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.
1. 1 Introduction - p. page2 2. 2 Problem statement and notation - p. 11 3. 3 Holomorphic functions of infinitely many variables - p. 12 4. 4 Best s-term polynomial approximation - p. 17 5. 5 Limits of learnability from data - p. 22 6. 6 Learning sparse polynomial approximations from data - p. 24 7. 7 DNN existence theory - p. 34 8. 8 Practical existence theory: near-optimal DL - p. 38 9. 9 Epilogue - p. 42 10. References - p. 45 11. Chapter 2: Weak form-based data-driven modeling: Computationally efficient and noise robust equation learning and parameter inference - p. 53 12. Abstract - p. 53 13. 1 Introduction - p. 54 14. 2 Weak form-based equation discovery - p. 55 15. 3 Theoretical results - p. page63 16. 4 Weak form-based parameter estimation - p. page67 17. 5 Weak form-based reduced order modeling - p. 77 18. 6 Conclusions - p. 81 19. Acknowledgements - p. 81 20. References - p. 81 21. Chapter 3: A mathematical guide to operator learning - p. 83 22. Abstract - p. 83 23. 1 Introduction - p. 84 24. 2 From numerical linear algebra to operator learning - p. 88 25. 3 Neural operator architectures - p. 94 26. 4 Learning neural operators - p. 105 27. 5 Conclusions and future challenges - p. 116 28. Acknowledgements - p. 119 29. References - p. 119 30. Chapter 4: The multiverse of dynamic mode decomposition algorithms - p. 127 31. Abstract - p. 128 32. 1 Introduction - p. 129 33. 2 The basics of DMD - p. 134 34. 3 Variants from the regression perspective - p. 153 35. 4 Variants from the Galerkin perspective - p. 174 36. 5 Variants that preserve structure - p. 195 37. 6 Further topics and open problems - p. 210 38. Acknowledgements - p. 215 39. References - p. 215 40. Chapter 5: Deep learning variational Monte Carlo for solving the electronic Schrödinger equation - p. 231 41. Abstract - p. 231 42. 1 Introduction - p. 232 43. 2 Mathematical preliminaries - p. 239 44. 3 Introduction to variational Monte Carlo (VMC) - p. 260 45. 4 Deep learning VMC - p. page274 46. 5 Results - p. 284 47. References - p. 289 48. Chapter 6: Theoretical foundations of physics-informed neural networks and deep neural operators: A brief review - p. 293 49. Abstract - p. 294 50. 1 Introduction - p. page294 51. 2 Neural networks - p. 295 52. 3 Mathematical formulations - p. 296 53. 4 Approximation error for PINN in strong formulations - p. 305 54. 5 Training/optimization methods - p. page306 55. 6 Approximation theory with small weights - p. 314 56. 7 PINN with observational data - p. 320 57. 8 Deep operator networks - p. 320 58. Acknowledgements - p. 333 59. References - p. 355 60. Chapter 7: Computability of optimizers for AI and data science - p. 359 61. Abstract - p. 359 62. 1 Introduction - p. 360 63. 2 Basic notions - p. 363 64. 3 Deep learning as a key technique of artificial intelligence - p. page369 65. 4 Computability of optimal values and existence of computable optimizers - p. 371 66. 5 Finding the optimizer is not effectively solvable - p. 378 67. Acknowledgements - p. 386 68. References - p. 386 69. Chapter 8: Neural Galerkin schemes for sequential-in-time solving of partial differential equations with deep networks - p. 389 70. Chapter 9: Operator learning: Algorithms and analysis - p. 419 71. Abstract - p. 419 72. 1 Introduction - p. 420 73. 2 Operator learning - p. 424 74. 3 Specific supervised learning architectures - p. 427 75. 4 Universal approximation - p. 434 76. 5 Quantitative error and complexity estimates - p. 441 77. 6 Conclusions - p. 459 78. Acknowledgements - p. 460 79. References - p. 460 80. Chapter 10: A structure-preserving domain decomposition method for data-driven modeling - p. 469 81. Abstract - p. 470 82. 1 Introduction - p. page470 83. 2 Relation to previous work - p. 472 84. 3 Local learning of Whitney form elements - p. 474 85. 4 Mortar method - p. 478 86. 5 Numerical results - p. 491 87. References - p. page512 88. Chapter 11: Two-layer neural networks for partial differential equations: optimization and generalization theory - p. 515 89. Abstract - p. 515 90. 1 Introduction - p. 516 91. 2 Deep learning-based PDE solvers - p. 519 92. 3 Main results - p. 527 93. 4 Global convergence of gradient descent - p. 529 94. 5 A priori estimates of generalization error for two-layer neural networks - p. 542 95. 6 Conclusion - p. 551 96. Acknowledgements - p. 551 97. References - p. 552 98. Index - p. 554
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.
1. 1 Introduction - p. page2 2. 2 Problem statement and notation - p. 11 3. 3 Holomorphic functions of infinitely many variables - p. 12 4. 4 Best s-term polynomial approximation - p. 17 5. 5 Limits of learnability from data - p. 22 6. 6 Learning sparse polynomial approximations from data - p. 24 7. 7 DNN existence theory - p. 34 8. 8 Practical existence theory: near-optimal DL - p. 38 9. 9 Epilogue - p. 42 10. References - p. 45 11. Chapter 2: Weak form-based data-driven modeling: Computationally efficient and noise robust equation learning and parameter inference - p. 53 12. Abstract - p. 53 13. 1 Introduction - p. 54 14. 2 Weak form-based equation discovery - p. 55 15. 3 Theoretical results - p. page63 16. 4 Weak form-based parameter estimation - p. page67 17. 5 Weak form-based reduced order modeling - p. 77 18. 6 Conclusions - p. 81 19. Acknowledgements - p. 81 20. References - p. 81 21. Chapter 3: A mathematical guide to operator learning - p. 83 22. Abstract - p. 83 23. 1 Introduction - p. 84 24. 2 From numerical linear algebra to operator learning - p. 88 25. 3 Neural operator architectures - p. 94 26. 4 Learning neural operators - p. 105 27. 5 Conclusions and future challenges - p. 116 28. Acknowledgements - p. 119 29. References - p. 119 30. Chapter 4: The multiverse of dynamic mode decomposition algorithms - p. 127 31. Abstract - p. 128 32. 1 Introduction - p. 129 33. 2 The basics of DMD - p. 134 34. 3 Variants from the regression perspective - p. 153 35. 4 Variants from the Galerkin perspective - p. 174 36. 5 Variants that preserve structure - p. 195 37. 6 Further topics and open problems - p. 210 38. Acknowledgements - p. 215 39. References - p. 215 40. Chapter 5: Deep learning variational Monte Carlo for solving the electronic Schrödinger equation - p. 231 41. Abstract - p. 231 42. 1 Introduction - p. 232 43. 2 Mathematical preliminaries - p. 239 44. 3 Introduction to variational Monte Carlo (VMC) - p. 260 45. 4 Deep learning VMC - p. page274 46. 5 Results - p. 284 47. References - p. 289 48. Chapter 6: Theoretical foundations of physics-informed neural networks and deep neural operators: A brief review - p. 293 49. Abstract - p. 294 50. 1 Introduction - p. page294 51. 2 Neural networks - p. 295 52. 3 Mathematical formulations - p. 296 53. 4 Approximation error for PINN in strong formulations - p. 305 54. 5 Training/optimization methods - p. page306 55. 6 Approximation theory with small weights - p. 314 56. 7 PINN with observational data - p. 320 57. 8 Deep operator networks - p. 320 58. Acknowledgements - p. 333 59. References - p. 355 60. Chapter 7: Computability of optimizers for AI and data science - p. 359 61. Abstract - p. 359 62. 1 Introduction - p. 360 63. 2 Basic notions - p. 363 64. 3 Deep learning as a key technique of artificial intelligence - p. page369 65. 4 Computability of optimal values and existence of computable optimizers - p. 371 66. 5 Finding the optimizer is not effectively solvable - p. 378 67. Acknowledgements - p. 386 68. References - p. 386 69. Chapter 8: Neural Galerkin schemes for sequential-in-time solving of partial differential equations with deep networks - p. 389 70. Chapter 9: Operator learning: Algorithms and analysis - p. 419 71. Abstract - p. 419 72. 1 Introduction - p. 420 73. 2 Operator learning - p. 424 74. 3 Specific supervised learning architectures - p. 427 75. 4 Universal approximation - p. 434 76. 5 Quantitative error and complexity estimates - p. 441 77. 6 Conclusions - p. 459 78. Acknowledgements - p. 460 79. References - p. 460 80. Chapter 10: A structure-preserving domain decomposition method for data-driven modeling - p. 469 81. Abstract - p. 470 82. 1 Introduction - p. page470 83. 2 Relation to previous work - p. 472 84. 3 Local learning of Whitney form elements - p. 474 85. 4 Mortar method - p. 478 86. 5 Numerical results - p. 491 87. References - p. page512 88. Chapter 11: Two-layer neural networks for partial differential equations: optimization and generalization theory - p. 515 89. Abstract - p. 515 90. 1 Introduction - p. 516 91. 2 Deep learning-based PDE solvers - p. 519 92. 3 Main results - p. 527 93. 4 Global convergence of gradient descent - p. 529 94. 5 A priori estimates of generalization error for two-layer neural networks - p. 542 95. 6 Conclusion - p. 551 96. Acknowledgements - p. 551 97. References - p. 552 98. Index - p. 554
