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Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

Machine learning

Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

The Nelder--Mead simplex algorithm, first published in 1965, is an enormously popular direct search method for multidimensional unconstrained minimization. Despite its widespread use, essentially no theoretical results have been proved explicitly for the Nelder--Mead algorithm. This paper presents convergence properties of the Nelder--Mead algorithm applied to strictly convex functions in dimensions 1 and 2. We prove convergence to a minimizer for dimension 1, and various limited convergence results for dimension 2. A counterexample of McKinnon gives a family of strictly convex functions in two dimensions and a set of initial conditions for which the Nelder--Mead algorithm converges to a nonminimizer. It is not yet known whether the Nelder--Mead method can be proved to converge to a minimizer for a more specialized class of convex functions in two dimensions.

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Convergence Properties of the Nelder--Mead Simplex Method in Low Dimensions

The subject of inverse problems in differential equations is of enormous practical importance, and has also generated substantial mathematical and computational innovation. Typically some form of regularization is required to ameliorate ill-posed behaviour. In this article we review the Bayesian approach to regularization, developing a function space viewpoint on the subject. This approach allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion. Although expensive to implement, this approach is starting to lie within the range of the available computational resources in many application areas. It also allows for the quantification of uncertainty and risk, something which is increasingly demanded by these applications. Furthermore, the approach is conceptually important for the understanding of simpler, computationally expedient approaches to inverse problems.

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Inverse problems: A Bayesian perspective

This introduction to multiscale methods gives readers a broad overview of the many uses and applications of the methods. The book begins by setting the theoretical foundations of the subject area, and moves on to develop a unified approach to the simplification of a wide range of problems which possess multiple scales, via perturbation expansions; differential equations and stochastic processes are studied in one unified framework. The book concludes with an overview of a range of theoretical tools used to justify the simplified models derived via the perturbation expansions.

The presentation of the material is particularly suited to the range of mathematicians, scientists and engineers who want to exploit multiscale methods in applications. Extensive use of examples shows how to apply multiscale methods to solving a variety of problems. Exercises then enable readers to build their own skills and put them into practice.

Extensions and generalizations of the results presented in the book, as well as references to the literature, are provided in the Discussion and Bibliography section at the end of each chapter. All of the twenty-one chapters are supplemented with exercises.

Multiscale Methods: Averaging and Homogenization

The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of PDEs, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. The Fourier neural operator is the first ML-based method to successfully model turbulent flows with zero-shot super-resolution. It is up to three orders of magnitude faster compared to traditional PDE solvers. Additionally, it achieves superior accuracy compared to previous learning-based solvers under fixed resolution.

Fourier Neural Operator for Parametric Partial Differential Equations

This book unites the study of dynamical systems and numerical solution of differential equations. The first three chapters contain the elements of the theory of dynamical systems and the numerical solution of initial-value problems. In the remaining chapters, numerical methods are formulted as dynamical systems and the convergence and stability properties of the methods are examined. Topics studied include the stability of numerical methods for contractive, dissipative, gradient and Hamiltonian systems together with the convergence properties of equilibria, periodic solutions and strage attractors under numerical approximation. This book will be an invaluable tool for graduate students and researchers in the fields of numerical analysis and dynamical systems.

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Dynamical systems and numerical analysis

http://homepages.warwick.ac.uk/~masdr/journalpubs/stuart50.pdf

Andrew M. Stuart

Papers

Inverse problems: A Bayesian perspective

Multiscale Methods: Averaging and Homogenization

Fourier Neural Operator for Parametric Partial Differential Equations

Dynamical systems and numerical analysis

Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise