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Umfang: XIV, 230 p. 75 illus., 70 illus. in color. , online resource.

Ausgabe: 1st ed. 2024.

ISBN: 9783031620294

Serie: Interdisciplinary Applied Mathematics, 60

Inhalt: 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.

Anmerkung: Part I. Introduction and Background in Continuum Materials Physics -- Introduction -- Nonlinear Elasticity -- Phase Field Methods -- Part II. Solving Partial Differential Equations -- Finite Element Methods -- Part III. Data-driven Modelling and Scientific Machine Learning -- Reduced Order Models: Numerical Homogenization for the Elastic Response of Material Microstructures -- Surrogate Optimization -- Graph Theoretic Methods -- Scale Bridging -- Inverse Modeling and System Inference from Data -- Machine Learning Solvers of Partial Differential Equations -- An Outlook on Scientific Machine Learning in Continuum Physics -- References.

In: Springer Nature eBook

Weitere Ausg.: Printed edition: ISBN 9783031620287

Weitere Ausg.: Printed edition: ISBN 9783031620300

Weitere Ausg.: Printed edition: ISBN 9783031633645

Sprache: Englisch

DOI: 10.1007/978-3-031-62029-4

URL: https://doi.org/10.1007/978-3-031-62029-4

URL: lizenzpflichtig

URL: Volltext  (URL des Erstveröffentlichers)

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Umfang: 1 online resource (233 pages)

Ausgabe: 1st ed.

ISBN: 9783031620294

Serie: Interdisciplinary Applied Mathematics Series ; v.60

Anmerkung: Intro -- Preface -- Acknowledgments -- Contents -- Part I Introduction and Background in Continuum Materials Physics -- 1 Introduction -- 1.1 Data in Computational Continuum Physics -- 1.2 Room for Learning -- 1.3 Differential Equation Models of Continuum Physics -- 1.4 Computing Solutions -- 1.5 Learning from Computed Solutions -- 1.6 Better Computed Solutions -- 1.7 Equations in the Data -- 1.8 New, Faster Solvers -- 1.9 Meanwhile -- 1.10 The Road Ahead -- 2 Nonlinear Elasticity -- 2.1 Mathematical Preliminaries -- 2.2 Kinematics -- 2.2.1 Inelastic Deformation -- 2.3 Balance Laws -- 2.4 Constitutive Relations -- 2.4.1 Frame Invariance of the Strain Energy DensityFunction -- 2.4.2 Material Symmetry -- 2.4.3 Constitutive Relation for Stress in a Hyperelastic Solid -- 2.5 Some Constitutive Models of Importance -- 2.5.1 The St. Venant-Kirchhoff Model -- 2.5.2 The St. Venant-Kirchhoff Model Extended to Cubic Crystals -- 2.5.3 The Quadratic-Logarithmic Model -- 2.5.4 The Compressible Neo-Hookean Model -- 2.6 A Variational Treatment of Nonlinear Elasticity -- 3 Phase Field Methods -- 3.1 Continuum Mass Transport -- 3.2 The Chemical Potential -- 3.3 Non-convex Free Energy Density and Phase Separation -- 3.4 Interface Energy -- 3.5 The Cahn-Hilliard Equation -- 3.6 The Allen-Cahn Equation -- 3.7 Free Energy Dissipation in Phase Field Models -- Part II Solving Partial Differential Equations -- 4 Finite Element Methods -- 4.1 Finite Element Basics -- 4.2 Nonlinear, Elliptic Partial Differential Equations for Vector Fields: Steady-State Nonlinear Elasticity -- 4.3 Linear, Parabolic Differential Equations for Scalars: Transient Mass Transport -- 4.3.1 Time Integration -- 4.4 Nonlinear, Parabolic Differential Equations for Scalars: Transient Phase Field Equations -- Part III Data-Driven Modeling and Scientific Machine Learning. , 5 Reduced-Order Models: Numerical Homogenization for the Elastic Response of Material Microstructures -- 5.1 Background -- 5.2 Mechanochemical Spinodal Decomposition: A Summary -- 5.2.1 The Free Energy Density Function -- 5.2.2 Governing Equations -- 5.2.3 Homogenized Mechanical Properties for Heterogeneous Microstructures -- 5.3 Neural Networks -- 5.3.1 Deep Neural Networks -- 5.3.2 Convolutional Neural Networks -- 5.3.3 Multi-Resolution Learning: Knowledge-Based Neural Networks -- 5.4 Data Generation, Feature Choice, and Hyperparameter Optimization -- 5.4.1 Data Generation: Direct Numerical Simulation of Microstructures -- 5.4.2 Selection of Microstructural Features -- 5.4.3 Data Generation -- 5.4.4 Hyperparameter Optimization -- 5.5 Training and Testing of Neural Networks -- 5.5.1 Representation of the Baseline Mechanical Free Energy for a Single DNS -- 5.5.1.1 DNNs for the Baseline Mechanical Free Energy -- 5.5.1.2 CNNs for the Baseline Mechanical Free Energy -- 5.5.2 Baseline Elastic Free Energies from Multiple DNS -- 5.5.3 Neural Network Representations for Homogenization of the Mechanical Behavior of a Single Microstructure -- 5.5.3.1 DNN-Based KBNN Representations -- 5.5.3.2 CNN-Based KBNN Representations -- 5.6 Summary -- 6 Surrogate Optimization -- 6.1 Background -- 6.2 Phase Field Models of Precipitate Evolution -- 6.2.1 Local Free Energy -- 6.2.2 Gradient Energy in Terms of Non-conserved Order Parameters -- 6.2.3 Strain Energy Density -- 6.2.4 Direct Numerical Simulation: Coupled Mass Transport, Phase Field, and Elasticity -- 6.2.5 Taking Stock -- 6.3 Surrogate Optimization of Precipitate Morphologies via Active Learning and Sensitivity Analysis -- 6.3.1 Reduced-Order Geometric Representation of Precipitate Shapes -- 6.3.2 Free Energy Computed by Direct NumericalSimulation -- 6.3.2.1 Strain Energy -- 6.3.2.2 Interfacial Energy. , 6.3.2.3 Bulk Chemical Free Energy -- 6.3.3 Optimization with a Surrogate -- 6.3.4 Sobol Sampling -- 6.3.5 Knowledge-Based Neural Networks for Multifidelity Modelling -- 6.3.6 Minimization on the Surrogate Energy Surface and Sensitivity Analysis -- 6.4 The DNS-ML Algorithm for Precipitate Evolution Compared with Phase Field Models -- 6.5 Summary -- 7 Graph Theoretic Methods -- 7.1 A Primer on the Use of Graphs in Computational Physics -- 7.2 Graph Representations of Stationary and Steady-State Systems -- 7.2.1 The Graph Properties of Stationary Problems -- 7.3 Non-dissipative Dynamics -- 7.3.1 The Graph Properties of Non-dissipative Dynamics -- 7.4 Dissipative Dynamics -- 7.4.1 Graph Properties of Dissipative Dynamics -- 7.5 Representation of Computed Solutions by Graphs and Their Exploration -- 7.5.1 Graphs Representing Non-dissipative Linear Elastodynamics and Stationary Elasticity -- 7.5.2 Graphs Representing the Stationary States of a Problem of Non-convex Elasticity at Finite Strain with Gradient Regularization -- 7.5.2.1 Graph Embedding of Strain States and Transitions -- 7.5.2.2 Graph Layouts, Centrality, and Traversal of Strain States -- 7.5.2.3 Eigenvector Centrality of a Vertex in Relation to the Strain State -- 7.5.2.4 Strain State Cliques -- 7.5.2.5 Cyclic Deformation in a Graph -- 7.5.2.6 Shortest Paths -- 7.5.3 Graph Embedding of Time Series Trajectories of a Dissipative Dynamical System -- 7.5.3.1 First-Order Dynamics of a Two-Species, Phase-Separating System -- 7.5.3.2 A Comparison of Organizing Principles for Graph Embedding -- 7.5.4 Graph Embedding of a Dissipative Dynamical System Without Time Series Data -- 7.5.4.1 Graph Chemical Potentials and Definition of Edges -- 7.5.4.2 Most and Least Favored Energy Minimization Paths -- 7.5.4.3 Graph Time for Edge Transition -- 7.6 Closing Remarks -- 8 Scale Bridging. , 8.1 Scale Bridging with the Free Energy -- 8.2 First Principles Statistical Mechanics -- 8.3 Integrating a Neural Network -- 8.4 Effective Sampling via Active Learning -- 8.4.1 Accounting for Crystal Symmetry -- 8.5 The Elastic Strain Energy Density Representation -- 8.6 Coupled Phase Field Models for Conserved and Non-conserved Variables -- 8.7 Computation: Implementation, Results -- 8.8 Closing Remarks -- 9 Inverse Modeling and System Inference from Data -- 9.1 Model Inference -- 9.1.1 The PDEs of Interest in Galerkin Weak Form -- 9.2 The Governing Parabolic PDEs Identified in Weak Form -- 9.2.1 A Library of Candidate Operators in Weak Form -- 9.2.2 NURBS -- 9.2.3 Identifying Candidate Operators -- 9.2.3.1 The Idea of Stepwise Regression -- 9.2.4 Noisy and Low-Fidelity Data -- 9.3 Identifying Example Systems of PDEs -- 9.3.1 Data Generation by Direct Numerical Simulation -- 9.3.2 System Identification with Noise-Free Data of Varying Fidelity -- 9.3.3 System Identification with Noisy Data -- 9.4 Closing Remarks -- 10 Machine Learning Solvers of Partial Differential Equations -- 10.1 A Perspective on PDE Solvers -- 10.2 Discretization-Based Solvers and Machine Learning -- 10.2.1 Elliptic PDEs -- 10.2.1.1 Fickian Diffusion at Steady State -- 10.2.1.2 Linearized Elasticity at Steady State -- 10.2.1.3 Nonlinear Elasticity at Steady State -- 10.2.2 The Loss Function for Deterministic Problems -- 10.2.3 The Loss Function for Probabilistic Problems in a Bayesian Setting -- 10.2.4 The Neural Network Architecture and Evaluation of the Loss -- 10.2.5 Bayesian Uncertainty Quantification -- 10.3 Results with Networks That Have Learnt Solvers -- 10.3.1 The Form of the (Bayesian) NN-Based PDE Solver -- 10.3.2 Steady-State Diffusion Solver Learnt from a Small Dataset -- 10.3.3 Linearized Elasticity Solver Learnt froma Small Dataset. , 10.3.4 Nonlinear Elasticity Solver Learnt from aSmall Dataset -- 10.3.5 A Steady-State Diffusion Solver That Learns to Generalize over a Large Dataset -- 10.3.6 A Comparison with Other Machine Learning Approaches to PDE Solvers -- 10.4 Closing Remarks -- 11 An Outlook on Scientific Machine Learning in Continuum Physics -- 11.1 Automated Discovery of Constitutive Models in Continuum Mechanics -- 11.2 Operator Networks -- 11.3 Generative Artificial Intelligence and Foundation Models -- 11.4 Others -- References -- Index.

Weitere Ausg.: Print version: Garikipati, Krishna Data-Driven Modelling and Scientific Machine Learning in Continuum Physics Cham : Springer International Publishing AG,c2024 ISBN 9783031620287

Sprache: Englisch