Departure question
Find the edge of solvabilityPort 4 of 5Cross mechanical procedures, proof, quantum computation, learning, and strategy to explore the limits of calculation and choice.
This voyage is an editorial path for understanding, not a claim of direct historical influence or sole invention.
"A single hidden-layer neural network can approximate any continuous function to arbitrary precision." Cybenko's theorem (1989). The first mathematical guarantee for why deep learning works. Given enough width, expressiveness is unbounded — but how wide you actually need is still open (one of deep learning theory's biggest gaps).
A neural network with one hidden layer can approximate any continuous function.
Ports in time
Follow the scenes to see problems, notation, standards of proof, and applications changing across different times and places.
George Cybenko proved that a neural network with one hidden layer and a sigmoidal activation can approximate any continuous function on a compact domain arbitrarily well.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearKurt Hornik showed that universal approximation depends less on a particular sigmoid and holds for broad classes of activation functions.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearA growing body of theory demonstrated that some functions can be represented exponentially more efficiently by deep networks than by shallow ones.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearA theoretical foundation for neural networks, general function approximation, value-function estimation in reinforcement learning, and physics-informed neural networks for differential equations.
Curated sources and problems. Bring one discovery back from OEIS, Project Euler, MathOverflow, or arXiv.
No concept belongs to one person
These are not inventor credits. They are different ports: opening a problem, sharpening a language, or carrying it into another world.
Number lenses
These numbers are editorial lenses for the voyage, not required prerequisites.
Cross mechanical procedures, proof, quantum computation, learning, and strategy to explore the limits of calculation and choice.
Open the number voyage
Cross mechanical procedures, proof, quantum computation, learning, and strategy to explore the limits of calculation and choice.
Open the number voyage
Cross mechanical procedures, proof, quantum computation, learning, and strategy to explore the limits of calculation and choice.
Open the number voyage
Concept genealogy
Concepts arriving from before
Current port
Universal Approximation Theorem
Concepts opened from here
No direct successor port is curated yet.
Only direct editorial links are shown; this is not a complete learning order or historical influence line.
