An Introduction to Neural Network Methods for Differential Equations

An Introduction to Neural Network Methods for Differential Equations
Author: Neha Yadav
Publisher: Springer
Total Pages: 124
Release: 2015-02-26
Genre: Mathematics
ISBN: 9401798168
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This book introduces a variety of neural network methods for solving differential equations arising in science and engineering. The emphasis is placed on a deep understanding of the neural network techniques, which has been presented in a mostly heuristic and intuitive manner. This approach will enable the reader to understand the working, efficiency and shortcomings of each neural network technique for solving differential equations. The objective of this book is to provide the reader with a sound understanding of the foundations of neural networks and a comprehensive introduction to neural network methods for solving differential equations together with recent developments in the techniques and their applications. The book comprises four major sections. Section I consists of a brief overview of differential equations and the relevant physical problems arising in science and engineering. Section II illustrates the history of neural networks starting from their beginnings in the 1940s through to the renewed interest of the 1980s. A general introduction to neural networks and learning technologies is presented in Section III. This section also includes the description of the multilayer perceptron and its learning methods. In Section IV, the different neural network methods for solving differential equations are introduced, including discussion of the most recent developments in the field. Advanced students and researchers in mathematics, computer science and various disciplines in science and engineering will find this book a valuable reference source.

Artificial Neural Networks for Engineers and Scientists

Artificial Neural Networks for Engineers and Scientists
Author: S. Chakraverty
Publisher: CRC Press
Total Pages: 157
Release: 2017-07-20
Genre: Mathematics
ISBN: 1351651315
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Differential equations play a vital role in the fields of engineering and science. Problems in engineering and science can be modeled using ordinary or partial differential equations. Analytical solutions of differential equations may not be obtained easily, so numerical methods have been developed to handle them. Machine intelligence methods, such as Artificial Neural Networks (ANN), are being used to solve differential equations, and these methods are presented in Artificial Neural Networks for Engineers and Scientists: Solving Ordinary Differential Equations. This book shows how computation of differential equation becomes faster once the ANN model is properly developed and applied.

Deep Learning in Computational Mechanics

Deep Learning in Computational Mechanics
Author: Stefan Kollmannsberger
Publisher: Springer Nature
Total Pages: 108
Release: 2021-08-05
Genre: Technology & Engineering
ISBN: 3030765873
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This book provides a first course on deep learning in computational mechanics. The book starts with a short introduction to machine learning’s fundamental concepts before neural networks are explained thoroughly. It then provides an overview of current topics in physics and engineering, setting the stage for the book’s main topics: physics-informed neural networks and the deep energy method. The idea of the book is to provide the basic concepts in a mathematically sound manner and yet to stay as simple as possible. To achieve this goal, mostly one-dimensional examples are investigated, such as approximating functions by neural networks or the simulation of the temperature’s evolution in a one-dimensional bar. Each chapter contains examples and exercises which are either solved analytically or in PyTorch, an open-source machine learning framework for python.

Applied Artificial Neural Network Methods For Engineers And Scientists: Solving Algebraic Equations

Applied Artificial Neural Network Methods For Engineers And Scientists: Solving Algebraic Equations
Author: Snehashish Chakraverty
Publisher: World Scientific
Total Pages: 192
Release: 2021-01-26
Genre: Computers
ISBN: 9811230226
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The aim of this book is to handle different application problems of science and engineering using expert Artificial Neural Network (ANN). As such, the book starts with basics of ANN along with different mathematical preliminaries with respect to algebraic equations. Then it addresses ANN based methods for solving different algebraic equations viz. polynomial equations, diophantine equations, transcendental equations, system of linear and nonlinear equations, eigenvalue problems etc. which are the basic equations to handle the application problems mentioned in the content of the book. Although there exist various methods to handle these problems, but sometimes those may be problem dependent and may fail to give a converge solution with particular discretization. Accordingly, ANN based methods have been addressed here to solve these problems. Detail ANN architecture with step by step procedure and algorithm have been included. Different example problems are solved with respect to various application and mathematical problems. Convergence plots and/or convergence tables of the solutions are depicted to show the efficacy of these methods. It is worth mentioning that various application problems viz. Bakery problem, Power electronics applications, Pole placement, Electrical Network Analysis, Structural engineering problem etc. have been solved using the ANN based methods.

Artificial Neural Networks for Modelling and Control of Non-Linear Systems

Artificial Neural Networks for Modelling and Control of Non-Linear Systems
Author: Johan A.K. Suykens
Publisher: Springer Science & Business Media
Total Pages: 242
Release: 2012-12-06
Genre: Technology & Engineering
ISBN: 1475724934
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Artificial neural networks possess several properties that make them particularly attractive for applications to modelling and control of complex non-linear systems. Among these properties are their universal approximation ability, their parallel network structure and the availability of on- and off-line learning methods for the interconnection weights. However, dynamic models that contain neural network architectures might be highly non-linear and difficult to analyse as a result. Artificial Neural Networks for Modelling and Control of Non-Linear Systems investigates the subject from a system theoretical point of view. However the mathematical theory that is required from the reader is limited to matrix calculus, basic analysis, differential equations and basic linear system theory. No preliminary knowledge of neural networks is explicitly required. The book presents both classical and novel network architectures and learning algorithms for modelling and control. Topics include non-linear system identification, neural optimal control, top-down model based neural control design and stability analysis of neural control systems. A major contribution of this book is to introduce NLq Theory as an extension towards modern control theory, in order to analyze and synthesize non-linear systems that contain linear together with static non-linear operators that satisfy a sector condition: neural state space control systems are an example. Moreover, it turns out that NLq Theory is unifying with respect to many problems arising in neural networks, systems and control. Examples show that complex non-linear systems can be modelled and controlled within NLq theory, including mastering chaos. The didactic flavor of this book makes it suitable for use as a text for a course on Neural Networks. In addition, researchers and designers will find many important new techniques, in particular NLq emTheory, that have applications in control theory, system theory, circuit theory and Time Series Analysis.

Opportunities for the Deep Neural Network Method of Solving Partial Differential Equations in the Computational Study of Biomolecules Driven Through Periodic Geometries

Opportunities for the Deep Neural Network Method of Solving Partial Differential Equations in the Computational Study of Biomolecules Driven Through Periodic Geometries
Author: Martin Magill
Publisher:
Total Pages: 0
Release: 2022
Genre:
ISBN:
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As deep learning emerged in the 2010s to become a groundbreaking technology in machine vision and natural language processing, it also ushered in many new algorithms for use in scientific research. Among these is the neural network method, in which the solution to a differential equation is approximated by varying the parameters of a deep neural network trial function. Although this idea has been explored with shallow neural networks since the 1990s, it has experienced a resurgence of interest in recent years now that it can be implemented with deep neural networks. A series of empirical and theoretical studies have acclaimed the deep variants of the neural network method for being able to solve many classes of traditionally challenging partial differential equations. These early works emphasized its potential to solve high-dimensional, highly parameterized, and nonlinear equations in arbitrary geometries, all without requiring the discretization of the geometry into a mesh. Problems exhibiting these challenging features abound in computational biophysics, and this thesis presents recent efforts to adapt the neural network method for use in this _eld. The investigations in this thesis center on models of biomolecular motion in periodic geometries. Such models arise, for example, in the study of microfluidic and nanofluidic devices used for the separation of free-draining molecules. These problems exhibit many of the characteristics for which the neural network method is appealing, and serve here as non-trivial test problems on which to characterize its performance. Perspectives from biophysics, numerical analysis, and deep learning are combined to elucidate the true potential of the neural network method as a technique for studying such differential equations. Altogether, these works have moved the neural network method closer to being another reliable numerical method in the computational biophysicist's toolkit.

Semi-empirical Neural Network Modeling and Digital Twins Development

Semi-empirical Neural Network Modeling and Digital Twins Development
Author: Dmitriy Tarkhov
Publisher: Academic Press
Total Pages: 288
Release: 2019-11-23
Genre: Science
ISBN: 012815652X
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Semi-empirical Neural Network Modeling presents a new approach on how to quickly construct an accurate, multilayered neural network solution of differential equations. Current neural network methods have significant disadvantages, including a lengthy learning process and single-layered neural networks built on the finite element method (FEM). The strength of the new method presented in this book is the automatic inclusion of task parameters in the final solution formula, which eliminates the need for repeated problem-solving. This is especially important for constructing individual models with unique features. The book illustrates key concepts through a large number of specific problems, both hypothetical models and practical interest. Offers a new approach to neural networks using a unified simulation model at all stages of design and operation Illustrates this new approach with numerous concrete examples throughout the book Presents the methodology in separate and clearly-defined stages

Artificial Intelligence and Soft Computing

Artificial Intelligence and Soft Computing
Author: Leszek Rutkowski
Publisher: Springer Nature
Total Pages: 536
Release: 2021-10-04
Genre: Computers
ISBN: 3030879860
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The two-volume set LNAI 12854 and 12855 constitutes the refereed proceedings of the 20th International Conference on Artificial Intelligence and Soft Computing, ICAISC 2021, held in Zakopane, Poland, in June 2021. Due to COVID 19, the conference was held virtually. The 89 full papers presented were carefully reviewed and selected from 195 submissions. The papers included both traditional artificial intelligence methods and soft computing techniques as well as follows: · Neural Networks and Their Applications · Fuzzy Systems and Their Applications · Evolutionary Algorithms and Their Applications · Artificial Intelligence in Modeling and Simulation · Computer Vision, Image and Speech Analysis · Data Mining · Various Problems of Artificial Intelligence · Bioinformatics, Biometrics and Medical Applications

Advances in Neural Networks – ISNN 2016

Advances in Neural Networks – ISNN 2016
Author: Long Cheng
Publisher: Springer
Total Pages: 751
Release: 2016-07-01
Genre: Computers
ISBN: 3319406639
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This book constitutes the refereed proceedings of the 13th International Symposium on Neural Networks, ISNN 2016, held in St. Petersburg, Russia in July 2016. The 84 revised full papers presented in this volume were carefully reviewed and selected from 104 submissions. The papers cover many topics of neural network-related research including signal and image processing; dynamical behaviors of recurrent neural networks; intelligent control; clustering, classification, modeling, and forecasting; evolutionary computation; and cognition computation and spiking neural networks.

On Efficient Training & Inference of Neural Differential Equations

On Efficient Training & Inference of Neural Differential Equations
Author: Avik Pal
Publisher:
Total Pages: 0
Release: 2023
Genre:
ISBN:
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The democratization of machine learning requires architectures that automatically adapt to new problems. Neural Differential Equations have emerged as a popular modeling framework, enabling ML practitioners to design neural networks that can adaptively modify their depth based on the input problem. Neural Differential Equations combine differential equations with neural networks and rely on adaptive differential equation solvers for the forward process. The flexibility of automatically adapting the depths comes with the cost of expensive training and slower predictions. Several prior works have tried to accelerate training and inference. However, almost all of them have severe tradeoffs. Either these works rely on expensive training methods to accelerate predictions or use algorithms that are harder to integrate into existing workflows. This thesis will discuss two methods to accelerate Neural Differential Equations. We propose an Infinite Time Neural ODE, which paradoxically can be trained faster than integrating a Neural ODE to a fixed time-point. We also build upon prior works on regularized Neural ODEs and propose a stochastic local regularization scheme that can be used as a drop-in replacement for Neural ODEs.