Contents for Volume 1

Preface xvii

1 Artificial Intelligence and Complex Systems: What It Can and Cannot Do 1
1.1 Introduction 1
1.2 A Glance at History 2
1.3 Complex Media and Systems 4
1.4 Three Types of Complex Systems 5
1.5 Physics-informed and Data-driven Approach to Complex Media and Phenomena 6
1.6 What Artificial Intelligence Cannot Do 7

2 Neural Networks and Other Machine-learning Algorithms 9
2.1 Introduction 9
2.2 Training of Neural Networks: Backpropagation 11
2.3 Classification of Learning 15
2.4 Weak Learners and Boosting Algorithms 21
2.5 Activation Functions 23
2.6 Types of Neural Networks 25
2.7 Regularization of Neural Networks 43
2.8 Training of Large Neural Networks 44
2.9 Other Machine-learning Algorithms 45
2.10 Methods for Minimizing the Loss Function 48
2.11 Challenges and Future Directions 50

3 Solving Differential and Partial Differential Equations 59
3.1 Introduction 59
3.2 Solving Ordinary Differential Equations 61
3.3 Solving Partial Differential Equations 62
3.4 Solving High-dimensional Partial Differential Equations: Deep BSDE Algorithm 66
3.5 Feynman-Kac Solution for Backward Kolmogorov Equation of Stochastic Processes 69
3.6 Data-driven Discretization of Partial Differential Equations 72
3.7 Other Methods 77
3.8 Space-time Fractional Partial Differential Equations 79
3.9 Challenges and Future Directions 80

4 Fluid Mechanics: Single-phase Flow 85
4.1 Introduction 85
4.2 The Microscopic Conservation Laws 85Contents for Volume 1 ix
4.3 A Glance at History 88
4.4 Kinematics of Fluid Flow 89
4.5 Dynamics of Fluid Flow 94
4.6 Modeling Flow Systems of Type I 96
4.7 Data-driven Neural Networks for Flow Systems of Type I 99
4.8 Physics-informed and Data-driven Machine-learning Approach 109
4.9 Turbulent Flows 127
4.10 Control of a Flow Field 139
4.11 Aerodynamic Systems 141
4.12 Machine Learning for Accelerating Direct Numerical Simulations 143
4.13 Challenges and Future Directions 143

5 Fluid Mechanics: Multiphase Flows 155
5.1 Introduction 155
5.2 Physics-informed Simulation of Two-phase Flows 156
5.3 Data-driven Approach to Simulating Two-phase Flows 170
5.4 Multiphase Flow in Heterogeneous Porous Materials and Media 173
5.5 Challenges and Future Directions 174

6 Heat and Mass Transfer Processes 179
6.1 Introduction 179
6.2 Heat and Mass Transfer Processes 180
6.3 Applications of Neural Networks to Heat Transfer Processes 182
6.4 Mass Transfer 224
6.5 Challenges and Future Directions 229

7 Porous Materials and Media 241
7.1 Introduction 241
7.2 Characterization of Core-scale Porous Media 243
7.3 Characterization of Large-scale Porous Media 258
7.4 Reconstruction of Porous Media 261
7.5 Data-driven Neural Networks for Simulating Single-phase Flow and Transport Processes 268
7.6 Physics-informed Neural Networks for Simulating Single-phase Flow and Transport 280
7.7 Two-phase Flow 286
7.8 Thermo-hydro-mechanical Processes 298
7.9 Data-driven Neural Networks for Two-phase Flow 299
7.10 Challenges and Future Directions 302

8 Density-functional Theory and Molecular Simulation 313
8.1 Introduction 313
8.2 Quantum Monte Carlo Method 313
8.3 First-principle Simulation: Density-functional Theory Calculations 316
8.4 Molecular Dynamics Simulation 326
8.5 Active Learning 339
8.6 Other Aspects of Development of Force Fields by Machine-learning Algorithms 341
8.7 Challenges and Future Directions 343

9 Membranes for Separation of Fluid Mixtures 351
9.1 Introduction 351
9.2 Data-driven Neural Networks for Separation Processes 353
9.3 Da