CFNN: Continued Fraction Neural Network

TL;DR

CFNNs introduce a novel neural network architecture combining continued fractions with gradient optimization, enabling efficient and accurate modeling of complex non-linear functions with fewer parameters and enhanced robustness.

Contribution

The paper presents CFNNs, a new neural network model that incorporates continued fractions to improve approximation of complex functions with fewer parameters and better stability.

Findings

CFNNs outperform MLPs in precision with fewer parameters.

CFNNs show up to 47-fold improvement in noise robustness.

CFNNs demonstrate exponential convergence and stability guarantees.

Abstract

Accurately characterizing non-linear functional manifolds with singularities is a fundamental challenge in scientific computing. While Multi-Layer Perceptrons (MLPs) dominate, their spectral bias hinders resolving high-curvature features without excessive parameters. We introduce Continued Fraction Neural Networks (CFNNs), integrating continued fractions with gradient-based optimization to provide a ``rational inductive bias.'' This enables capturing complex asymptotics and discontinuities with extreme parameter frugality. We provide formal approximation bounds demonstrating exponential convergence and stability guarantees. To address recursive instability, we develop three implementations: CFNN-Boost, CFNN-MoE, and CFNN-Hybrid. Benchmarks show CFNNs consistently outperform MLPs in precision with one to two orders of magnitude fewer parameters, exhibiting up to a 47-fold improvement in noise robustness and physical consistency. By bridging black-box flexibility and white-box transparency, CFNNs establish a reliable ``grey-box'' paradigm for AI-driven scientific research.