximation and the recently proposed log-ODE scheme. First, we prove that, in contrast to the Euler–Maruyama and Milstein schemes, the splitting and ODE schemes preserve the boundary properties of the process, independently of the choice of the time discretisation step. Second, we prove that the limit distribution of the splitting and ODE methods exists for all stepsize values and parameters. Third, we derive closed-form expressions for the conditional and asymptotic means and variances of all considered schemes and analyse the resulting biases. While the Euler–Maruyama and Milstein schemes are the only methods which may have an asymptotically unbiased mean, the splitting and ODE schemes perform better in terms of variance preservation. The Strang schemes outperform the Lie–Trotter splittings, and the log-ODE scheme the classical ODE method. The mean and variance biases of the log-ODE scheme are very small for many relevant parameter settings. However, in some situations the two derived Strang splittings may be a better alternative, one of them requiring considerably less computational effort than the log-ODE method. The proposed analysis may be carried out in a similar fashion on other numerical methods and stochastic differential equations with comparable features.

ChemNODE: A neural ordinary differential equations framework for efficient chemical kinetic solvers

Solving for detailed chemical kinetics remains one of the major bottlenecks for computational fluid dynamics simulations of reacting flows using a finite-rate-chemistry approach. This has motivated the use of neural networks to predict stiff chemical source terms as functions of the thermochemical state of the combustion system. However, due to the nonlinearities and multi-scale nature of combustion, the predicted solution often diverges from the true solution when these machine learning models are coupled with a computational fluid dynamics solver. This is because these approaches minimize the error during training without guaranteeing successful integration with ordinary differential equation solvers. In the present work, a novel neural ordinary differential equations approach to modeling chemical kinetics, termed as ChemNODE, is developed. In this machine learning framework, the chemical source terms predicted by the neural networks are integrated during training, and by computing the required derivatives, the neural network weights are adjusted accordingly to minimize the difference between the predicted and ground-truth solution. A proof-of-concept study is performed with ChemNODE for homogeneous autoignition of hydrogen-air mixture over a range of composition and thermodynamic conditions. It is shown that ChemNODE accurately captures the chemical kinetic behavior and reproduces the results obtained using the detailed kinetic mechanism at a fraction of the computational cost.

The Green's Function Method Solutions to Solve of nth-order Linear Differential Equations

في هذا المقال البحثي هذا، نقدم مقاربة دالة جرين للمعادلات التفاضلية العادية المتضمنة شروطاً ابتدائية وكذلك المتضمنة شروطاً حدية وتم تمثيل المعادلات التفاضلية بمعادلات تكاملية يوفر النص أساسًا نظريًا كافيًا لفهم طريقة دالة جرين، والتي تُستخدم لحل المشكلات ; القيمة الأولية والحدودية التي تتضمن المعادلات التفاضلية العادية والمعادلات التفاضلية الجزئية الخطية. النتيجة الرئيسية هي بناء حزمة ماثيمتكا صالحة لحساب التعبير الصريح لدالة جرين المتعلقة بمشكلة القيمة الحدية للنقطتين في المعادلة (2. 3)، حيث يكون للمؤثر الخطي من الدرجة النونية المحدد في المعادلة (2. 1) ذات المعاملات الثابتة.

Observer-Based Decentralized Predictor Control for Large-Scale Interconnected Systems With Large Delays

This article proposes a decentralized predictor-based output feedback to compensate for large delays in large-scale interconnected systems under the premise that the interconnections between subsystems are not strong. The full state of each subsystem is assumed to be unmeasurable and the observer-based output feedback is employed. Two methods are used to tackle the large delays: the backstepping-based partial differential equation (PDE) approach and the reduction-based ordinary differential equation (ODE) approach. The PDE method is used for continuous-time control and manages with larger delays, whereas the ODE method is applicable to sampled-data implementation under discrete-time measurement.

A sparse FFT approach for ODE with random coefficients

The paper presents a general strategy to solve ordinary differential equations (ODE), where some coefficient depend on the spatial variable and on additional random variables. The approach is based on the application of a recently developed dimension-incremental sparse fast Fourier transform. Since such algorithms require periodic signals, we discuss periodization strategies and associated necessary deperiodization modifications within the occurring solution steps. The computed approximate solutions of the ODE depend on the spatial variable and on the random variables as well. Certainly, one of the crucial challenges of the high-dimensional approximation process is to rate the influence of each variable on the solution as well as the determination of the relations and couplings within the set of variables. The suggested approach meets these challenges in a full automatic manner with reasonable computational costs, i.e., in contrast to already existing approaches, one does not need to seriously restrict the used set of ansatz functions in advance.

On- and off-shell Jost functions for the Manning-Rosen potential

The s-wave on-shell Jost solution for the Manning-Rosen potential is constructed via the ordinary differential equation approach to the problem. The corresponding off-shell Jost function is calculated by using the integral representation of the off-shell Jost function in a single channel scattering. The half off-shell T- matrix is obtained from the knowledge of the off-shell Jost function. Our constructed expressions for the Jost functions are applied to compute T-matrix, quasi-elastic phase shifts and the scattering phase shifts for the nuclear systems.

Predictor methods for decentralized control of large-scale systems with input delays

This paper develops a decentralized predictor-based control for large-scale systems with large input delays under the premise that the interconnections between subsystems are not strong. The local controller operates independently. Given any large delays, the predictor which exponentially stabilizes each uncoupled system, will stabilize the coupled one provided that the coupling is not solid. We propose two methods for the delay compensation: the backstepping-based partial differential equation (PDE) approach and the reduction-based ordinary differential equation (ODE) approach. We present decentralized Lyapunov-based analysis under the two predictor methods. It appears that the first predictor method leads to simpler conditions and manages with larger delays, whereas the second is easily applied to decentralized asynchronous sampled-data implementation, both under continuous and under discrete-time measurements. Through a benchmark example of two coupled cart–pendulum systems, the proposed methods are demonstrated to be effective when the input delays are too large for the system to be stabilized without a predictor.

An ordinary differential equation approach for nonlinear programming and nonlinear complementary problem

We consider an ordinary differential equation (ODE) approach for solving non- linear programming (NLP) and nonlinear complementary problem (NCP). The Karush- Kuhn Tucker (KKT) optimality conditions can be converted to NCP. Based on the Fischer-Burmeister (FB) function and the Natural-Residual (NR) function are obtained the new NCP-functions. A special technique is employed to reformulate of the NCP as the system of nonlinear algebraic equations (NAEs) later on reformulated once more by force. of an original time-like function into an ODE. Afterwards, a group preserving scheme (GPS) is a package to reformulate an ODE into the new numerical equation in a way the ODEs system is designed into a nonlinear dynamical system (NDS) and is continued to a discovery the new numerical equation through activating the Lorentz group SO0(n, 1) and its Lie algebra so(n, 1). Lastly, the fictitious time integration method (FTIM) is utilized into this new numerical equation to determine an approximation solution at the numerical experiments area.

Null space gradient flows for constrained optimization with applications to shape optimization

The purpose of this article is to introduce a gradient-flow algorithm for solving equality and inequality constrained optimization problems, which is particularly suited for shape optimization applications. We rely on a variant of the Ordinary Differential Equation (ODE) approach proposed by Yamashita (Math. Program. 18 (1980) 155–168) for equality constrained problems: the search direction is a combination of a null space step and a range space step, aiming to decrease the value of the minimized objective function and the violation of the constraints, respectively. Our first contribution is to propose an extension of this ODE approach to optimization problems featuring both equality and inequality constraints. In the literature, a common practice consists in reducing inequality constraints to equality constraints by the introduction of additional slack variables. Here, we rather solve their local combinatorial character by computing the projection of the gradient of the objective function onto the cone of feasible directions. This is achieved by solving a dual quadratic programming subproblem whose size equals the number of active or violated constraints. The solution to this problem allows to identify the inequality constraints to which the optimization trajectory should remain tangent. Our second contribution is a formulation of our gradient flow in the context of – infinite-dimensional – Hilbert spaces, and of even more general optimization sets such as sets of shapes, as it occurs in shape optimization within the framework of Hadamard’s boundary variation method. The cornerstone of this formulation is the classical operation of extension and regularization of shape derivatives. The numerical efficiency and ease of implementation of our algorithm are demonstrated on realistic shape optimization problems.

HybridModels: An R Package for the Stochastic Simulation of Disease Spreading in Dynamic Networks

Disease spreading simulations are traditionally performed using coupled differential equations. However, in the setting of metapopulations, most of the solutions provided by this method do not account for the dynamic topography of subpopulations. Conversely, the alternative approach of individual-based modeling (IBM) may add computational cost and complexity. Hybrid models allow for the study of disease spreading because they combine both aforementioned approaches by separating them across different scales: a local scale that addresses subpopulation dynamics using coupled differential equations and a global scale that addresses the contact between these subpopulations using IBM. We present a simple way of simulating the spread of disease in dynamic networks using the high-level statistical computational language R and the hybridModels package. We built four examples using disease spread models at the local scale in several different networks: an animal movement network; a three-node network, whose model solution using a stochastic simulation algorithm is compared with the ordinary differential equations approach; the commuting of individuals between patches, which we compare with the permanent migration of individuals; and the commuting of individuals within the metropolitan area of São Paulo.