Generating functionals for guided self-organization

TL;DR

This paper introduces the concept of generating functionals for modeling self-organizing complex systems, highlighting their advantages in systematic formulation, minimalism, and flexibility, with applications to neural networks.

Contribution

It proposes using generating functionals as a systematic and flexible approach to model complex self-organizing systems, emphasizing their advantages over traditional methods.

Findings

Generating functionals enable systematic formulation of complex systems.

Multiple generating functionals can define a system without combining into one.

Applications demonstrated in neural network adaptation.

Abstract

Time evolution equations for dynamical systems can often be derived from generating functionals. Examples are Newton's equations of motion in classical dynamics which can be generated within the Lagrange or the Hamiltonian formalism. We propose that generating functionals for self-organizing complex systems offer several advantages. Generating functionals allow to formulate complex dynamical systems systematically and the results obtained are typically valid for classes of complex systems, as defined by the type of their respective generating functionals. The generated dynamical systems tend, in addition, to be minimal, containing only few free and undetermined parameters. We point out that two or more generating functionals may be used to define a complex system and that multiple generating function may not, and should not, be combined into a single overall objective function. We provide and discuss examples in terms of adapting neural networks.