Automatic Implementation of Neural Networks through Reaction Networks--Part II: Error Analysis

TL;DR

This paper develops an error analysis framework for biochemical neural networks, establishing bounds on implementation errors and demonstrating exponential convergence, with numerical validation on classification tasks.

Contribution

It introduces a comprehensive error analysis method for biochemical neural networks, including error bounds and convergence properties, enhancing the reliability of automated biochemical neural implementations.

Findings

Error bounds for biochemical neural network modules

Exponential convergence of error with phase length

Numerical validation on classification examples

Abstract

This paired article aims to develop an automated and programmable biochemical fully connected neural network (BFCNN) with solid theoretical support. In Part I, a concrete design for BFCNN is presented, along with the validation of the effectiveness and exponential convergence of computational modules. In this article, we establish the framework for specifying the realization errors by monitoring the errors generated from approaching equilibrium points in individual modules, as well as their vertical propagation from upstream modules and horizontal accumulation from previous iterations. Ultimately, we derive the general error upper bound formula for any iteration and illustrate its exponential convergence order with respect to the phase length of the utilized chemical oscillator. The numerical experiments, based on the classification examples, reveal the tendency of total errors related to both the phase length and iteration number.