CFNN: Continued Fraction Neural Network
arXiv:2603.20634v1 Announce Type: new 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 bridgi
arXiv:2603.20634v1 Announce Type: new 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.
Executive Summary
This article introduces Continued Fraction Neural Networks (CFNNs) as a novel approach to address the limitations of Multi-Layer Perceptrons (MLPs) in characterizing non-linear functional manifolds with singularities. By integrating continued fractions with gradient-based optimization, CFNNs provide a "rational inductive bias" enabling the capture of complex asymptotics and discontinuities with extreme parameter frugality. The authors provide formal approximation bounds demonstrating exponential convergence and stability guarantees, and present three implementations to address recursive instability. Benchmarks show CFNNs consistently outperform MLPs in precision, robustness, and physical consistency, establishing a reliable "grey-box" paradigm for AI-driven scientific research.
Key Points
- ▸ Introduction of Continued Fraction Neural Networks (CFNNs) as a novel approach to address the limitations of MLPs
- ▸ CFNNs provide a "rational inductive bias" enabling the capture of complex asymptotics and discontinuities
- ▸ Formal approximation bounds demonstrate exponential convergence and stability guarantees
Merits
Rational Inductive Bias
The integration of continued fractions with gradient-based optimization enables the capture of complex asymptotics and discontinuities with extreme parameter frugality.
Exponential Convergence and Stability Guarantees
Formal approximation bounds demonstrate exponential convergence and stability guarantees, ensuring the reliability and robustness of CFNNs.
Grey-Box Paradigm
CFNNs establish a reliable "grey-box" paradigm for AI-driven scientific research, bridging black-box flexibility and white-box transparency.
Demerits
Limited Exploratory Analysis
The article primarily focuses on the performance of CFNNs in comparison to MLPs, with limited exploratory analysis of the potential applications and limitations of CFNNs in various scientific domains.
Lack of Interpretability Techniques
While CFNNs provide a "rational inductive bias", the article does not discuss techniques for interpreting and understanding the decision-making processes of CFNNs, which may limit their adoption in certain scientific applications.
Expert Commentary
The introduction of CFNNs represents a significant advancement in the field of deep learning, addressing the limitations of MLPs and offering a novel approach to characterizing non-linear functional manifolds with singularities. The authors' rigorous formal analysis and experimental evaluations demonstrate the effectiveness and reliability of CFNNs. However, further research is needed to explore the potential applications and limitations of CFNNs in various scientific domains and to develop techniques for interpreting and understanding the decision-making processes of CFNNs.
Recommendations
- ✓ Further research should focus on exploring the potential applications of CFNNs in various scientific domains, including physics, engineering, and biology.
- ✓ The development of interpretability techniques for CFNNs is essential to facilitate their adoption in scientific applications and to ensure transparency and accountability in AI-driven scientific research.
Sources
Original: arXiv - cs.LG
