Numerical analysis meets machine learning:
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
Gespeichert in:
| Weitere Verfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Amsterdam, Netherlands
Elsevier North-Holland
[2024]
|
| Schriftenreihe: | Handbook of numerical analysis
volume 25 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | 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 |
| Beschreibung: | xviii, 570 Seiten Illustrationen, Diagramme |
| ISBN: | 9780443239847 0443239843 |
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Contents Contributors Preface 1. xiii xv Learning smooth functions in high dimensions Ben Adcock, Simone Brugiapaglia, Nick Dexter, and Sebastian Moraga 1 Introduction 1.1 Motivations and challenges 1.2 Overview 1.3 Further literature 2 Problem statement and notation 3 Holomorphic functions of infinitely many variables 3.1 (b, s)-holomorphic functions 3.2 Holomorphy and parametric DEs 3.3 Known and unknown anisotropy 3.4 ^-summability and the Ή(ρ) and Ή(ρ, Μ) classes 4 Best s-term polynomial approximation 4.1 Orthogonal polynomials 4.2 Orthogonal polynomial expansions 4.3 Best s-term polynomial approximation 4.4 Rates of best s-term polynomial approximation 4.5 How high is high dimensional? 5 Limits of learnability from data 5.1 Adaptive m-widths 5.2 Lower bounds for adaptive m-widths 5.3 Towards methods 6 Learning sparse polynomial approximations from data 6.1 Setup 6.2 Sampling discretizations for multivariatepolynomials 6.3 Resolving Challenge 1 : lower and anchored sets 6.4 Resolving Challenge 2: weighted λ-termapproximation 6.5 Weighted e1-minimization 6.6 Theoretical guarantee 6.7 Discussion and extensions 7 DNN existence theory 7.1 Review 7.2 Neural network architectures 7.3 Emulating polynomials with DNNs: typicalresult 2 3 3 9 11 12 12 14 16 17 17 17 18 19 19 20 22 22 23 24 24 25 26 28 29 31 32 34 34 34 36 36 V
vi 7.4 Elements of the proof of Theorem 7.1 7.5 Existence theorem for (b, s)-holomorphic functions 8 Practical existence theory: near-optimal DL 8.1 Setup 8.2 Practical existence theorem 8.3 The mechanism of practicalexistence theorems 9 Epilogue 9.1 Scientific computing and data scarcity 9.2 Potential benefits to DNNs over sparse polynomials 9.3 Theorem 8.1 does not eliminate the theory-practice gap 9.4 Practical insights 9.5 Eliminating the gap: beating the Monte Carlo rate is key 9.6 Conclusion References 2. Contents Contents 4. Weak form-based data-driven modeling David Μ. Bortz, Daniel A. Messenger, and April Tran 1 Introduction 2 Weak form-based equation discovery 2.1 The sparse identification of nonlinear dynamics (SINDy) method for learning governing equations 2.2 Weak form SINDy (WSINDy) 2.3 WSINDy for ordinary differential equations 2.4 WSINDy for partial differential equations 3 Theoretical results 3.1 Assumptions 4 Weak form-based parameter estimation 4.1 Ordinary differential equations 4.2 Partial differential equations 4.3 Stochastic differential equations 5 Weak form-based reduced order modeling 5.1 WLaSDI algorithm 5.2 WLaSDI example 6 Conclusions Acknowledgments References 3. 54 55 55 56 57 59 64 65 67 68 71 72 77 77 79 81 81 81 A mathematical guide to operator learning Nicolas Boullé and Alex Townsend 1 Introduction 1.1 What is a neural operator? 1.2 Where is operator learning relevant? 1.3 Organization of the paper 2 From numerical linear algebra to operator learning 2.1 Low rank matrix recovery 2.2 Circulant matrix recovery 2.3 Banded matrix recovery 2.4
Hierarchical low rank matrix recovery 3 Neural operator architectures 3.1 Deep operator networks 3.2 Fourier neural operators 3.3 Deep Green networks 3.4 Graph neural operators 3.5 Multipole graph neural operators 4 Learning neural operators 4.1 Data acquisition 4.2 Optimization 5 Conclusions and future challenges Acknowledgments References 37 38 38 39 39 40 42 42 43 43 43 44 45 45 θθ g$ 91 vii 92 93 94 95 97 100 102 103 105 106 112 116 119 119 The multiverse of dynamic mode decomposition algorithms Matthew J. Colbrook 1 Introduction 2 The basics of DMD 2.1 The underlying theory: Koopman operators and spectra 2.2 The fundamental DMD algorithm 2.3 Three canonical examples 2.4 The goals and challenges of DMD 3 Variants from theregression perspective 3.1 Increasing robustness to noise 3.2 Compression and randomized linear algebra 3.3 Multiresolution dynamic mode decomposition (mrDMD) 3.4 Control 4 Variants fromthe Galerkin perspective 4.1 Nonlinear observables: extended dynamic mode decomposition (EDMD) 4.2 Time-delay embedding 4.3 Controlling projection errors: residual dynamic mode decomposition (ResDMD) 5 Variants that preserve structure 5.1 Physics-informed dynamic mode decomposition (piDMD) 196 5.2 Measure-preserving extended dynamic mode decomposition (mpEDMD) 5.3 Compactification methods for continuous-time systems 5.4 Further methods 6 Further topics and open problems 6.1 Transfer operators 6.2 Continuous spectra and spectral measures 6.3 Stochastic dynamical systems 6.4 Some open problems 129 134 134 141 145 152 153 154 160 166 170 174 175 185 189 195 199 205 209 210
210 211 212 213
viii Acknowledgments References 5. Contents Contents Deep learning variational Monte Carlo for solving the electronic Schrödinger equation i Leon Gerard, Philipp Grohs, and Michael Scherbeia 1 Introduction 1.1 The molecular and electronic Schrödinger equations 1.2 Outline 1.3 Notation 2 Mathematical preliminaries 2.1 Basic mathematical setting 2.2 The Hamiltonian of the electronic Schrödinger equation 2.3 Spin and the Pauli exclusion principle 3 Introduction to variational Monte Carlo (VMC) 3.1 Slater determinants 3.2 Sampling using the Metropolis-Hastings algorithm 3.3 Optimization 4 Deep learning VMC 4.1 Multilayer perceptrons 4.2 Overall structure of neural networkwavefunctions 4.3 Input features 4.4 Embedding 4.5 Orbitals 4.6 Jastrow factor 4.7 Architectures for transferable wavefunctions 5 Results 5.1 Highly accurate variational energies 5.2 Transfer learning for ground-state energy predictions 5.3 Literature overview References 6. 5 Training/optimization methods 5.1 Initialization schemes 5.2 Generic methods: stochastic gradient descent 5.3 Quasi-Newton methods of 1,5-order 6 Approximation theory with small weights 7 PINN with observational data 8 Deep operator networks 8.1 Introduction 8.2 Vanilla DeepONets 8.3 Approximation rates for general Hôlder operators 8.4 Error estimates for solution operators from PDEs 8.5 Training DeepONets 8.6 Extending DeepONets 8.7 Benchmark test: Burgers'equation Acknowledgments Appendix 6.A Approximation of elementary functions with ReLU NNs Appendix 6.B Approximation of piecewise polynomials Appendix 6.C Approximation of horizon
functions Appendix 6.D Proof of Theorem 6.1 References 215 215 232 234 238 238 239 239 247 254 280 260 285 288 274 275 275 276 277 281 282 284 284 285 288 289 Theoretical foundations of physics-informed neural networks and deep neural operators Yeonjong Shin, Zhongqiang Zhang, and George Em Karniadakis 1 Introduction 2 Neural networks 3 Mathematical formulations 3,1 Stability 3.2 Strong formulation 3.3 Weak/variational formulations 3.4 Extended PINN: domain decomposition 3.5 Useful techniques 4 Approximation error for PINN in strong formulations 4.1 A posteriori estimate 4.2 A priori estimate 294 295 296 297 299 300 302 303 305 305 306 . ; 7. ix 306 307 309 312 314 319 320 320 321 322 324 328 330 331 333 333 347 350 354 355 Computability of optimizers for Al and data science Yunseok Lee, Holger Boche, and Gitta Kutyniok 1 Introduction 360 1 -1 Overview 361 1-2 Numerical computations on digital hardware 362 1-3 Computability and hardware 362 2 Basic notions 363 2.1 Search for the optimal value 364 2.2 Computability 365 3 Deep learning as a key technique of artificial intelligence 369 3.1 Essence of deep learning 369 3.2 Drawbacks of deep learning 370 4 Computability of optimal values and existence of computable optimizers 371 4.1 One-dimensional optimization 371 4.2 Computability of convex optimizers in higher dimensions 374 4.3 Example of multidimensional optimization in information theory 375 4.4 Other computable and noncomputable functions 377 5 Finding the optimizer is not effectively solvable 378 5.1 General noncomputability theorem 378 5.2 Noncomputability of neural
networks and other optimizers 379 5.3 Wasserstein distance 382 5.4 Lattice problem for cryptographic applications 384
Contents X Contents 5.5 Inverse problems Acknowledgments References 385 386 386 Neural Galerkin schemes for sequential-in-time solving of partial differential equations with deep networks Jules Berman, Paul Schwerdtner, and Benjamin Peherstorfer 1 Introduction 1.1 Linear parametrizations in numerical analysis 1.2 Nonlinear parametrizations for discretizing PDE solution fields 1.3 Neural Galerkin schemes 1A Literature overview 1.5 Outline 2 The need for nonlinear parametrizations in approximating solution fields of PDEs 2.1 Setup 2.2 Linear parametrizations of solution fields 2.3 The Kolmogorov barrier 2.4 Numerical illustrations of limitations of linear parametrizations 2.5 Nonlinear parametrizations 3 Neural Galerkin schemes based on the Dirac-Frenkel variational principle and deep networks 3.1 The Dirac-Frenkel variational principle 3.2 An optimization perspective of the Dirac-Frenkel variational principle 3.3 Least-squares formulation and discretization in time 4 Adaptive sampling in Neural Galerkin schemes 4.1 The sampling challenge 4.2 Objectives with time-dependent measures 4.3 A computational procedure for adaptive sampling based on particles and Stein variational gradient descent 4.4 Using adaptive samples in least-squares formulations of Neural Galerkin schemes 4.5 Example: Fokker-Planck equations in moderately high dimensions 5 Randomized sparse Neural Galerkin schemes 5.1 The i mportance of the tangent spaces of the parametrization manifold 5.2 Tangent space col lapse 5.3 Leveraging low-rankness of batch gradients with subsampling 5.4 Numerical experiments with
randomized sparse Neural Galerkin schemes 6 Conclusions References 390 390 391 391 392 393 393 393 394 395 396 397 399 400 402 402 403 403 404 405 406 407 408 409 410 410 412 412 413 9. xi Operator learning Nikola B. Kovachki, Samuel Lanthaler, and Andrew Μ. Stuart 1 Introduction 1.1 High dimensional vectors versus functions 1.2 Literature review 1.3 Overview of paper 2 Operator learning 2.1 Supervised learning 2.2 Supervised learning of operators 2.3 Training and testing 2.4 Finding latent structure 2.5 Example (fluid flow in a porous medium) 3 Specific supervised learning architectures 3.1 PCA-Net 3.2 DeepONet 3.3 FNO 3.4 Random features method 3.5 Discussion 4 Universal approximation 4.1 Encoder-decoder-nets 4.2 Neural operators 5 Quantitative error and complexity estimates 5.1 Linear operators 5.2 Holomorphic operators 5.3 General (Lipschitz) operators 5.4 Structure beyond smoothness 6 Conclusions Acknowledgments References 420 420 421 424 424 424 425 425 426 426 427 427 428 429 431 432 434 435 438 441 442 443 446 457 459 460 460 10. A structure-preserving domain decomposition method for data-driven modeling Shuai Jiang, Jonas Actor, Scott Roberts, and Nathaniel Trask 1 Introduction 2 Relation to previous work 2.1 Data-driven DEC/FEEC and Dirichlet-to-Neumann maps 2.2 Structure-preserving ML vs. physics-informed ML 2.3 Choice of mortar scheme 3 Local learning of Whitney form elements 4 Mortar method 4.1 Stability analysis for continuous case 4.2 Discretized case 4.3 Data-driven elements with mortar method 4.4 Neumann boundary conditions and conservation 5 Numerical
results 5.1 Example 1 : pure finite elements 470 472 473 473 474 474 478 480 483 487 488 491 491
xii Contents Example 2: pure FEEC and pure FEM elements comparison 5.3 Example 3: hybrid methods 5.4 Example 4: subdomainrefinement with FEEC Appendix 10.A Technical details 10. A.1 Technical proofs 10. A.2 FEEC element training References 5.2 492 493 497 504 504 511 512 11. Two-layer neural networks for partial differential equations: optimization and generalization theory Tao Luo and Haizhao Yang 1 Introduction 2 Deep learning-based PDE solvers 2.1 Notations, definitions, and basic lemmas 2.2 Expectation minimization 2.3 Scope of analysis and applications 3 Main results 4 Global convergence of gradient descent 4.1 Notations and main ideas 4.2 Proofs of lemmas for Theorem 3.1 4.3 Proof of Theorem 3.1 5 A priori estimates of generalization error for two-layer neural networks 5.1 Preliminary lemmas of Rademacher complexity 5.2 Proofs of generalization bounds 6 Conclusion Acknowledgments References Index 516 519 519 522 524 527 529 529 532 541 543 543 546 551 551 552 555 |
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