Alphabeta MathNeural ODEs and continuous normalizing flows
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1 · Prerequisites
Abstract Algebra
Differential Geometry
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
Numerical Differential Equations and Stochastic Dynamics
Numerical Optimization Inverse Problems and Uncertainty Quantification
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Areas of Elementary Plane Figures
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Inverse and Implicit Function Theorems
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
A parameterized neural ordinary differential equation
Theorem
A globally Lipschitz neural ODE has a global smooth flow
Theorem
Instantaneous change of variables along a neural ODE flow
Algorithm
Exact continuous adjoint sensitivity for a neural ODE
Sources
- Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud, Neural Ordinary Differential Equations
- George Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende, Shakir Mohamed, and Balaji Lakshminarayanan, Normalizing Flows for Probabilistic Modeling and Inference
- Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, Chapter 2, §§2.2–2.5
- Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud, Neural Ordinary Differential Equations, §4 and Appendix A
- George Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende, Shakir Mohamed, and Balaji Lakshminarayanan, Normalizing Flows for Probabilistic Modeling and Inference, §4.1
- Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud, Neural Ordinary Differential Equations, §2 and Appendix B
- George Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende, Shakir Mohamed, and Balaji Lakshminarayanan, Normalizing Flows for Probabilistic Modeling and Inference, §4.2.2