虽然《Deterministic， Stochastic， and Deep Learning Methods for Computational Electromagnetics（计算电磁学：确定性、随机和深度学习方法）》的重点是各种数值方法/算法模拟的电磁现象，还包括涉及的基本物理学。由于数值计算方法研究受到包括生物、物理、化学和工程等许多不同领域的相互作用和强烈影响，因此非常需要用一种平衡的方法来解决数学算法与物理基础和应用之间的相互联系，以便为应用数学、科学和工程领域的研究生和研究人员做好准备，在许多应用领域进行创新的高级计算研究。如溶剂中的生物分子溶剂化、雷达波散射、金属材料中光与电子的相互作用、纳米电子学中的电流流动等。《Deterministic， Stochastic， and Deep Learning Methods for Computational Electromagnetics（计算电磁学：确定性、随机和深度学习方法）》的目标是提供一个良好的平衡和全面的平台，基于清晰的物理和严谨的数学公式，并通过确定性、随机和机器学习的方法为电磁和输运过程的计算机模拟生物学、微波和光波器件、纳米电子学提供有效的数值方法。
目錄：
ContentsPart I Electrostatics1 Dielectric Constant and Fluctuation Formulae for Molecular Dynamics 31.1 Electrostatics of Charges and Dipoles 31.2 Polarization P and Displacement Flux D 51.2.1 Bound Charges Induced by Polarization 61.2.2 Electric Field Epoi(r) of a Polarization Density P(r) 81.2.3 Singular Integral Expressions of Epo/ (r) Inside Dielectrics 101.3 Clausius-Mossotti and Onsager Formulae for Dielectric Constant 111.3.1 Clausius-Mossotti Formula for Non-polar Dielectrics 111.3.2 Onsager Dielectric Theory for Dipolar Liquids 131.4 Statistical Molecular Theory and Dielectric Fluctuation Formulae 181.4.1 Statistical Methods for Polarization Density Change AP 201.4.2 Classical Electrostatics for Polarization Density Change AP 221.4.3 Fluctuation Formulae for Dielectric Constant 241.5 Appendices 261.5.1 Appendix 1: Average Field of a Charge in a Dielectric Sphere 261.5.2 Appendix 2: Electric Field Due to a Uniformly Polarized Sphere 261.6 Summary 28References 282 Poisson-Boltzmann Electrostatics and Analytical Approximations 312.1 Poisson-Boltzmann (PB) Model for Electrostatic Solvation 312.1.1 Debye—Hiickel Poisson-Boltzmann Theory 332.1.2 Helmholtz Double Layer and Ion-Size Effect 362.1.3 Electrostatic Solvation Energy 402.2 Generalized Bom (GB) Approximations of Solvation Energy 422.2.1 Still’s Generalized Bom Formulism 422.2.2 Integral Expression for Born Radii 432.2.3 FFT-Based Algorithm for the Bom Radii 452.3 Method of Images for Reaction Fields 512.3.1 Methods of Images for Simple Geometries 522.3.2 Image Methods for Dielectric Spheres 542.3.3 Image Methods for Dielectric Spheres in Ionic Solvent 612.3.4 Image Methods for Multi-layered Media 632.4 Summary 68References 683 Numerical Methods for Poisson-Boltzmann Equations 733.1 Boundary Element Methods (BEMs) 733.1.1 Cauchy Principal Value (CPV) and Hadamard Finite Part (p.f.) 743.1.2 Surface Integral Equations for the PB Equation 793.1.3 Computations of CPV and Hadamard p.f.and Collocation BEMs 863.2 Finite Element Methods (FEMs) 993.3 Immersed Interface Methods (IIMs) 1023.4 Summary 105References 1064 Random Walk Stochastic Methods for PDE Boundary Value Problems 1094.1 Brownian Motion， It6 Integral， and It6，s Formula 1094.1.1 Probability Space， Random Variable，and Conditional Expectation 1094.1.2 Brownian Motion， Ito Integral of Stochastic Processes， and It6，s Formula 1134.2 Feynman-Kac Formula for Mixed BVPs 1174.2.1 Dynkin’s Formula and Feynman-Kac Formula for Dirichlet BVPs of Elliptic PDEs 1174.2.2 Local Time L{t) and Feynman-Kac Formulae for Neumann and Robin BVPs of the Schrodinger Equation 1214.3 Pardoux-Peng s Nonlinear Feynman-Kac Formula for Quasilinear Parabolic PDEs 1234.4 Walk-On-Spheres (WOS) and Local Time L(t) for Reflecting Brownian Motion 1254.4.1 WOS Method and Dirichlet BVP1254.4.2 Reflecting Brownian Motion and Its Local Time L(t) 1274.5 Electrical Impedance Tomography (EIT).1294.5.1 Feynman-Kac Formula for Mixed BVPs 1294.5.2 Local EIT Solution with the WOS Method 1304.6 Boundary Integral Equation and WOS (BIE-WOS) Method for the Dirichlet BVP of the Laplace Equation 1324.7 Iterative BIE-WOS Method for the Mixed BVP of the Laplace Equation 1344.8 Summary 136References 1365 Stochastic Spectral Methods and Uncertainty Quantification of Microchip Interconnects 1395.1 Deterministic Spectral Methods 1395.1.1 Fourier， Chebyshev， Legendre， and Hermite Spectral Approximations 1395.1.2 Spectral Methods for BVPs of Differential Equations 1455.2 Stochastic Spectral Methods for Stochastic Differential and Integral Equations 1465.2.1 Wiener Polynomial Chaos (PC) and Generalized PC (gPC) Expansions 1475.2.2 Random Variable Dimension Reduction 1495.2.3 Spectral Galerkin Methods 1505.2.4 Spectral Collocation Methods and Sparse Grids 1535.3 Uncertainty Quantification of Rough Microchip Interconnects 1555.4 Summary 159References 1596 Deep Neural Network Learning for PDE Solutions 1616.1 Deep Neural Network (DNN).1616.2 Optimization with Stochastic Gradient Descent Method 1636.3 DeepRitz Method for the Poisson-Boltzmann Equation of Biomolecular Solvation 1646.4 Physics-Informed Neural Network (PINN) 1676.5 Galerkin Weak Form Learning 1676.6 Stochastic Differential Equation (SDE) Based DNN Methods 1686.6.1 DeepBSDE for Quasilinear Parabolic Equations via a Terminal Condition Loss 1696.6.2 Backward SDE Based Algorithms for Quasilinear Parabolic Equations with a Path wise Loss 1696.6.3 Martingale Based Learning Algorithm for BVPs 1716.6.4 Diffusion Monte Carlo DNN Eigensolver 1746.7 DNNs for Oscillatory Problems and Their Spectral Bias 1756.7.1 Phase-Shift DNN (PhaseDNN) 1756.7.2 Multiscale D
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