[GitHub - yuzhimanhua/Awesome-Scientific-Language-Models: A Curated List of Language Models in Scientific Domains](https://github.com/yuzhimanhua/Awesome-Scientific-Language-Models)
[GitHub - dave-esch/learn-biology: Repository of educational resources for molecular biology.](https://github.com/dave-esch/learn-biology)
[GitHub - tilde-lab/awesome-materials-informatics: Curated list of known efforts in materials informatics = modern materials science](https://github.com/tilde-lab/awesome-materials-informatics)
[GitHub - yuzhimanhua/Awesome-Scientific-Language-Models: A Curated List of Language Models in Scientific Domains](https://github.com/yuzhimanhua/Awesome-Scientific-Language-Models)
[GitHub - ossu/computer-science: :mortar_board: Path to a free self-taught education in Computer Science!](https://github.com/ossu/computer-science)
[GitHub - ChessMax/awesome-programming-languages: The list of an awesome programming languages that you might be interested in](https://github.com/ChessMax/awesome-programming-languages)
[GitHub - mbrukman/awesome-computer-engineering: A curated list of awesome Computer Engineering resources [WIP - contributions are welcome]](https://github.com/mbrukman/awesome-computer-engineering)
Create a gigantic detailed map of evolutionary machine learning/artificial intelligence!
Make it more detailed and bigger!
Add theory!
Create a gigantic detailed map of meta-learning in machine learning/artificial intelligence
Here is an even more comprehensive map of the mathematics used in various fields of machine learning, including theoretical foundations:
1. Supervised Learning
a. Regression
- ...
- Learning Theory: PAC learning, VC dimension, Rademacher complexity, algorithmic stability, generalization bounds, sample complexity, uniform convergence, concentration inequalities (Hoeffding's inequality, Chernoff bounds, Azuma-Hoeffding inequality, McDiarmid's inequality)
b. Classification
- ...
- Learning Theory: PAC learning, VC dimension, Rademacher complexity, algorithmic stability, generalization bounds, sample complexity, uniform convergence, concentration inequalities (Hoeffding's inequality, Chernoff bounds, Azuma-Hoeffding inequality, McDiarmid's inequality), margin bounds, covering numbers, entropy numbers
2. Unsupervised Learning
a. Clustering
- ...
- Learning Theory: algorithmic stability, generalization bounds, sample complexity, uniform convergence, concentration inequalities (Hoeffding's inequality, Chernoff bounds, Azuma-Hoeffding inequality, McDiarmid's inequality), U-statistics, empirical process theory
b. Dimensionality Reduction
- ...
- Learning Theory: algorithmic stability, generalization bounds, sample complexity, uniform convergence, concentration inequalities (Hoeffding's inequality, Chernoff bounds, Azuma-Hoeffding inequality, McDiarmid's inequality), Johnson-Lindenstrauss lemma, random projection, compressed sensing
3. Reinforcement Learning
- ...
- Learning Theory: PAC learning, VC dimension, Rademacher complexity, algorithmic stability, generalization bounds, sample complexity, uniform convergence, concentration inequalities (Hoeffding's inequality, Chernoff bounds, Azuma-Hoeffding inequality, McDiarmid's inequality), regret bounds, multi-armed bandit theory, online learning theory, adversarial learning, no-regret learning
- Control Theory: optimal control, adaptive control, robust control, stochastic control, Lyapunov stability, Bellman equations, Hamilton-Jacobi-Bellman equations, Pontryagin's maximum principle
4. Deep Learning
a. Neural Networks
- ...
- Learning Theory: PAC learning, VC dimension, Rademacher complexity, algorithmic stability, generalization bounds, sample complexity, uniform convergence, concentration inequalities (Hoeffding's inequality, Chernoff bounds, Azuma-Hoeffding inequality, McDiarmid's inequality), margin bounds, covering numbers, entropy numbers, approximation theory, universal approximation theorem
b. Convolutional Neural Networks (CNNs)
- ...
- Learning Theory: PAC learning, VC dimension, Rademacher complexity, algorithmic stability, generalization bounds, sample complexity, uniform convergence, concentration inequalities (Hoeffding's inequality, Chernoff bounds, Azuma-Hoeffding inequality, McDiarmid's inequality), margin bounds, covering numbers, entropy numbers, approximation theory, universal approximation theorem, scattering networks
c. Recurrent Neural Networks (RNNs)
- ...
- Learning Theory: PAC learning, VC dimension, Rademacher complexity, algorithmic stability, generalization bounds, sample complexity, uniform convergence, concentration inequalities (Hoeffding's inequality, Chernoff bounds, Azuma-Hoeffding inequality, McDiarmid's inequality), margin bounds, covering numbers, entropy numbers, approximation theory, universal approximation theorem, reservoir computing
d. Generative Models (GANs, VAEs)
- ...
- Learning Theory: PAC learning, VC dimension, Rademacher complexity, algorithmic stability, generalization bounds, sample complexity, uniform convergence, concentration inequalities (Hoeffding's inequality, Chernoff bounds, Azuma-Hoeffding inequality, McDiarmid's inequality), margin bounds, covering numbers, entropy numbers, approximation theory, universal approximation theorem, optimal transport theory, Wasserstein distances, Sinkhorn distances
5. Natural Language Processing (NLP)
- ...
- Learning Theory: PAC learning, VC dimension, Rademacher complexity, algorithmic stability, generalization bounds, sample complexity, uniform convergence, concentration inequalities (Hoeffding's inequality, Chernoff bounds, Azuma-Hoeffding inequality, McDiarmid's inequality), margin bounds, covering numbers, entropy numbers, approximation theory, universal approximation theorem, distributional semantics, compositional semantics, formal languages and automata theory, computational complexity theory
6. Computer Vision
- ...
- Learning Theory: PAC learning, VC dimension, Rademacher complexity, algorithmic stability, generalization bounds, sample complexity, uniform convergence, concentration inequalities (Hoeffding's inequality, Chernoff bounds, Azuma-Hoeffding inequality, McDiarmid's inequality), margin bounds, covering numbers, entropy numbers, approximation theory, universal approximation theorem, scale-space theory, visual attention models, perceptual organization, Gestalt principles
- Differential Geometry: Riemannian geometry, Lie groups, Lie algebras, homogeneous spaces, symmetric spaces, principal bundles, fiber bundles, gauge theory, contact geometry, symplectic geometry
- Algebraic Geometry: varieties, schemes, sheaves, cohomology, Hodge theory, intersection theory, Chow groups, moduli spaces, toric varieties, tropical geometry
7. Recommender Systems
- ...
- Learning Theory: PAC learning, VC dimension, Rademacher complexity, algorithmic stability, generalization bounds, sample complexity, uniform convergence, concentration inequalities (Hoeffding's inequality, Chernoff bounds, Azuma-Hoeffding inequality, McDiarmid's inequality), margin bounds, covering numbers, entropy numbers, approximation theory, universal approximation theorem, collaborative filtering theory, matrix completion theory, low-rank approximation, compressed sensing
- Game Theory: cooperative game theory, non-cooperative game theory, Nash equilibrium, correlated equilibrium, Stackelberg equilibrium, mechanism design, auction theory, social choice theory, voting theory, stable matching, fair division
8. Theoretical Foundations
- Computational Learning Theory: PAC learning, agnostic learning, online learning, active learning, semi-supervised learning, unsupervised learning, reinforcement learning, multi-task learning, transfer learning, domain adaptation, lifelong learning, curriculum learning, meta-learning, learning to learn
- Statistical Learning Theory: empirical risk minimization, structural risk minimization, regularization, kernel methods, support vector machines, Gaussian processes, reproducing kernel Hilbert spaces, feature selection, model selection, cross-validation, bootstrap, ensemble methods
- Information Theory: source coding, channel coding, rate-distortion theory, network information theory, algorithmic information theory, Kolmogorov complexity, minimum description length, information geometry, information bottleneck, information theoretic learning
- Optimization Theory: convex optimization, non-convex optimization, combinatorial optimization, integer programming, quadratic programming, semidefinite programming, stochastic optimization, online optimization, distributed optimization, federated optimization, multi-objective optimization, Pareto optimality
- Game Theory and Decision Theory: game theory, decision theory, utility theory, prospect theory, risk measures, coherent risk measures, stochastic dominance, regret theory, social choice theory, mechanism design, auction theory, voting theory, stable matching, fair division
- Dynamical Systems and Control Theory: nonlinear dynamics, chaos theory, bifurcation theory, stability theory, Lyapunov theory, optimal control, adaptive control, robust control, stochastic control, model predictive control, observer design, system identification, fault detection and diagnosis
- Complexity Theory: computational complexity, communication complexity, information complexity, query complexity, sample complexity, algorithmic complexity, parameterized complexity, approximation algorithms, hardness of approximation, inapproximability, randomized algorithms, derandomization
- Cryptography and Security: cryptographic primitives, symmetric-key cryptography, public-key cryptography, hash functions, digital signatures, key exchange protocols, secure multiparty computation, homomorphic encryption, differential privacy, adversarial machine learning, robustness, interpretability, fairness, accountability, transparency
This further expanded map incorporates theoretical foundations and additional mathematical disciplines that underpin and intersect with machine learning, such as learning theory, optimization theory, game theory, information theory, dynamical systems and control theory, complexity theory, and cryptography and security. These theoretical frameworks provide a rigorous basis for analyzing and understanding the properties, limitations, and capabilities of various machine learning algorithms and models.
However, it is important to note that even this highly comprehensive map is not exhaustive, as machine learning is a rapidly evolving field that draws upon and contributes to a wide range of mathematical and computational disciplines. Researchers and practitioners in machine learning continue to develop new theories, algorithms, and applications, pushing the boundaries of what is possible and expanding the mathematical landscape of the field.
Creating a gigantic, detailed map of the mathematics used in modern physics is a significant undertaking that would require a substantial amount of time and space. However, I can provide you with a high-level overview of the main branches of mathematics that are essential to modern physics:
1. Calculus
- Differential calculus
- Integral calculus
- Vector calculus
- Tensor calculus
2. Linear Algebra
- Inner products and orthogonality
3. Differential Equations
- Partial differential equations (PDEs)
- Fourier series and transforms
- Laplace transforms
4. Complex Analysis
- Complex numbers and functions
- Cauchy's theorem and residue theorem
5. Probability and Statistics
- Expectation values and variances
- Statistical mechanics
6. Group Theory
- Symmetry groups
- Lie groups and Lie algebras
- Representations of groups
- Applications in particle physics and crystallography
7. Topology
- Homotopy and homology
- Fiber bundles and gauge theories
8. Functional Analysis
- Operators and their spectra
- Applications in quantum mechanics
9. Differential Geometry
- Geodesics and curvature
- Connections and covariant derivatives
- Applications in general relativity
10. Numerical Methods
- Monte Carlo methods
- Computational fluid dynamics
These branches of mathematics are interconnected and often used in combination to tackle various problems in modern physics, such as quantum mechanics, statistical mechanics, electromagnetism, general relativity, and particle physics. Each of these branches could be further expanded to include more detailed concepts and techniques.
"Here is a detailed map of the mathematics used in various fields of modern physics:
Classical Mechanics:
- Calculus (derivatives, integrals)
- Differential equations
- Linear algebra (vectors, matrices)
- Variational principles
- Lagrangian and Hamiltonian mechanics
Electromagnetism:
- Vector calculus (gradient, divergence, curl)
- Partial differential equations (Maxwell's equations)
- Tensor analysis
Thermodynamics and Statistical Mechanics:
- Differential equations
- Multivariate calculus
Quantum Mechanics:
- Linear algebra (Hilbert spaces, operators)
- Differential equations (Schrödinger equation)
- Group theory
Quantum Field Theory:
- Tensor analysis
- Group theory (Lie groups, Lie algebras)
- Topology
- Functional analysis (path integrals, operator algebras)
- Differential geometry (manifolds, tensors, curvature)
- Partial differential equations (Einstein field equations)
- Variational principles
- Tensor analysis
- Topology
Particle Physics:
- Group theory (representation theory, Lie groups)
- Topology
Condensed Matter Physics:
- Differential equations
- Topology
- Group theory
Fluid Dynamics:
- Partial differential equations (Navier-Stokes equations)
- Vector calculus
- Perturbation theory
Plasma Physics:
- Partial differential equations (magnetohydrodynamic equations)
- Kinetic theory
Astrophysics and Cosmology:
- General relativity
- Nuclear physics
- Particle physics
Mathematical Physics (general tools):
- Differential equations (ordinary and partial)
- Operator theory
- Topology
- Lie groups and algebras
- Variational principles
This map highlights the main mathematical tools and concepts used in each field of modern physics. However, it's important to note that there is significant overlap between these fields, and many mathematical techniques find applications across multiple areas of physics.
Here is an expanded and more detailed map of the mathematics used in various fields of modern physics:
Classical Mechanics:
- Calculus: derivatives, integrals, differential equations (ordinary and partial), variational calculus
- Linear algebra: vectors, matrices, eigenvalues, eigenvectors
- Variational principles: Euler-Lagrange equations, Hamilton's principle
- Lagrangian and Hamiltonian mechanics: generalized coordinates, phase space, Poisson brackets
- Dynamical systems: stability analysis, bifurcations, chaos theory
- Perturbation theory: regular and singular perturbations
- Numerical methods: Runge-Kutta, finite difference, finite element
Electromagnetism:
- Vector calculus: gradient, divergence, curl, Stokes' theorem, Green's theorem
- Partial differential equations: Maxwell's equations, wave equation, Helmholtz equation
- Complex analysis: analytic functions, Cauchy-Riemann equations, residue theorem
- Tensor analysis: electromagnetic field tensor, stress-energy tensor
- Differential forms: exterior derivative, Hodge star operator
- Gauge theory: gauge transformations, fiber bundles, connections
- Numerical methods: finite difference time domain (FDTD), method of moments (MoM)
Thermodynamics and Statistical Mechanics:
- Probability theory: random variables, probability distributions, central limit theorem
- Differential equations: transport equations, Fokker-Planck equation
- Multivariate calculus: Jacobians, Hessians, Lagrange multipliers
- Linear algebra: matrix exponentials, eigenvalue problems
- Information theory: entropy, mutual information, Kullback-Leibler divergence
- Stochastic processes: Markov chains, Brownian motion, Langevin equation
- Monte Carlo methods: Metropolis-Hastings algorithm, Gibbs sampling
Quantum Mechanics:
- Linear algebra: Hilbert spaces, linear operators, eigenvalues, eigenvectors
- Differential equations: Schrödinger equation, Dirac equation, Klein-Gordon equation
- Probability theory: Born rule, density matrices, quantum measurement
- Group theory: symmetries, representations, Lie groups, Lie algebras
- Functional analysis: self-adjoint operators, spectral theory, Banach spaces
- Perturbation theory: time-independent and time-dependent perturbation theory
- Numerical methods: finite difference, variational methods, quantum Monte Carlo
Quantum Field Theory:
- Tensor analysis: Lorentz transformations, spinors, Clifford algebras
- Group theory: Lie groups (U(1), SU(2), SU(3)), Lie algebras, representation theory
- Differential geometry: fiber bundles, connections, curvature, characteristic classes
- Topology: homotopy groups, homology, cohomology, index theorems
- Complex analysis: dispersion relations, Feynman integrals, renormalization
- Functional analysis: path integrals, operator algebras, BRST quantization
- Perturbation theory: Feynman diagrams, renormalization group, effective field theories
- Lattice field theory: lattice gauge theory, lattice QCD
- Differential geometry: manifolds, tangent spaces, differential forms, Riemannian geometry
- Tensor analysis: Riemann curvature tensor, Ricci tensor, energy-momentum tensor
- Partial differential equations: Einstein field equations, wave equations in curved spacetime
- Variational principles: Einstein-Hilbert action, Palatini formalism
- Topology: causal structure, singularity theorems, black hole thermodynamics
- Numerical relativity: ADM formalism, BSSN formalism, pseudospectral methods
Particle Physics:
- Group theory: representation theory, Lie groups (U(1), SU(2), SU(3)), grand unification
- Differential geometry: gauge theories, spontaneous symmetry breaking, Higgs mechanism
- Topology: instantons, monopoles, solitons, anomalies
- Complex analysis: dispersion relations, analytic S-matrix theory
- Functional analysis: path integrals, operator product expansion, conformal field theory
- Perturbative QCD: Feynman diagrams, parton model, factorization theorems
- Lattice QCD: discretization of QCD, numerical simulations
Condensed Matter Physics:
- Differential equations: Schrödinger equation, Ginzburg-Landau theory, Bogoliubov-de Gennes equations
- Linear algebra: tight-binding models, Wannier functions, Bloch functions
- Probability theory: stochastic processes, master equations, fluctuation-dissipation theorem
- Topology: topological insulators, Berry phase, Chern numbers
- Group theory: space groups, point groups, representation theory
- Many-body theory: Green's functions, Feynman diagrams, renormalization group
- Numerical methods: density functional theory (DFT), quantum Monte Carlo, tensor networks
Fluid Dynamics:
- Partial differential equations: Navier-Stokes equations, Euler equations, Boltzmann equation
- Vector calculus: vorticity, circulation, Kelvin's theorem, Helmholtz decomposition
- Complex analysis: conformal mapping, potential flow, Joukowsky transform
- Perturbation theory: boundary layer theory, Kolmogorov's theory of turbulence
- Numerical methods: finite volume methods, spectral methods, lattice Boltzmann methods
Plasma Physics:
- Partial differential equations: magnetohydrodynamic (MHD) equations, Vlasov equation
- Kinetic theory: Boltzmann equation, Fokker-Planck equation, quasilinear theory
- Statistical mechanics: BBGKY hierarchy, kinetic equations
- Fluid dynamics: ideal MHD, resistive MHD, two-fluid models
- Numerical methods: particle-in-cell (PIC) methods, gyrokinetic simulations
Astrophysics and Cosmology: