A Scientific Machine Learning Approach To Learning Reduced Models For Nonlinear Partial Differential Equations
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A Scientific Machine Learning Approach To Learning Reduced Models For Nonlinear Partial Differential Equations
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Author : Elizabeth Yi Qian
language : en
Publisher:
Release Date : 2021
A Scientific Machine Learning Approach To Learning Reduced Models For Nonlinear Partial Differential Equations written by Elizabeth Yi Qian and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021 with categories.
This thesis presents a new scientific machine learning method which learns from data a computationally inexpensive surrogate model for predicting the evolution of a system governed by a time-dependent nonlinear partial differential equation (PDE), an enabling technology for many computational algorithms used in engineering settings. The proposed approach generalizes to the PDE setting an Operator Inference method previously developed for systems of ordinary differential equations (ODEs) with polynomial nonlinearities. The method draws on ideas from traditional physics-based modeling to explicitly parametrize the learned model by low-dimensional polynomial operators which reflect the known form of the governing PDE. This physics-informed parametrization is then united with tools from supervised machine learning to infer from data the reduced operators. The Lift & Learn method extends Operator Inference to systems whose governing PDEs contain more general (non-polynomial) nonlinearities through the use of lifting variable transformations which expose polynomial structure in the PDE. The proposed approach achieves a number of desiderata for scientific machine learning formulations, including analyzability, interpretability, and making underlying modeling assumptions explicit and transparent. This thesis therefore provides analysis of the Operator Inference and Lift & Learn methods in both the spatially continuous PDE and spatially discrete ODE settings. Results are proven regarding the mean square errors of the learned models, the impact of spatial and temporal discretization, and the recovery of traditional reduced models via the learning method. Sensitivity analysis of the operator inference problem to model misspecifications and perturbations in the data is also provided. The Lift & Learn method is demonstrated on the compressible Euler equations, the FitzHugh-Nagumo reaction-diffusion equations, and a large-scale three-dimensional simulation of a rocket combustion experiment with over 18 million degrees of freedom. For the first two examples, the Lift & Learn models achieve 2–3 orders of magnitude dimension reduction and match the generalization performance of traditional reduced models based on Galerkin projection of the PDE operators, predicting the system evolution with errors between 0.01% and 1% relative to the original nonlinear simulation. For the combustion application, the Lift & Learn models accurately predict the amplitude and frequency of pressure oscillations as well as the large-scale structures in the flow field’s temperature and chemical variables, with 5–6 orders of magnitude dimension reduction and 6–7 orders of magnitude computational savings. The demonstrated ability of the Lift & Learn models to accurately approximate the system evolution with orders-of-magnitude dimension reduction and computational savings makes the learned models suitable for use in many-query computations used to support scientific discovery and engineering decision-making.
Data Driven Modelling And Scientific Machine Learning In Continuum Physics
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Author : Krishna Garikipati
language : en
Publisher: Springer Nature
Release Date : 2024-07-29
Data Driven Modelling And Scientific Machine Learning In Continuum Physics written by Krishna Garikipati and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-07-29 with Mathematics categories.
This monograph takes the reader through recent advances in data-driven methods and machine learning for problems in science—specifically in continuum physics. It develops the foundations and details a number of scientific machine learning approaches to enrich current computational models of continuum physics, or to use the data generated by these models to infer more information on these problems. The perspective presented here is drawn from recent research by the author and collaborators. Applications drawn from the physics of materials or from biophysics illustrate each topic. Some elements of the theoretical background in continuum physics that are essential to address these applications are developed first. These chapters focus on nonlinear elasticity and mass transport, with particular attention directed at descriptions of phase separation. This is followed by a brief treatment of the finite element method, since it is the most widely used approach to solve coupled partial differential equations in continuum physics. With these foundations established, the treatment proceeds to a number of recent developments in data-driven methods and scientific machine learning in the context of the continuum physics of materials and biosystems. This part of the monograph begins by addressing numerical homogenization of microstructural response using feed-forward as well as convolutional neural networks. Next is surrogate optimization using multifidelity learning for problems of phase evolution. Graph theory bears many equivalences to partial differential equations in its properties of representation and avenues for analysis as well as reduced-order descriptions--all ideas that offer fruitful opportunities for exploration. Neural networks, by their capacity for representation of high-dimensional functions, are powerful for scale bridging in physics--an idea on which we present a particular perspective in the context of alloys. One of the most compelling ideas in scientific machine learning is the identification of governing equations from dynamical data--another topic that we explore from the viewpoint of partial differential equations encoding mechanisms. This is followed by an examination of approaches to replace traditional, discretization-based solvers of partial differential equations with deterministic and probabilistic neural networks that generalize across boundary value problems. The monograph closes with a brief outlook on current emerging ideas in scientific machine learning.
Knowledge Guided Machine Learning
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Author : Anuj Karpatne
language : en
Publisher: CRC Press
Release Date : 2022-08-15
Knowledge Guided Machine Learning written by Anuj Karpatne and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-08-15 with Business & Economics categories.
Given their tremendous success in commercial applications, machine learning (ML) models are increasingly being considered as alternatives to science-based models in many disciplines. Yet, these "black-box" ML models have found limited success due to their inability to work well in the presence of limited training data and generalize to unseen scenarios. As a result, there is a growing interest in the scientific community on creating a new generation of methods that integrate scientific knowledge in ML frameworks. This emerging field, called scientific knowledge-guided ML (KGML), seeks a distinct departure from existing "data-only" or "scientific knowledge-only" methods to use knowledge and data at an equal footing. Indeed, KGML involves diverse scientific and ML communities, where researchers and practitioners from various backgrounds and application domains are continually adding richness to the problem formulations and research methods in this emerging field. Knowledge Guided Machine Learning: Accelerating Discovery using Scientific Knowledge and Data provides an introduction to this rapidly growing field by discussing some of the common themes of research in KGML using illustrative examples, case studies, and reviews from diverse application domains and research communities as book chapters by leading researchers. KEY FEATURES First-of-its-kind book in an emerging area of research that is gaining widespread attention in the scientific and data science fields Accessible to a broad audience in data science and scientific and engineering fields Provides a coherent organizational structure to the problem formulations and research methods in the emerging field of KGML using illustrative examples from diverse application domains Contains chapters by leading researchers, which illustrate the cutting-edge research trends, opportunities, and challenges in KGML research from multiple perspectives Enables cross-pollination of KGML problem formulations and research methods across disciplines Highlights critical gaps that require further investigation by the broader community of researchers and practitioners to realize the full potential of KGML
Data Driven Science And Engineering
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Author : Steven L. Brunton
language : en
Publisher: Cambridge University Press
Release Date : 2022-05-05
Data Driven Science And Engineering written by Steven L. Brunton and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-05-05 with Computers categories.
A textbook covering data-science and machine learning methods for modelling and control in engineering and science, with Python and MATLAB®.
Reduced Basis Methods For Partial Differential Equations
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Author : Alfio Quarteroni
language : en
Publisher: Springer
Release Date : 2015-08-19
Reduced Basis Methods For Partial Differential Equations written by Alfio Quarteroni and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-08-19 with Mathematics categories.
This book provides a basic introduction to reduced basis (RB) methods for problems involving the repeated solution of partial differential equations (PDEs) arising from engineering and applied sciences, such as PDEs depending on several parameters and PDE-constrained optimization. The book presents a general mathematical formulation of RB methods, analyzes their fundamental theoretical properties, discusses the related algorithmic and implementation aspects, and highlights their built-in algebraic and geometric structures. More specifically, the authors discuss alternative strategies for constructing accurate RB spaces using greedy algorithms and proper orthogonal decomposition techniques, investigate their approximation properties and analyze offline-online decomposition strategies aimed at the reduction of computational complexity. Furthermore, they carry out both a priori and a posteriori error analysis. The whole mathematical presentation is made more stimulating by the use of representative examples of applicative interest in the context of both linear and nonlinear PDEs. Moreover, the inclusion of many pseudocodes allows the reader to easily implement the algorithms illustrated throughout the text. The book will be ideal for upper undergraduate students and, more generally, people interested in scientific computing. All these pseudocodes are in fact implemented in a MATLAB package that is freely available at https://github.com/redbkit
The Proceedings Of 2023 4th International Symposium On Insulation And Discharge Computation For Power Equipment Idcompu2023
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Author : Xuzhu Dong
language : en
Publisher: Springer Nature
Release Date : 2023-12-30
The Proceedings Of 2023 4th International Symposium On Insulation And Discharge Computation For Power Equipment Idcompu2023 written by Xuzhu Dong and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-12-30 with Technology & Engineering categories.
This book includes original, peer-reviewed research papers from the 2023 4th International Symposium on Insulation and Discharge Computation for Power Equipment (IDCOMPU2023), held in Wuhan, China. The topics covered include but are not limited to: insulation, discharge computations, electric power equipment, and electrical materials. The papers share the latest findings in the field of insulation and discharge computations of electric power equipment, making the book a valuable asset for researchers, engineers, university students, etc.
Proceedings Of Fluid Mechanics And Fluid Power Fmfp 2023 Vol 2
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Author : Hardik Kothadia
language : en
Publisher: Springer Nature
Release Date : 2025-05-02
Proceedings Of Fluid Mechanics And Fluid Power Fmfp 2023 Vol 2 written by Hardik Kothadia and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-05-02 with Technology & Engineering categories.
This book presents select proceedings of the 10th International and 50th National Conference on Fluid Mechanics and Fluid Power. It covers recent research developments in the area of fluid mechanics, measurement techniques in fluid flows, and computational fluid dynamics. The key research topics discussed in this book are fundamental studies in flow instability and transition, fluid-structure interaction, multiphase flows, solidification, melting, cavitation, porous media flows, bubble and droplet dynamics, bio-mems, micro-scale experimental techniques, flow control devices, underwater vehicles, bluff body, bio-fluid mechanics, aerodynamics, turbomachinery, propulsion and power, heat transfer and thermal engineering, fluids engineering, advances in aerospace and defence technology, micro- and nano-systems engineering, acoustics, structures and fluids, advanced theory and simulations, novel experimental techniques in thermo-fluids engineering and many more. The book is a valuable reference for researchers and professionals interested in thermo-fluids engineering.
Backward Stochastic Differential Equations
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Author : N El Karoui
language : en
Publisher: CRC Press
Release Date : 1997-01-17
Backward Stochastic Differential Equations written by N El Karoui and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-01-17 with Mathematics categories.
This book presents the texts of seminars presented during the years 1995 and 1996 at the Université Paris VI and is the first attempt to present a survey on this subject. Starting from the classical conditions for existence and unicity of a solution in the most simple case-which requires more than basic stochartic calculus-several refinements on the hypotheses are introduced to obtain more general results.
Numerical Analysis Meets Machine Learning
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Author :
language : en
Publisher: Elsevier
Release Date : 2024-06-13
Numerical Analysis Meets Machine Learning written by and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-06-13 with Mathematics categories.
Numerical Analysis Meets Machine Learning series, highlights new advances in the field, with this new volume presenting interesting chapters. Each chapter is written by an international board of authors. - Provides the authority and expertise of leading contributors from an international board of authors - Presents the latest release in the Handbook of Numerical Analysis series - Updated release includes the latest information on the Numerical Analysis Meets Machine Learning
Multiscale Nonlinear And Adaptive Approximation Ii
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Author : Ronald DeVore
language : en
Publisher: Springer Nature
Release Date : 2024-12-03
Multiscale Nonlinear And Adaptive Approximation Ii written by Ronald DeVore and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-12-03 with Mathematics categories.
This book presents a collection of high-quality papers in applied and numerical mathematics, as well as approximation theory, all closely related to Wolfgang Dahmen’s scientific contributions. Compiled in honor of his 75th birthday, the papers are written by leading experts and cover topics including nonlinear approximation theory, numerical analysis of partial differential equations, learning theory, and electron microscopy. A unifying theme throughout the collection is the emphasis on a solid mathematical foundation, which serves as the basis for the most efficient numerical algorithms used to simulate complex phenomena.