A framework for solving functional equations with neural networks

摘要

In this "essay towards a calculus of functions" from 1815 Charles Babbage introduced a branch of mathematics now known as the theory of functional equations [1]. But since then finding concrete solutions for a given functional equation remained a hard task in many cases. For one of his examples, the now famous "Babbage equation" Φ(Φ(x)) = x, which solutions Φ are called "the roots of identity" and the more general equation Φ(Φ(x)) = f(x) which defines kind of a "square root" of some given function f we have previously shown that this type of equation can be solved approximately by neural networks with a special topology and learning rule. Here we extend that method towards a wider range of functional equations which can be mapped in similar ways to neural networks too. The method is demonstrated on - but not limited to - multilayer perceptrons. We present a first sketch of this ideas here on some important equations.