A framework for Solving Functional Equations with Neural Networks

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 concrete solutions for a given function-al 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 func-tion 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.