A framework for solving functional equations with neural networks
Lars Kindermann, Achim Lewandowski, Peter Protzel · 2001
In his "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 specific 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.