Prerequisites for Physics-Informed Neural Networks

Direct prerequisites (9)

1. The Jacobian Matrixlayer 0A, tier 1
2. Automatic Differentiationlayer 1, tier 1
3. Feedforward Networks and Backpropagationlayer 2, tier 1
4. Gradient Descent Variantslayer 1, tier 1
5. Classical ODEs: Existence, Stability, and Numerical Methodslayer 1, tier 1
6. Divergence, Curl, and Line Integralslayer 0A, tier 2
7. Kolmogorov-Arnold Networks (KANs)layer 4, tier 2
8. PDE Fundamentals for Machine Learninglayer 1, tier 2
9. Symbolic Regression and Equation Discoverylayer 4, tier 3

Reachable through the chain (131)

These topics are not directly cited as prerequisites but are reached transitively by following the chain upward. Working through the direct prerequisites pulls these in.

• Differentiation in Rⁿlayer 0A, tier 1
• Sets, Functions, and Relationslayer 0A, tier 1
• Basic Logic and Proof Techniqueslayer 0A, tier 2
• Vectors, Matrices, and Linear Mapslayer 0A, tier 1
• Continuity in Rⁿlayer 0A, tier 1
• Metric Spaces, Convergence, and Completenesslayer 0A, tier 1
• Matrix Calculuslayer 1, tier 1
• The Hessian Matrixlayer 0A, tier 1
• Matrix Operations and Propertieslayer 0A, tier 1
• Linear Independencelayer 0A, tier 1
• Eigenvalues and Eigenvectorslayer 0A, tier 1
• Inner Product Spaces and Orthogonalitylayer 0A, tier 1
• Matrix Normslayer 0A, tier 1
• Vector Calculus Chain Rulelayer 0A, tier 1
• Taylor Expansionlayer 0A, tier 1
• Activation Functionslayer 1, tier 1
• Convex Optimization Basicslayer 1, tier 1
• Common Inequalitieslayer 0A, tier 1
• Common Probability Distributionslayer 0A, tier 1
• Exponential Function Propertieslayer 0A, tier 1
• Integration and Change of Variableslayer 0A, tier 2
• Measure-Theoretic Probabilitylayer 0B, tier 1
• Cardinality and Countabilitylayer 0A, tier 2
• Kolmogorov Probability Axiomslayer 0A, tier 1
• Random Variableslayer 0A, tier 1
• Zermelo-Fraenkel Set Theorylayer 0A, tier 2
• Dynamic Programminglayer 0A, tier 1
• Graph Algorithms Essentialslayer 0A, tier 2
• Greedy Algorithmslayer 0A, tier 2
• GraphSLAM and Factor Graphslayer 3, tier 2
• Inverse and Implicit Function Theoremlayer 0A, tier 2
• Positive Semidefinite Matriceslayer 0A, tier 1
• Submodular Optimizationlayer 3, tier 3
• Decision Trees and Ensembleslayer 2, tier 2
• Empirical Risk Minimizationlayer 2, tier 1
• Concentration Inequalitieslayer 1, tier 1
• Expectation, Variance, Covariance, and Momentslayer 0A, tier 1
• Joint, Marginal, and Conditional Distributionslayer 0A, tier 1
• Triangular Distributionlayer 0A, tier 2
• Central Limit Theoremlayer 0B, tier 1
• Law of Large Numberslayer 0B, tier 1
• Borel-Cantelli Lemmaslayer 0B, tier 1
• Modes of Convergence of Random Variableslayer 0B, tier 1
• Characteristic Functionslayer 1, tier 1
• Moment Generating Functionslayer 0A, tier 2
• Martingale Theorylayer 0B, tier 2
• Radon-Nikodym and Conditional Expectationlayer 0B, tier 1
• Skewness, Kurtosis, and Higher Momentslayer 1, tier 1
• High-Dimensional Probability (Vershynin)layer 2, tier 1
• Cramér-Wold Theoremlayer 1, tier 2
• Loss Functions Cataloglayer 1, tier 1
• Logistic Regressionlayer 1, tier 1
• Maximum Likelihood Estimation: Theory, Information Identity, and Asymptotic Efficiencylayer 0B, tier 1
• KL Divergencelayer 1, tier 1
• Information Theory Foundationslayer 0B, tier 2
• Distance Metrics Comparedlayer 1, tier 2
• Non-Euclidean and Hyperbolic Geometrylayer 1, tier 2
• Total Variation Distancelayer 1, tier 1
• Method of Momentslayer 0B, tier 2
• Data Preprocessing and Feature Engineeringlayer 1, tier 1
• Linear Regressionlayer 1, tier 1
• The Elements of Statistical Learning (Hastie, Tibshirani, Friedman)layer 0B, tier 1
• Naive Bayeslayer 1, tier 2
• Robust Statistics and M-Estimatorslayer 3, tier 2
• Minimax and Saddle Pointslayer 2, tier 2
• Convex Dualitylayer 2, tier 1
• Subgradients and Subdifferentialslayer 1, tier 1
• Winsorizationlayer 1, tier 3
• Order Statisticslayer 1, tier 2
• Sequences and Series of Functionslayer 0A, tier 2
• Understanding Machine Learning (Shalev-Shwartz, Ben-David)layer 1, tier 1
• Bias-Variance Tradeofflayer 2, tier 2
• Elastic Netlayer 2, tier 2
• Ridge Regressionlayer 1, tier 1
• Shrinkage Estimation and the James-Stein Estimator: Inadmissibility, SURE, and Brown's Characterizationlayer 0B, tier 1
• Cramér-Rao Bound: Information Inequality, Achievability, and Sharper Variantslayer 0B, tier 1
• Fisher Information: Curvature, KL Geometry, and the Natural Gradientlayer 0B, tier 1
• Basu's Theoremlayer 0B, tier 3
• Sufficient Statistics and Exponential Familieslayer 0B, tier 2
• Minimax Lower Bounds: Le Cam, Fano, Assouad, and the Reduction to Testinglayer 3, tier 1
• Empirical Processes and Chaininglayer 3, tier 2
• Rademacher Complexitylayer 3, tier 1
• VC Dimensionlayer 2, tier 1
• Counting and Combinatoricslayer 0A, tier 2
• Hypothesis Classes and Function Spaceslayer 2, tier 1
• PAC Learning Frameworklayer 1, tier 1
• Uniform Convergencelayer 2, tier 1
• Adaptive Learning Is Not IIDlayer 3, tier 2
• Bernstein Inequalitylayer 2, tier 1
• Bennett's Inequalitylayer 2, tier 1
• Chernoff Boundslayer 1, tier 1
• Hoeffding's Lemmalayer 1, tier 1
• Realizability Assumptionlayer 2, tier 1
• Loss Functionslayer 1, tier 2
• Slud's Inequalitylayer 2, tier 2
• Bias-Complexity Tradeofflayer 2, tier 2
• No-Free-Lunch Theoremlayer 2, tier 2
• Glivenko-Cantelli Theoremlayer 2, tier 2
• McDiarmid's Inequalitylayer 3, tier 1
• Sub-Gaussian Random Variableslayer 2, tier 1
• Epsilon-Nets and Covering Numberslayer 3, tier 1
• Contraction Inequalitylayer 3, tier 2
• Sub-Exponential Random Variableslayer 2, tier 1
• Chi-Squared Concentrationlayer 2, tier 1
• Symmetrization Inequalitylayer 3, tier 1
• Asymptotic Statistics: M-Estimators, Delta Method, LANlayer 0B, tier 1
• Measure Concentration and Geometric Functional Analysislayer 3, tier 1
• Stochastic Processes for MLlayer 2, tier 2
• Gauss-Markov Theoremlayer 2, tier 1
• The Multivariate Normal Distributionlayer 0B, tier 1
• Maximum A Posteriori (MAP) Estimationlayer 0B, tier 1
• Bayesian Estimationlayer 0B, tier 2
• Lasso Regressionlayer 2, tier 1
• Generalized Additive Modelslayer 2, tier 2
• MARS (Multivariate Adaptive Regression Splines)layer 2, tier 3
• K-Nearest Neighborslayer 1, tier 2
• Deep Learning (Goodfellow, Bengio, Courville)layer 0B, tier 1
• Gradient Boostinglayer 2, tier 1
• AdaBoostlayer 2, tier 2
• Cubist and Model Treeslayer 2, tier 3
• Perceptronlayer 1, tier 2
• Tensors and Tensor Operationslayer 0A, tier 1
• Pandas and NumPy Fundamentalslayer 4, tier 3
• Universal Approximation Theoremlayer 2, tier 1
• Fast Fourier Transformlayer 1, tier 2
• Complex Numbers for Fourierlayer 0A, tier 2
• Stochastic Differential Equationslayer 3, tier 2
• Ito's Lemmalayer 3, tier 2
• Stochastic Calculus for MLlayer 3, tier 3
• Functional Analysis Corelayer 0B, tier 2
• Sparse Recovery and Compressed Sensinglayer 4, tier 3