用连分数构造高效神经网络，精准捕捉奇异非线性特征。
CFNN: Continued Fraction Neural Network
- 将连分数融入神经网络，引入有理数归纳偏置提升建模能力。
- 参数减少1~2个数量级，噪声鲁棒性提升47倍，物理一致性更强。
- 适合需要高精度与可解释性的科学计算场景，如复杂系统建模。
在科学计算中，准确刻画带有奇点的非线性函数流形是一项基本挑战。尽管多层感知机（MLPs）占主导地位，但其谱偏差限制了对高曲率特征的解析能力，需大量参数支持。本文提出连续分数神经网络（CFNN），将连分数与梯度优化相结合，引入“有理归纳偏置”，实现对复杂渐近行为和不连续性的高效捕捉，且参数极省。我们提供了形式化逼近界，证明其指数收敛性和稳定性。为解决递归不稳定性，设计了三种实现：CFNN-Boost、CFNN-MoE 和 CFNN-Hybrid。基准测试显示，CFNN 在精度上持续优于 MLP，参数量少1~2个数量级，噪声鲁棒性最高提升47倍，物理一致性显著增强。该方法融合黑箱灵活性与白箱透明性，为人工智能驱动的科学研究建立了可靠的“灰箱”范式。
原文摘要 · Abstract (English)
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.
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