Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Chemical Reaction Networks Learn Better than Spiking Neural Networks

Sophie Jaffard, Ivo F. Sbalzarini

Abstract

We mathematically prove that chemical reaction networks without hidden layers can solve tasks for which spiking neural networks require hidden layers. Our proof uses the deterministic mass-action kinetics formulation of chemical reaction networks. Specifically, we prove that a certain reaction network without hidden layers can learn a classification task previously proved to be achievable by a spiking neural network with hidden layers. We provide analytical regret bounds for the global behavior of the network and analyze its asymptotic behavior and Vapnik-Chervonenkis dimension. In a numerical experiment, we confirm the learning capacity of the proposed chemical reaction network for classifying handwritten digits in pixel images, and we show that it solves the task more accurately and efficiently than a spiking neural network with hidden layers. This provides a motivation for machine learning in chemical computers and a mathematical explanation for how biological cells might exhibit more efficient learning behavior within biochemical reaction networks than neuronal networks.

Chemical Reaction Networks Learn Better than Spiking Neural Networks

Abstract

We mathematically prove that chemical reaction networks without hidden layers can solve tasks for which spiking neural networks require hidden layers. Our proof uses the deterministic mass-action kinetics formulation of chemical reaction networks. Specifically, we prove that a certain reaction network without hidden layers can learn a classification task previously proved to be achievable by a spiking neural network with hidden layers. We provide analytical regret bounds for the global behavior of the network and analyze its asymptotic behavior and Vapnik-Chervonenkis dimension. In a numerical experiment, we confirm the learning capacity of the proposed chemical reaction network for classifying handwritten digits in pixel images, and we show that it solves the task more accurately and efficiently than a spiking neural network with hidden layers. This provides a motivation for machine learning in chemical computers and a mathematical explanation for how biological cells might exhibit more efficient learning behavior within biochemical reaction networks than neuronal networks.
Paper Structure (32 sections, 12 theorems, 74 equations, 4 figures)

This paper contains 32 sections, 12 theorems, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Network architecture
  4. Description of the chemical reaction network
  5. Selection phase
  6. Learning phase
  7. Inference phase
  8. Network complexity
  9. Analogy with spiking neural networks
  10. Links to artificial neural networks
  11. Performance measures
  12. Optimal weight family
  13. Theoretical results
  14. Selection phase
  15. Average learning
  16. ...and 17 more sections

Key Result

Proposition 9

There exist positive constants $C_1$ and $C_2$, independent of $\lvert I \rvert, \lvert \Bar{J}_n \rvert, \lvert K \rvert$, $T_{\text{renorm}}$, and $M$, such that for every $j\in \Bar{J}_n$ the concentration $w^j(T^0_1)$ satisfies Consequently, for $T_{\text{renorm}}\to \infty$, we have $w^j(T_1^0) \to \frac{b_1}{b_2}.$

Figures (4)

  • Figure 1: Schematic representation of the time evolution of the chemical reaction network during the selection and learning phases.
  • Figure 2: Analogy between the structure of the SNN proposed by jaffard2026chani (left) and the present CRN (right).
  • Figure 3: Example images from the handwritten digits data set. Each image shows a digit (ground-truth labels above the images) in 8$\times$8 grayscale pixels.
  • Figure 4: Numerical results on the handwritten digits data set. We plot the CRN's classification accuracy on the test set as a function of network complexity, quantified by the number of selected input sets $\lvert \bar{J}_n \rvert$ during the selection phase. We show results for network depths $n=1$ (no hidden layer, left panel) and $n=2$ (one hidden layer, right panel). The shaded bands are the 10% confidence intervals over 100 independent repetitions of each configuration. All rate constants are equal to 1, except $c_k$ and $d_k$ for $k\in K$, which are equal to $\frac{10}{9}\eta$ and $10\eta$, respectively. The initial species concentrations are $A=1$ and $S=3$, and the noise variance is set to $\sigma^2=0.00001$. For network depth $n=1$, we chose $\eta = 0.0005$ and for network depth $n=2$, we chose $\eta = 0.0001$.

Theorems & Definitions (19)

  • Definition 1: Flux of a set $j$
  • Definition 2: Species discrepancy
  • Definition 3: Species discrepancy with constant weights
  • Definition 4: Network discrepancy
  • Definition 5: Network discrepancy with constant weights
  • Definition 7: Flux discrepancy
  • Definition 8: Optimal weight family
  • Proposition 9
  • Proposition 11
  • Proposition 13: Regret bound of species $X^k$
  • ...and 9 more