Kolmogorov-Arnold Representation Theorem

In real analysis and approximation theory, the Kolmogorov–Arnold representation theorem (or superposition theorem) states that every multivariate continuous function  It can be represented as a superposition of two-parameter additions of continuous functions of one variable. It solves a more constrained form of Hilbert's 13th problem, so the original Hilbert's 13th problem is a corollary. The Kolmogorov-Arnold representation theorem makes it easier to analyze complex dynamical systems because it allows us to map nonlinear systems into linear systems, which are generally easier to analyze and understand.

The theorem was first proposed by Soviet mathematician Andrey Kolmogorov and further developed by his student Vladimir Arnold in 1957. The theorem was originally motivated by the question of how multivariate functions can be represented by a set of simpler functions, a fundamental problem in mathematics and theoretical computer science, and in fact partially answers the 13th of the famous 23 problems of mathematician Hilbert: whether the seventh-degree equation can be solved using addition, subtraction, multiplication, division, and combinations of algebraic functions of up to two variables. The Kolmogorov-Arnold theorem is in the context of a wider range of continuous functions, rather than the framework of algebraic equations originally proposed by Hilbert, and is therefore only a partial solution.

References

【1】https://juejin.cn/post/7364964796988932105

【2】https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold_representation_theorem