Showing papers on "Mathematical model published in 2020"

Systems biology informed deep learning for inferring parameters and hidden dynamics.

Abstract: Mathematical models of biological reactions at the system-level lead to a set of ordinary differential equations with many unknown parameters that need to be inferred using relatively few experimental measurements. Having a reliable and robust algorithm for parameter inference and prediction of the hidden dynamics has been one of the core subjects in systems biology, and is the focus of this study. We have developed a new systems-biology-informed deep learning algorithm that incorporates the system of ordinary differential equations into the neural networks. Enforcing these equations effectively adds constraints to the optimization procedure that manifests itself as an imposed structure on the observational data. Using few scattered and noisy measurements, we are able to infer the dynamics of unobserved species, external forcing, and the unknown model parameters. We have successfully tested the algorithm for three different benchmark problems.

PySINDy: A Python package for the sparse identification of nonlinear dynamical systems from data

Abstract: Scientists have long quantified empirical observations by developing mathematical models that characterize the observations, have some measure of interpretability, and are capable of making predictions. Dynamical systems models in particular have been widely used to study, explain, and predict system behavior in a wide range of application areas, with examples ranging from Newton’s laws of classical mechanics to the Michaelis-Menten kinetics for modeling enzyme kinetics. While governing laws and equations were traditionally derived by hand, the current growth of available measurement data and resulting emphasis on data-driven modeling motivates algorithmic approaches for model discovery. A number of such approaches have been developed in recent years and have generated widespread interest, including Eureqa (Schmidt & Lipson, 2009), sure independence screening and sparsifying operator (Ouyang, Curtarolo, Ahmetcik, Scheffler, & Ghiringhelli, 2018), and the sparse identification of nonlinear dynamics (SINDy) (Brunton, Proctor, & Kutz, 2016). Maximizing the impact of these model discovery methods requires tools to make them widely accessible to scientists across domains and at various levels of mathematical expertise.

Training deep neural density estimators to identify mechanistic models of neural dynamics.

Abstract: Computational neuroscientists use mathematical models built on observational data to investigate what’s happening in the brain. Models can simulate brain activity from the behavior of a single neuron right through to the patterns of collective activity in whole neural networks. Collecting the experimental data is the first step, then the challenge becomes deciding which computer models best represent the data and can explain the underlying causes of how the brain behaves. Researchers usually find the right model for their data through trial and error. This involves tweaking a model’s parameters until the model can reproduce the data of interest. But this process is laborious and not systematic. Moreover, with the ever-increasing complexity of both data and computer models in neuroscience, the old-school approach of building models is starting to show its limitations. Now, Goncalves, Lueckmann, Deistler et al. have designed an algorithm that makes it easier for researchers to fit mathematical models to experimental data. First, the algorithm trains an artificial neural network to predict which models are compatible with simulated data. After initial training, the method can rapidly be applied to either raw experimental data or selected data features. The algorithm then returns the models that generate the best match. This newly developed machine learning tool was able to automatically identify models which can replicate the observed data from a diverse set of neuroscience problems. Importantly, further experiments showed that this new approach can be scaled up to complex mechanisms, such as how a neural network in crabs maintains its rhythm of activity. This tool could be applied to a wide range of computational investigations in neuroscience and other fields of biology, which may help bridge the gap between ‘data-driven’ and ‘theory-driven’ approaches.

Robust identification for fault detection in the presence of non-Gaussian noises: application to hydraulic servo drives

Abstract: Intensive research in the field of mathematical modeling of hydraulic servo systems has shown that their mathematical models have many important details which cannot be included in the model. Due to impossibility of direct measurement or calculation of dimensions of certain components, leakage coefficients or friction coefficients, it was supposed that parameters of the hydraulic servo system are random. On the other side, it has been well known that the hydraulic servo system can be approximated by a linear model with time-varying parameters. An estimation of states and time-varying parameters of linear state-space models is of practical importance for fault diagnosis and fault-tolerant control. Previous works on this topic consider estimation in Gaussian noise environment, but not in the presence of outliers. The known fact is that the measurements have inconsistent observations with the largest part of the observation population (outliers). They can significantly make worse the properties of linearly recursive algorithms which are designed to work in the presence of Gaussian noises. This paper proposes the strategy of parameter–state robust estimation of linear state-space models in the presence of all possible faults and non-Gaussian noises. Because of its good features in robust filtering, Masreliez–Martin filter represents a cornerstone for realization of the robust algorithm. The good features of the proposed robust algorithm to identification of the hydraulic servo system are illustrated by intensive simulations.

Systems biology informed deep learning for inferring parameters and hidden dynamics

Abstract: Mathematical models of biological reactions at the system-level lead to a set of ordinary differential equations with many unknown parameters that need to be inferred using relatively few experimental measurements. Having a reliable and robust algorithm for parameter inference and prediction of the hidden dynamics has been one of the core subjects in systems biology, and is the focus of this study. We have developed a new systems-biology-informed deep learning algorithm that incorporates the system of ordinary differential equations into the neural networks. Enforcing these equations effectively adds constraints to the optimization procedure that manifests itself as an imposed structure on the observational data. Using few scattered and noisy measurements, we are able to infer the dynamics of unobserved species, external forcing, and the unknown model parameters. We have successfully tested the algorithm for three different benchmark problems. Author summary The dynamics of systems biological processes are usually modeled using ordinary differential equations (ODEs), which introduce various unknown parameters that need to be estimated efficiently from noisy measurements of concentration for a few species only. In this work, we present a new “systems-informed neural network” to infer the dynamics of experimentally unobserved species as well as the unknown parameters in the system of equations. By incorporating the system of ODEs into the neural networks, we effectively add constraints to the optimization algorithm, which makes the method robust to noisy and sparse measurements.

Measurement of CO2 concentration for occupancy estimation in educational buildings with energy efficiency purposes

Abstract: The measurement of CO2 concentration is a relevant indicator for defining the occupation of indoor spaces. The real-time knowledge of occupation of such spaces is relevant both for maintaining indoor air quality standards and for energy efficiency purposes connected with the operation of heating, ventilation, and air-conditioning (HVAC) systems. The exact knowledge of occupation allows for rapid feedback from and the regulation of an HVAC system and the ventilation rate. Interesting applications include educational buildings and other buildings of the civil sector (e.g., shopping centres and hospitals). This paper provides the results of an experimental analysis in different classrooms of a university campus under real operating conditions, in different periods of the year, and with different kinds of activities. The correlation between the CO2 concentration and occupancy profiles of the spaces is then analysed. Some graphical trends of the CO2 concentrations in these indoor spaces are provided to determine the most important variables affecting such concentrations. The basic elements of the mathematical models for estimating the occupation of classrooms in relation to increases in CO2 concentration are also discussed and analysed.

Generalized Homogeneity in Systems and Control

Abstract: This monograph introduces the theory of generalized homogeneous systems governed by differential equations in both Euclidean (finite-dimensional) and Banach/Hilbert (infinite-dimensional) spaces. It develops methods of stability and robustness analysis, control design, state estimation and discretization of homogeneous control systems. Generalized Homogeneity in Systems and Control is structured in two parts. Part I discusses various models of control systems and related tools for their analysis, including Lyapunov functions. Part II deals with the analysis and design of homogeneous control systems. Some of the key features of the text include: - mathematical models of dynamical systems in finite-dimensional and infinite-dimensional spaces; - the theory of linear dilations in Banach spaces; - homogeneous control and estimation; - simple methods for an "upgrade" of existing linear control laws; - numerical schemes for a consistent digital implementation of homogeneous algorithms; and - experiments confirming an improvement of PID controllers. The advanced mathematical material will be of interest to researchers, mathematicians working in control theory and mathematically oriented control engineers.

Numerical simulation of time-fractional partial differential equations arising in fluid flows via reproducing Kernel method

Abstract: The subject of the fractional calculus theory has gained considerable popularity and importance due to their attractive applications in widespread fields of physics and engineering. The purpose of this paper is to present results on the numerical simulation for time-fractional partial differential equations arising in transonic multiphase flows, which are described by the Tricomi and the Keldysh equations of Robin functions types.,Those resulting mathematical models are solved by using the reproducing kernel method, which provide appropriate solutions in term of infinite series formula. Convergence analysis, error estimations and error bounds under some hypotheses, which provide the theoretical basis of the proposed method are also discussed.,The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the prospects of the gained results and the method are discussed through academic validations.,In this paper and for the first time: the authors presented results on the numerical simulation for classes of time-fractional PDEs such as those found in the transonic multiphase flows. The authors applied the reproducing kernel method systematically for the numerical solutions of time-fractional Tricomi and Keldysh equations subject to Robin functions types.

Physics-Informed Neural Networks for Power Systems

Abstract: This paper introduces for the first time, to our knowledge, a framework for physics-informed neural networks in power system applications. Exploiting the underlying physical laws governing power systems, and inspired by recent developments in the field of machine learning, this paper proposes a neural network training procedure that can make use of the wide range of mathematical models describing power system behavior, both in steady-state and in dynamics. Physics-informed neural networks require substantially less training data and can result in simpler neural network structures, while achieving high accuracy. This work unlocks a range of opportunities in power systems, being able to determine dynamic states, such as rotor angles and frequency, and uncertain parameters such as inertia and damping at a fraction of the computational time required by conventional methods. This paper focuses on introducing the framework and showcases its potential using a single-machine infinite bus system as a guiding example. Physics-informed neural networks are shown to accurately determine rotor angle and frequency up to 87 times faster than conventional methods.

Different types analytic solutions of the (1+1)-dimensional resonant nonlinear Schrödinger’s equation using (G′/G)-expansion method

Abstract: In this paper, an alternative method has been studied for traveling wave solutions of mathematical models which have an important place in applied sciences and balance term is not integer. With thi...

A Bayesian machine scientist to aid in the solution of challenging scientific problems.

Abstract: Closed-form, interpretable mathematical models have been instrumental for advancing our understanding of the world; with the data revolution, we may now be in a position to uncover new such models for many systems from physics to the social sciences. However, to deal with increasing amounts of data, we need “machine scientists” that are able to extract these models automatically from data. Here, we introduce a Bayesian machine scientist, which establishes the plausibility of models using explicit approximations to the exact marginal posterior over models and establishes its prior expectations about models by learning from a large empirical corpus of mathematical expressions. It explores the space of models using Markov chain Monte Carlo. We show that this approach uncovers accurate models for synthetic and real data and provides out-of-sample predictions that are more accurate than those of existing approaches and of other nonparametric methods.

A Bayesian machine scientist to aid in the solution of challenging scientific problems

Abstract: Closed-form, interpretable mathematical models have been instrumental for advancing our understanding of the world; with the data revolution, we may now be in a position to uncover new such models for many systems from physics to the social sciences. However, to deal with increasing amounts of data, we need "machine scientists" that are able to extract these models automatically from data. Here, we introduce a Bayesian machine scientist, which establishes the plausibility of models using explicit approximations to the exact marginal posterior over models and establishes its prior expectations about models by learning from a large empirical corpus of mathematical expressions. It explores the space of models using Markov chain Monte Carlo. We show that this approach uncovers accurate models for synthetic and real data and provides out-of-sample predictions that are more accurate than those of existing approaches and of other nonparametric methods.

Biophysically detailed mathematical models of multiscale cardiac active mechanics.

Abstract: We propose four novel mathematical models, describing the microscopic mechanisms of force generation in the cardiac muscle tissue, which are suitable for multiscale numerical simulations of cardiac electromechanics. Such models are based on a biophysically accurate representation of the regulatory and contractile proteins in the sarcomeres. Our models, unlike most of the sarcomere dynamics models that are available in the literature and that feature a comparable richness of detail, do not require the time-consuming Monte Carlo method for their numerical approximation. Conversely, the models that we propose only require the solution of a system of PDEs and/or ODEs (the most reduced of the four only involving 20 ODEs), thus entailing a significant computational efficiency. By focusing on the two models that feature the best trade-off between detail of description and identifiability of parameters, we propose a pipeline to calibrate such parameters starting from experimental measurements available in literature. Thanks to this pipeline, we calibrate these models for room-temperature rat and for body-temperature human cells. We show, by means of numerical simulations, that the proposed models correctly predict the main features of force generation, including the steady-state force-calcium and force-length relationships, the length-dependent prolongation of twitches and increase of peak force, the force-velocity relationship. Moreover, they correctly reproduce the Frank-Starling effect, when employed in multiscale 3D numerical simulation of cardiac electromechanics.

Modelling longitudinal vehicle dynamics with neural networks

Abstract: This paper studies neural network models of vehicle dynamics. We consider both models with a generic layer architecture and models with specialised topologies that hard-wire physics principles. Net...

Parameter Redundancy and Identifiability

Abstract: Statistical and mathematical models are defined by parameters that describe different characteristics of those models. Ideally it would be possible to find parameter estimates for every parameter in that model, but, in some cases, this is not possible. For example, two parameters that only ever appear in the model as a product could not be estimated individually; only the product can be estimated. Such a model is said to be parameter redundant, or the parameters are described as non-identifiable. This book explains why parameter redundancy and non-identifiability is a problem and the different methods that can be used for detection, including in a Bayesian context. Key features of this book: Detailed discussion of the problems caused by parameter redundancy and non-identifiability Explanation of the different general methods for detecting parameter redundancy and non-identifiability, including symbolic algebra and numerical methods Chapter on Bayesian identifiability Throughout illustrative examples are used to clearly demonstrate each problem and method. Maple and R code are available for these examples More in-depth focus on the areas of discrete and continuous state-space models and ecological statistics, including methods that have been specifically developed for each of these areas This book is designed to make parameter redundancy and non-identifiability accessible and understandable to a wide audience from masters and PhD students to researchers, from mathematicians and statisticians to practitioners using mathematical or statistical models.

Evaluation of solar module equivalent models under real operating conditions—A review

Abstract: A number of mathematical models are available to model the performance of solar modules under varying operating conditions. Most commonly recognized and used models include (a) the basic three-parameter model, (b) the five-parameter model, and (c) the seven-parameter model. The basic three-parameter model does not incorporate series and shunt resistance for IV curves. The five-parameter model incorporates the effect of series and shunt resistance, and the seven-parameter model further includes the additional effect of temperature and irradiance variation on solar cell parameters. While all these models reasonably predict IV profiles of solar modules at small variations from standard testing conditions (STCs), their performance in modeling the module performance at low irradiances and high temperatures is far from ideal. This work primarily reviews the accuracy of available models for various module technologies not only under STC conditions but also over a wide range of operating conditions. The accuracy of modeled results is quantified (with datasheet results) for 10 crystalline silicon (c-Si) based modules as well as 9 thin film module (TF) samples (commercial modules) at multiple irradiance conditions. The results show that the three-parameter model generally overestimates the power output both for c-Si and TF modules. The five-parameter model predicts TF technology more accurately compared to the other two available models, whereas the seven-parameter model is most accurate for c-Si module modeling under varying operations.

Mathematical Models of Inverse Problems for Finding the Main Characteristics of Air Pollution Sources

Abstract: The paper describes optimization of mathematical models for determining the main characteristics of the source of environmental pollution. A modification of the classical Newton’s method for finding a numerical solution of the constructed mathematical models for identifying the parameters of an environmental pollutant has been developed. A modification of the classical Newton’s method is obtained, which makes it possible to reduce the total number of calculations in the process of determining the main characteristics of the pollution source. A number of software-implemented computational experiments have been carried out for the model for determining the height of the pipe of the pollution source and the concentration of emissions on it, the model for determining the full location of the pipe of the pollution source and the concentration of emissions from the source. The possibility of complete localization of the pollution source in less than 40 measurement iterations using 1 post of the air pollution monitoring system has been established. The proposed method makes it possible to reduce by 3 times the number of simulation iterations for detecting a source of pollution in comparison with classical methods for solving inverse problems during monitoring of air pollution.

Seepage Analysis in Short Embankments Using Developing a Metaheuristic Method Based on Governing Equations

Abstract: Seepage is one of the most challenging issues in some procedures such as design, construction, and operation of embankment or earth fill dams. The purpose of this research is to develop a new solution based on governing equations to solve the seepage problem in an effective way. Therefore, by implementing the equations in the programming environment, more than 24,000 models were designed to be applicable to different conditions. Input data included different parameters such as slopes in upstream and downstream, embankment width, soil permeability coefficient, height, and freeboard. With the use of this big data, a new process was developed to provide simple mathematical models for the seepage rate analysis. The study first used intelligent models to simulate the seepage behavior. Finally, the accuracy of the models was optimized using a new metaheuristic algorithm. This led to the ultimate flexibility of the final model presented as a new solution capable of evaluating different conditions. Finally, using the best model, new mathematical relationships were developed based on this methodology. This new solution can be used as a proper alternative to the governing equations of seepage rate estimation. Another advantage of the proposed model is its high flexibility that can be well applied to engineering design in this field, which was not possible using the initial equations.

Finite Element Simulation of Ionic Electrodiffusion in Cellular Geometries.

Abstract: Mathematical models for excitable cells are commonly based on cable theory, which considers a homogenized domain and spatially constant ionic concentrations. Although such models provide valuable insight, the effect of altered ion concentrations or detailed cell morphology on the electrical potentials cannot be captured. In this paper, we discuss an alternative approach to detailed modeling of electrodiffusion in neural tissue. The mathematical model describes the distribution and evolution of ion concentrations in a geometrically-explicit representation of the intra- and extracellular domains. As a combination of the electroneutral Kirchhoff-Nernst-Planck (KNP) model and the Extracellular-Membrane-Intracellular (EMI) framework, we refer to this model as the KNP-EMI model. Here, we introduce and numerically evaluate a new, finite element-based numerical scheme for the KNP-EMI model, capable of efficiently and flexibly handling geometries of arbitrary dimension and arbitrary polynomial degree. Moreover, we compare the electrical potentials predicted by the KNP-EMI and EMI models. Finally, we study ephaptic coupling induced in an unmyelinated axon bundle and demonstrate how the KNP-EMI framework can give new insights in this setting.

Moving contact lines and dynamic contact angles: a ‘litmus test’ for mathematical models, accomplishments and new challenges

Abstract: After a brief overview of the ‘moving contact-line problem’ as it emerged and evolved as a research topic, a ‘litmus test’ allowing one to assess adequacy of the mathematical models proposed as solutions to the problem is described. Its essence is in comparing the contact angle, an element inherent in every model, with what follows from a qualitative analysis of some simple flows. It is shown that, contrary to a widely held view, the dynamic contact angle is not a function of the contact-line speed as for different spontaneous spreading flows one has different paths in the contact angle-versus-speed plane. In particular, the dynamic contact angle can decrease as the contact-line speed increases. This completely undermines the search for the ‘right’ velocity-dependence of the dynamic contact angle, actual or apparent, as a direction of research. With a reference to an earlier publication, it is shown that, to date, the only mathematical model passing the ‘litmus test’ is the model of dynamic wetting as an interface formation process. The model, which was originated back in 1993, inscribes dynamic wetting into the general physical context as a particular case in a wide class of flows, which also includes coalescence, capillary breakup, free-surface cusping and some other flows, all sharing the same underlying physics. New challenges in the field of dynamic wetting are discussed.

A Concise Review of Gradient Models in Mechanics and Physics

Abstract: The various mathematical models developed over the years to interpret the behavior of materials and corresponding processes they undergo, were based on observations and experiments made at that time. Classical laws for solids (Hooke) and fluids (Navier-Stokes) formed the basis of current technology. The discovery of new phenomena with the aid of novel developed experimental probes have led to various modifications of these laws, especially at small scales. The emergence of nanotechnology is ultimately connected with the design of novel tools for observation and measurements, as well as the development of new methods and approaches for quantification and understanding. The paper first reviews the author’s previously developed weakly nonlocal or gradient models for elasticity, diffusion and plasticity. It then proposes a similar extension for fluids and electrodynamics. Finally, it suggests a gradient modification of Newton’s law of gravity, with a possible connection to the strong force of elementary particle physics.

The Analytical Analysis of Time-Fractional Fornberg–Whitham Equations

Abstract: This article is dealing with the analytical solution of Fornberg–Whitham equations in fractional view of Caputo operator. The effective method among the analytical techniques, natural transform decomposition method, is implemented to handle the solutions of the proposed problems. The approximate analytical solutions of nonlinear numerical problems are determined to confirm the validity of the suggested technique. The solution of the fractional-order problems are investigated for the suggested mathematical models. The solutions-graphs are then plotted to understand the effectiveness of fractional-order mathematical modeling over integer-order modeling. It is observed that the derived solutions have a closed resemblance with the actual solutions. Moreover, using fractional-order modeling various dynamics can be analyzed which can provide sophisticated information about physical phenomena. The simple and straight-forward procedure of the suggested technique is the preferable point and thus can be used to solve other nonlinear fractional problems.