The Wavefunction of Continuous-Time Recurrent Neural Networks

TL;DR

This paper introduces a quantum wavefunction framework for continuous-time recurrent neural networks by deriving a Hamiltonian, quantizing it, and solving the Schrödinger equation, revealing conditions on network weights and hyperparameters.

Contribution

It presents a novel approach to analyze CTRNNs through quantum mechanics, linking neural dynamics to wavefunctions and boundary conditions.

Findings

Derived a Hamiltonian from classical CTRNN dynamics

Quantized the Hamiltonian using Weyl quantization

Obtained wavefunctions via Schrödinger equation with hypergeometric functions

Abstract

In this paper, we explore the possibility of deriving a quantum wavefunction for continuous-time recurrent neural network (CTRNN). We did this by first starting with a two-dimensional dynamical system that describes the classical dynamics of a continuous-time recurrent neural network, and then deriving a Hamiltonian. After this, we quantized this Hamiltonian on a Hilbert space $\mathbb{H} = L^2(\mathbb{R})$ using Weyl quantization. We then solved the Schrodinger equation which gave us the wavefunction in terms of Kummer's confluent hypergeometric function corresponding to the neural network structure. Upon applying spatial boundary conditions at infinity, we were able to derive conditions/restrictions on the weights and hyperparameters of the neural network, which could potentially give insights on the the nature of finding optimal weights of said neural networks.