Numerical Analysis meets Machine Learning

1st Edition, Volume 25 - June 13, 2024
Editors: Siddhartha Mishra, Alex Townsend
Language: English
Hardback ISBN:
9 7 8 - 0 - 4 4 3 - 2 3 9 8 4 - 7
eBook ISBN:
9 7 8 - 0 - 4 4 3 - 2 3 9 8 5 - 4

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 intern… Read more

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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.

Cover image
Title page
Table of Contents
Copyright
Contributors
Preface
Chapter 1: Learning smooth functions in high dimensions
Abstract
1. Introduction
2. Problem statement and notation
3. Holomorphic functions of infinitely many variables
4. Best s-term polynomial approximation
5. Limits of learnability from data
6. Learning sparse polynomial approximations from data
7. DNN existence theory
8. Practical existence theory: near-optimal DL
9. Epilogue
References
Chapter 2: Weak form-based data-driven modeling
Abstract
1. Introduction
2. Weak form-based equation discovery
3. Theoretical results
4. Weak form-based parameter estimation
5. Weak form-based reduced order modeling
6. Conclusions
Acknowledgements
References
Chapter 3: A mathematical guide to operator learning
Abstract
1. Introduction
2. From numerical linear algebra to operator learning
3. Neural operator architectures
4. Learning neural operators
5. Conclusions and future challenges
Acknowledgements
References
Chapter 4: The multiverse of dynamic mode decomposition algorithms
Abstract
1. Introduction
2. The basics of DMD
3. Variants from the regression perspective
4. Variants from the Galerkin perspective
5. Variants that preserve structure
6. Further topics and open problems
Acknowledgements
References
Chapter 5: Deep learning variational Monte Carlo for solving the electronic Schrödinger equation
Abstract
1. Introduction
2. Mathematical preliminaries
3. Introduction to variational Monte Carlo (VMC)
4. Deep learning VMC
5. Results
References
Chapter 6: Theoretical foundations of physics-informed neural networks and deep neural operators
Abstract
1. Introduction
2. Neural networks
3. Mathematical formulations
4. Approximation error for PINN in strong formulations
5. Training/optimization methods
6. Approximation theory with small weights
7. PINN with observational data
8. Deep operator networks
Acknowledgements
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
Chapter 7: Computability of optimizers for AI and data science
Abstract
1. Introduction
2. Basic notions
3. Deep learning as a key technique of artificial intelligence
4. Computability of optimal values and existence of computable optimizers
5. Finding the optimizer is not effectively solvable
Acknowledgements
References
Chapter 8: Neural Galerkin schemes for sequential-in-time solving of partial differential equations with deep networks
Abstract
1. Introduction
2. The need for nonlinear parametrizations in approximating solution fields of PDEs
3. Neural Galerkin schemes based on the Dirac-Frenkel variational principle and deep networks
4. Adaptive sampling in Neural Galerkin schemes
5. Randomized sparse Neural Galerkin schemes
6. Conclusions
References
Chapter 9: Operator learning
Abstract
1. Introduction
2. Operator learning
3. Specific supervised learning architectures
4. Universal approximation
5. Quantitative error and complexity estimates
6. Conclusions
Acknowledgements
References
Chapter 10: A structure-preserving domain decomposition method for data-driven modeling
Abstract
1. Introduction
2. Relation to previous work
3. Local learning of Whitney form elements
4. Mortar method
5. Numerical results
Appendix 10.A. Technical details
References
Chapter 11: Two-layer neural networks for partial differential equations: optimization and generalization theory
Abstract
1. Introduction
2. Deep learning-based PDE solvers
3. Main results
4. Global convergence of gradient descent
5. A priori estimates of generalization error for two-layer neural networks
6. Conclusion
Acknowledgements
References
Index