Showing papers on "Mathematical model published in 2018"

Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems

Abstract: The process of transforming observed data into predictive mathematical models of the physical world has always been paramount in science and engineering. Although data is currently being collected at an ever-increasing pace, devising meaningful models out of such observations in an automated fashion still remains an open problem. In this work, we put forth a machine learning approach for identifying nonlinear dynamical systems from data. Specifically, we blend classical tools from numerical analysis, namely the multi-step time-stepping schemes, with powerful nonlinear function approximators, namely deep neural networks, to distill the mechanisms that govern the evolution of a given data-set. We test the effectiveness of our approach for several benchmark problems involving the identification of complex, nonlinear and chaotic dynamics, and we demonstrate how this allows us to accurately learn the dynamics, forecast future states, and identify basins of attraction. In particular, we study the Lorenz system, the fluid flow behind a cylinder, the Hopf bifurcation, and the Glycoltic oscillator model as an example of complicated nonlinear dynamics typical of biological systems.

Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations

Abstract: A long-standing problem at the interface of artificial intelligence and applied mathematics is to devise an algorithm capable of achieving human level or even superhuman proficiency in transforming observed data into predictive mathematical models of the physical world. In the current era of abundance of data and advanced machine learning capabilities, the natural question arises: How can we automatically uncover the underlying laws of physics from high-dimensional data generated from experiments? In this work, we put forth a deep learning approach for discovering nonlinear partial differential equations from scattered and potentially noisy observations in space and time. Specifically, we approximate the unknown solution as well as the nonlinear dynamics by two deep neural networks. The first network acts as a prior on the unknown solution and essentially enables us to avoid numerical differentiations which are inherently ill-conditioned and unstable. The second network represents the nonlinear dynamics and helps us distill the mechanisms that govern the evolution of a given spatiotemporal data-set. We test the effectiveness of our approach for several benchmark problems spanning a number of scientific domains and demonstrate how the proposed framework can help us accurately learn the underlying dynamics and forecast future states of the system. In particular, we study the Burgers', Korteweg-de Vries (KdV), Kuramoto-Sivashinsky, nonlinear Schr\"{o}dinger, and Navier-Stokes equations.

A hierarchical least squares identification algorithm for Hammerstein nonlinear systems using the key term separation

Abstract: Mathematical models are basic for designing controller and system identification is the theory and methods for establishing the mathematical models of practical systems. This paper considers the parameter identification for Hammerstein controlled autoregressive systems. Using the key term separation technique to express the system output as a linear combination of the system parameters, the system is decomposed into several subsystems with fewer variables, and then a hierarchical least squares (HLS) algorithm is developed for estimating all parameters involving in the subsystems. The HLS algorithm requires less computation than the recursive least squares algorithm. The computational efficiency comparison and simulation results both confirm the effectiveness of the proposed algorithms.

Models for fast modelling of district heating and cooling networks

Abstract: Within the framework of AMBASSADOR, a collaborative project funded by European Commission under FP7, a Modelica® library for the modelling of thermal-energy transport in district heating systems has been developed. This library comprises detailed models of the distribution and consumption components commonly found in district heating systems. In this paper, the detailed models are discussed, along with their validation against Apros® and IDA-ICE® Software. The results show that, although most of the models perform similarly, they do not equally reproduce the dynamics. Some of the limitations detected from the simulation results are currently being solved in new developments within the EU-funded INDIGO project. Furthermore, with the aim of avoiding problems derived from the simulation of large models, the methodology for developing reduced mathematical models, implemented in Simulink®, is also presented in this research work. This methodology includes identifying the relevant includes identifying the relevant model dynamics. During the procedure, additional information about the models can be obtained. For instance, the mass flow rate and the temperature can be assumed to be decoupled, without losing accuracy in the case of the distribution pipe model.

Stochastic Geometry Methods for Modeling Automotive Radar Interference

Abstract: As the use of automotive radar increases, performance limitations associated with radar-to-radar interference will become more significant. In this paper, we employ tools from stochastic geometry to characterize the statistics of radar interference. Specifically, using two different models for the spatial distributions of vehicles, namely, a Poisson point process and a Bernoulli lattice process, we calculate for each case the interference statistics and obtain analytical expressions for the probability of successful range estimation. This paper shows that the regularity of the geometrical model appears to have limited effect on the interference statistics, and so it is possible to obtain tractable tight bounds for the worst case performance. A technique is proposed for designing the duty cycle for the random spectrum access, which optimizes the total performance. This analytical framework is verified using Monte Carlo simulations.

Design analysis of solar parabolic trough thermal collectors

Abstract: This paper presents a review of the design parameters, mathematical techniques and simulations used in the design of parabolic trough solar systems, along with a review on their applications. Recent studies that analyze the deployment of solar parabolic trough collectors (SPTC) in different countries and the operational SPTC plants are also presented and discussed. The paper also discusses the different kinds of software and test methods of solar collectors developed since 1981 which can be distinguished by their particular mathematical models or tracking techniques. In particular, since the mathematical models are especially required for the design, analysis, testing and validation of the systems results as they provide an approximation of the dynamic behavior of the physical properties of the system, they are discussed in depth. The mathematical models allow the calculation of different parameters of the solar parabolic trough system, the angle of inclination of the collecting surface and the forces acting on the system. The validity and experimental validation of the major mathematical models on practical solar parabolic trough concentrators, receivers and other components of different dimension are also reviewed. The paper showed the optical efficiency values are close to 63% and the theoretical peak optical efficiency reached 75%.

Mathematical Analysis of Chemical Reaction Systems

Abstract: The use of mathematical methods for the analysis of chemical reaction systems has a very long history, and involves many types of models: deterministic versus stochastic, continuous versus discrete, and homogeneous versus spatially distributed. Here we focus on mathematical models based on deterministic mass-action kinetics. These models are systems of coupled nonlinear differential equations on the positive orthant. We explain how mathematical properties of the solutions of mass-action systems are strongly related to key properties of the networks of chemical reactions that generate them, such as specific versions of reversibility and feedback interactions.

Mathematical Analysis of Chemical Reaction Systems

Abstract: The use of mathematical methods for the analysis of chemical reaction systems has a very long history, and involves many types of models: deterministic versus stochastic, continuous versus discrete, and homogeneous versus spatially distributed. Here we focus on mathematical models based on deterministic mass-action kinetics. These models are systems of coupled nonlinear differential equations on the positive orthant. We explain how mathematical properties of the solutions of mass-action systems are strongly related to key properties of the networks of chemical reactions that generate them, such as specific versions of reversibility and feedback interactions.

A Comparative Study of Different Curve Fitting Algorithms in Artificial Neural Network using Housing Dataset

Abstract: Representing a given dataset with a mathematical model is a very useful tool in many engineering applications. Several techniques exist to evolve a mathematical model for a given data set. Non-Linear regression being the most often used. The curves can be generated from these mathematical models, which provide a visualization of how that model fits the data. In this paper, three algorithms to train artificial neural networks are used to develop a good fit for the data. The three algorithms are Levenberg-Marquardt, Bayesian Regularization, and Scaled Conjugate Gradient. These algorithms were applied to the housing data set. The comparative performance of these algorithms was compared using Mean Square Error (MSE), which represents the best curve fitting for these data sets. The mean squared error was computed for each algorithm. Levenberg-Marquardt had MSE of 7.0902. Scaled Conjugate Gradient at 15.2932, and Bayesian Regularization at 5.3480. It was found that Bayesian Regularization gave the best accuracy at 96.78%, followed by Levenberg-Marquardt at 94.53% and Scaled Conjugate Gradient at 90.51%.

Bayesian uncertainty analysis for complex systems biology models: emulation, global parameter searches and evaluation of gene functions

Abstract: Many mathematical models have now been employed across every area of systems biology. These models increasingly involve large numbers of unknown parameters, have complex structure which can result in substantial evaluation time relative to the needs of the analysis, and need to be compared to observed data of various forms. The correct analysis of such models usually requires a global parameter search, over a high dimensional parameter space, that incorporates and respects the most important sources of uncertainty. This can be an extremely difficult task, but it is essential for any meaningful inference or prediction to be made about any biological system. It hence represents a fundamental challenge for the whole of systems biology. Bayesian statistical methodology for the uncertainty analysis of complex models is introduced, which is designed to address the high dimensional global parameter search problem. Bayesian emulators that mimic the systems biology model but which are extremely fast to evaluate are embeded within an iterative history match: an efficient method to search high dimensional spaces within a more formal statistical setting, while incorporating major sources of uncertainty. The approach is demonstrated via application to a model of hormonal crosstalk in Arabidopsis root development, which has 32 rate parameters, for which we identify the sets of rate parameter values that lead to acceptable matches between model output and observed trend data. The multiple insights into the model’s structure that this analysis provides are discussed. The methodology is applied to a second related model, and the biological consequences of the resulting comparison, including the evaluation of gene functions, are described. Bayesian uncertainty analysis for complex models using both emulators and history matching is shown to be a powerful technique that can greatly aid the study of a large class of systems biology models. It both provides insight into model behaviour and identifies the sets of rate parameters of interest.

Scientific and Methodological Approach for the Identification of Mathematical Models of Mechanical Systems by Using Artificial Neural Networks

Abstract: The article is aimed at developing the scientific and methodological approach of using artificial neural networks (ANN) for solving applied problems in the field of mechanical engineering. This approach is based on the comprehensive implementation of ANN with the modern methods of numerical analysis (e.g., the finite element method) and analytical methods of the research with the use of mathematical modeling of the dynamic state for mechanical systems. Conceptual schemes for the implementation of the abovementioned approach are proposed for solving a number of interdisciplinary problems, such as investigation of the dynamics for rotary machines and hydroaeroelastic interaction of gas-liquid mixtures with deformable structural elements, as well as the dynamic analysis of fixtures. The main advantages of the proposed approach in comparison with the traditional regression analysis are the ability to learn and improve the ANN architecture, and to solve nonlinear problems of the parameters’ identification for mathematical models by using data of the results of physical experiments and numerical simulations. This approach allows refining parameters of the linear and nonlinear mathematical models describing the complicated mechanical and hydro-mechanical interactions under the impossibility of determination of an absolutely precise solution of the equations describing the process, as well as the incompleteness of the initial data.

Development and Application of the Fourier Method for the Numerical Solution of Ito Stochastic Differential Equations

Abstract: This paper is devoted to the development and application of the Fourier method to the numerical solution of Ito stochastic differential equations. Fourier series are widely used in various fields of applied mathematics and physics. However, the method of Fourier series as applied to the numerical solution of stochastic differential equations, which are proper mathematical models of various dynamic systems affected by random disturbances, has not been adequately studied. This paper partially fills this gap.

Complex acoustic gravity wave behaviors to some mathematical models arising in fluid dynamics and nonlinear dispersive media

Abstract: This study acquires the wave solutions of the two well-known nonlinear models, namely; the modified Benjamin–Bona–Mahony and the coupled Klein–Gordon equations. The modified Benjamin–Bona–Mahony is a nonlinear model that describes the long surface gravity waves of small amplitude and the coupled Klein–Gordon equation describes the quantized version of the relativistic energy–momentum relation. We successfully acquire some new solutions to these models such as kink-type and soliton solutions in complex hyperbolic functions form. We plot the 3D and 2D surface of the all the obtained solutions in this study. The mathematical approach used in this study is the sine-Gordon expansion method.

Geometry characteristics modeling and process optimization in coaxial laser inside wire cladding

Abstract: Coaxial laser inside wire cladding method is very promising as it has a very high efficiency and a consistent interaction between the laser and wire. In this paper, the energy and mass conservation law, and the regression algorithm are used together for establishing the mathematical models to study the relationship between the layer geometry characteristics (width, height and cross section area) and process parameters (laser power, scanning velocity and wire feeding speed). At the selected parameter ranges, the predicted values from the models are compared with the experimental measured results, and there is minor error existing, but they reflect the same regularity. From the models, it is seen the width of the cladding layer is proportional to both the laser power and wire feeding speed, while it firstly increases and then decreases with the increasing of the scanning velocity. The height of the cladding layer is proportional to the scanning velocity and feeding speed and inversely proportional to the laser power. The cross section area increases with the increasing of feeding speed and decreasing of scanning velocity. By using the mathematical models, the geometry characteristics of the cladding layer can be predicted by the known process parameters. Conversely, the process parameters can be calculated by the targeted geometry characteristics. The models are also suitable for multi-layer forming process. By using the optimized process parameters calculated from the models, a 45 mm-high thin-wall part is formed with smooth side surfaces.

Mathematical models for magnetic particle imaging

Abstract: Magnetic particle imaging (MPI) is a relatively new imaging modality. The nonlinear magnetization behavior of nanoparticles in an applied magnetic field is employed to reconstruct an image of the concentration of nanoparticles. Finding a sufficiently accurate model for the particle behavior is still an open problem. For this reason the reconstruction is still computed using a measured forward operator which is obtained in a time-consuming calibration process. The state of the art model used for the imaging methodology and first model-based reconstructions relies on strong model simplifications which turned out to cause too large modeling errors. Neglecting particle-particle interactions, the forward operator can be expressed by a Fredholm integral operator of the first kind describing the inverse problem. In this article we give an overview of relevant mathematical models which have not been investigated theoretically in the context of inverse problems yet. We consider deterministic models which are based on the physical behavior including relaxation mechanisms affecting the particle magnetization. The behavior of the models is illustrated with numerical simulations for monodisperse as well as polydisperse tracer. We further motivate linear and nonlinear problems beyond the solely concentration reconstruction related to applications. This model survey complements a recent topical review on MPI [30] and builds the basis for upcoming theoretical as well as empirical investigations.

Real-time simulation of large-scale HTS systems: multi-scale and homogeneous models using T-A formulation

Abstract: The emergence of second-generation high temperature superconducting tapes has favored the development of large-scale superconductor systems. The mathematical models capable of estimating electromagnetic quantities in superconductors have evolved from simple analytical models to complex numerical models. The available analytical models are limited to the analysis of single wires or infinite arrays that, in general, do not represent real devices in real applications. The numerical models based on finite element method using the H formulation of the Maxwells equations are useful for the analysis of medium-size systems, but their application in large-scale systems is problematic due to the excessive computational cost in terms of memory and computation time. Then it is necessary to devise new strategies to make the computation more efficient. The homogenization and the multi-scale methods have successfully simplified the description of the systems allowing the study of large-scale systems. Also, efficient calculations have been achieved using the T-A formulation. In the present work, we propose a series of adaptations to the multi-scale and homogenization methods so that they can be efficiently used in conjunction with the T-A formulation to compute the distribution of current density and hysteresis losses in the superconducting layer of superconducting tapes. The computation time and the amount of memory are substantially reduced up to a point that it is possible to achieve real-time simulations of HTS large-scale systems under slow ramping cycles of practical importance on personal computers.

Representing model inadequacy: A stochastic operator approach

Abstract: Mathematical models of physical systems are subject to many uncertainties such as measurement errors and uncertain initial and boundary conditions. After accounting for these uncertainties, it is o...

Oblique Impact of Multi-Flexible-Link Systems:

Abstract: The main goal of this paper is to present an automatic approach for the dynamic modeling of the oblique impact of a multi-flexible-link robotic manipulator. The behavior of a multi-flexible-link system confined inside a closed environment with curved walls can be completely expressed by two distinct mathematical models. A set of differential equations is employed to model the system when it has no contact with the curved walls (Flight phase); and a set of algebraic equations is used whenever it collides with the confining surfaces (Impact phase). In this article, in addition to the Assumed Mode Method (AMM), the Euler-Bernoulli Beam Theory (EBBT), and the Newton’s kinematic impact law, the Gibbs-Appell (G-A) formulation has been employed to derive the governing equations in both phases. Also, instead of using 3 × 3 rotational matrices, which involves lengthy kinematic and dynamic formulations for deriving the governing equations, 4 × 4 transformation matrices have been used. Moreover, for the systematic m...

Simulation and Comparison of Mathematical Models of PV Cells with Growing Levels of Complexity

Abstract: The amount of energy generated from a photovoltaic installation depends mainly on two factors—the temperature and solar irradiance. Numerous maximum power point tracking (MPPT) techniques have been developed for photovoltaic systems. The challenge is what method to employ in order to obtain optimum operating points (voltage and current) automatically at the maximum photovoltaic output power in most conditions. This paper is focused on the structural analysis of mathematical models of PV cells with growing levels of complexity. The main objective is to simulate and compare the characteristic current-voltage (I-V) and power-voltage (P-V) curves of equivalent circuits of the ideal PV cell model and, with one and with two diodes, that is, equivalent circuits with five and seven parameters. The contribution of each parameter is analyzed in the particular context of a given model and then generalized through comparison to a more complex model. In this study the numerical simulation of the models is used intensively and extensively. The approach utilized to model the equivalent circuits permits an adequate simulation of the photovoltaic array systems by considering the compromise between the complexity and accuracy. By utilizing the Newton–Raphson method the studied models are then employed through the use of Matlab/Simulink. Finally, this study concludes with an analysis and comparison of the evolution of maximum power observed in the models.

Multidimensional Image Models and Processing

Abstract: The problems of developing mathematical models and statistical algorithms for processing of multidimensional images and their sequences are presented in this chapter. Different types of random fields are taken for the basic mathematical image model. This implies two main problems associated with image modeling, namely, model analysis and synthesis. The main attention is paid to the correlation aspect, i.e. evaluation of the correlation function of a random field generated by a given model and, vice versa, development of a model generating a random field with a predetermined correlation function. For this purpose, new models (tensor and wave) and new versions of autoregressive models (with multiple roots) are suggested. The problems of image simulation on the curved surfaces are considered. The suggested models are used to synthesize the algorithms of multidimensional image processing and their sequences. The tensor filtration of imaging sequences and recursive filtration of multidimensional images, as well as the asymptotic characteristics of efficiency of random field filtration on grids of arbitrary dimension are suggested. The problem of object and anomaly detection on the background of interfering images is considered for the images of any dimension, e.g. for multi-zone data. It is shown that four equivalent forms of the optimal decision rule, which reflect various aspects of detection procedure, exist. Potential efficiency of anomaly detection is analyzed. The problems of alignment and estimation of parameters for interframe geometric image transformations are considered for multidimensional image sequences. A tensor procedure of simultaneous filtration of multidimensional image sequence and their interframe displacements are suggested. A method based on a fixed point of a complex geometric image transformation was investigated in order to evaluate large interframe displacements. Options for adaptive image processing algorithms are also discussed in this chapter. In this context, pseudo-gradient procedures are taken as a basis, as they do not require preliminary evaluation of any characteristics of the processed data. This allows to develop the high-performance algorithms that can be implemented in real-time systems.

Parameter identification of a nonlinear model: replicating the motion response of an autonomous underwater vehicle for dynamic environments

Abstract: This study presents a system identification algorithm to determine the linear and nonlinear parameters of an autonomous underwater vehicle (AUV) motion response prediction mathematical model, utilising the recursive least squares optimisation method. The key objective of the model, which relies solely on propeller thrust, gyro measurements and parameters representing the vehicle hydrodynamic, hydrostatic and mass properties, is to calculate the linear velocities of the AUV in the x, y and z directions. Initially, a baseline mathematical model that represents the dynamics of a Gavia class AUV in a calm water environment was developed. Using a novel technique developed in this study, the parameters within the baseline model were calibrated to provide the motion response in different environmental conditions by conducting a calibration mission in the new environment. The accuracy of the velocity measurements from the calibrated model was substantially greater than those from the baseline model for the tested scenarios with a minimum velocity prediction improvement of 50%. The determined velocities will be used to aid the inertial navigation system (INS) position estimate using a Kalman filter data fusion algorithm when external aiding is unavailable. When an INS is not externally aided or constrained by a mathematical model such as that presented here, the positioning uncertainty can be more than 4% of the distance travelled (assuming a forward speed of 1.6 m s $$^{-1})$$ . The calibrated model is able to compute the position of the AUV within an uncertainty range of around 1.5% of the distance travelled, significantly improving the localisation accuracy.