Creating a Solver for Nonlinear Differential Equations Using Unsupervised Neural Networks

Creating a Solver for Nonlinear Differential Equations Using Unsupervised Neural Networks
Author: Jacob Frishman
Publisher:
Total Pages: 0
Release: 2024
Genre:
ISBN:

Differential equations are ubiquitous in science and engineering for describing the natural world and often appear as nonlinear differential equations. Unfortunately, there is no general method for solving all types of nonlinear differential equations. This work uses a machine learning process called deep neural networks (DNNs) to create a solver for the Ginzburg-Landau equation regardless of the boundary conditions or the right-hand side. This method overcomes challenges to previous methods that require recomputing the solution again for every change in the boundary conditions and right-hand side of the equation. The method develops a versatile solver capable of finding a solution using only the form of the differential equation without a predefined right-hand side or boundary conditions. Systematically varying the architecture of the network, the characteristics of the input data, the loss function optimized over, and the network's hyperparameters reveal that the method can find a general solution across a diverse range of boundary conditions and right-hand sides. The network can consistently find accurate approximations of slowly oscillating data and highly oscillating data built from many terms of the Fourier series. The model can generalize performance from training data to test data, indicating its success in creating a general inverse differential operator that solves the equation. For data with many oscillations and small magnitudes, the network suffers from the vanishing gradient problem. These challenges are addressed by implementing strategies such as batch normalization, varying initialization schemes, changing activation functions, modifying the network architecture, and altering the loss function. These changes help mitigate the problem, leading to more stable and robust solutions to the initial hyperparameters of the model. However, the vanishing gradient problem persists despite these changes. Developing a solver that works for nonlinear equations would be pivotal in developing a theory for solving differential equations, saving computational time and resources, and facilitating real-time applications of the network without retraining.

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

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.

An Introduction to Financial Option Valuation

An Introduction to Financial Option Valuation
Author: Desmond J. Higham
Publisher: Cambridge University Press
Total Pages: 300
Release: 2004-04-15
Genre: Mathematics
ISBN: 1139457896

This is a lively textbook providing a solid introduction to financial option valuation for undergraduate students armed with a working knowledge of a first year calculus. Written in a series of short chapters, its self-contained treatment gives equal weight to applied mathematics, stochastics and computational algorithms. No prior background in probability, statistics or numerical analysis is required. Detailed derivations of both the basic asset price model and the Black–Scholes equation are provided along with a presentation of appropriate computational techniques including binomial, finite differences and in particular, variance reduction techniques for the Monte Carlo method. Each chapter comes complete with accompanying stand-alone MATLAB code listing to illustrate a key idea. Furthermore, the author has made heavy use of figures and examples, and has included computations based on real stock market data.

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

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.

Fractional Order Analysis

Fractional Order Analysis
Author: Hemen Dutta
Publisher: John Wiley & Sons
Total Pages: 336
Release: 2020-08-06
Genre: Mathematics
ISBN: 1119654238

A guide to the new research in the field of fractional order analysis Fractional Order Analysis contains the most recent research findings in fractional order analysis and its applications. The authors—noted experts on the topic—offer an examination of the theory, methods, applications, and the modern tools and techniques in the field of fractional order analysis. The information, tools, and applications presented can help develop mathematical methods and models with better accuracy. Comprehensive in scope, the book covers a range of topics including: new fractional operators, fractional derivatives, fractional differential equations, inequalities for different fractional derivatives and fractional integrals, fractional modeling related to transmission of Malaria, and dynamics of Zika virus with various fractional derivatives, and more. Designed to be an accessible text, several useful, relevant and connected topics can be found in one place, which is crucial for an understanding of the research problems of an applied nature. This book: Contains recent development in fractional calculus Offers a balance of theory, methods, and applications Puts the focus on fractional analysis and its interdisciplinary applications, such as fractional models for biological models Helps make research more relevant to real-life applications Written for researchers, professionals and practitioners, Fractional Order Analysis offers a comprehensive resource to fractional analysis and its many applications as well as information on the newest research.

Deterministic Artificial Intelligence

Deterministic Artificial Intelligence
Author: Timothy Sands
Publisher: BoD – Books on Demand
Total Pages: 180
Release: 2020-05-27
Genre: Computers
ISBN: 1789841119

Kirchhoff’s laws give a mathematical description of electromechanics. Similarly, translational motion mechanics obey Newton’s laws, while rotational motion mechanics comply with Euler’s moment equations, a set of three nonlinear, coupled differential equations. Nonlinearities complicate the mathematical treatment of the seemingly simple action of rotating, and these complications lead to a robust lineage of research culminating here with a text on the ability to make rigid bodies in rotation become self-aware, and even learn. This book is meant for basic scientifically inclined readers commencing with a first chapter on the basics of stochastic artificial intelligence to bridge readers to very advanced topics of deterministic artificial intelligence, espoused in the book with applications to both electromechanics (e.g. the forced van der Pol equation) and also motion mechanics (i.e. Euler’s moment equations). The reader will learn how to bestow self-awareness and express optimal learning methods for the self-aware object (e.g. robot) that require no tuning and no interaction with humans for autonomous operation. The topics learned from reading this text will prepare students and faculty to investigate interesting problems of mechanics. It is the fondest hope of the editor and authors that readers enjoy the book.

Nonlinear System Identification

Nonlinear System Identification
Author: Oliver Nelles
Publisher: Springer Science & Business Media
Total Pages: 814
Release: 2001
Genre: Computers
ISBN: 9783540673699

Written from an engineering point of view, this book covers the most common and important approaches for the identification of nonlinear static and dynamic systems. The book also provides the reader with the necessary background on optimization techniques, making it fully self-contained. The new edition includes exercises.

Driving Scientific and Engineering Discoveries Through the Convergence of HPC, Big Data and AI

Driving Scientific and Engineering Discoveries Through the Convergence of HPC, Big Data and AI
Author: Jeffrey Nichols
Publisher: Springer Nature
Total Pages: 555
Release: 2020-12-22
Genre: Computers
ISBN: 3030633934

This book constitutes the revised selected papers of the 17th Smoky Mountains Computational Sciences and Engineering Conference, SMC 2020, held in Oak Ridge, TN, USA*, in August 2020. The 36 full papers and 1 short paper presented were carefully reviewed and selected from a total of 94 submissions. The papers are organized in topical sections of computational applications: converged HPC and artificial intelligence; system software: data infrastructure and life cycle; experimental/observational applications: use cases that drive requirements for AI and HPC convergence; deploying computation: on the road to a converged ecosystem; scientific data challenges. *The conference was held virtually due to the COVID-19 pandemic.

Deep Learning with PyTorch

Deep Learning with PyTorch
Author: Luca Pietro Giovanni Antiga
Publisher: Simon and Schuster
Total Pages: 518
Release: 2020-07-01
Genre: Computers
ISBN: 1638354073

“We finally have the definitive treatise on PyTorch! It covers the basics and abstractions in great detail. I hope this book becomes your extended reference document.” —Soumith Chintala, co-creator of PyTorch Key Features Written by PyTorch’s creator and key contributors Develop deep learning models in a familiar Pythonic way Use PyTorch to build an image classifier for cancer detection Diagnose problems with your neural network and improve training with data augmentation Purchase of the print book includes a free eBook in PDF, Kindle, and ePub formats from Manning Publications. About The Book Every other day we hear about new ways to put deep learning to good use: improved medical imaging, accurate credit card fraud detection, long range weather forecasting, and more. PyTorch puts these superpowers in your hands. Instantly familiar to anyone who knows Python data tools like NumPy and Scikit-learn, PyTorch simplifies deep learning without sacrificing advanced features. It’s great for building quick models, and it scales smoothly from laptop to enterprise. Deep Learning with PyTorch teaches you to create deep learning and neural network systems with PyTorch. This practical book gets you to work right away building a tumor image classifier from scratch. After covering the basics, you’ll learn best practices for the entire deep learning pipeline, tackling advanced projects as your PyTorch skills become more sophisticated. All code samples are easy to explore in downloadable Jupyter notebooks. What You Will Learn Understanding deep learning data structures such as tensors and neural networks Best practices for the PyTorch Tensor API, loading data in Python, and visualizing results Implementing modules and loss functions Utilizing pretrained models from PyTorch Hub Methods for training networks with limited inputs Sifting through unreliable results to diagnose and fix problems in your neural network Improve your results with augmented data, better model architecture, and fine tuning This Book Is Written For For Python programmers with an interest in machine learning. No experience with PyTorch or other deep learning frameworks is required. About The Authors Eli Stevens has worked in Silicon Valley for the past 15 years as a software engineer, and the past 7 years as Chief Technical Officer of a startup making medical device software. Luca Antiga is co-founder and CEO of an AI engineering company located in Bergamo, Italy, and a regular contributor to PyTorch. Thomas Viehmann is a Machine Learning and PyTorch speciality trainer and consultant based in Munich, Germany and a PyTorch core developer. Table of Contents PART 1 - CORE PYTORCH 1 Introducing deep learning and the PyTorch Library 2 Pretrained networks 3 It starts with a tensor 4 Real-world data representation using tensors 5 The mechanics of learning 6 Using a neural network to fit the data 7 Telling birds from airplanes: Learning from images 8 Using convolutions to generalize PART 2 - LEARNING FROM IMAGES IN THE REAL WORLD: EARLY DETECTION OF LUNG CANCER 9 Using PyTorch to fight cancer 10 Combining data sources into a unified dataset 11 Training a classification model to detect suspected tumors 12 Improving training with metrics and augmentation 13 Using segmentation to find suspected nodules 14 End-to-end nodule analysis, and where to go next PART 3 - DEPLOYMENT 15 Deploying to production

Machine Learning Refined

Machine Learning Refined
Author: Jeremy Watt
Publisher: Cambridge University Press
Total Pages: 597
Release: 2020-01-09
Genre: Computers
ISBN: 1108480721

An intuitive approach to machine learning covering key concepts, real-world applications, and practical Python coding exercises.