用几何流统一神经网络参数空间与量子纠缠，提升学习效率与稳定性。
Geometric Meta-Learning via Coupled Ricci Flow: Unifying Knowledge Representation and Quantum Entanglement
- 通过耦合里奇流动态调整参数空间以匹配损失曲面结构。
- 实现2.1倍加速收敛，拓扑简化63%，复杂度保持O(N log N)。
- 适用于需要高效小样本学习和理论可解释性的研究者。
本文提出一种融合几何流与深度学习的统一框架，包含三项核心创新：首先，设计热力学耦合里奇流，动态适应参数空间几何以匹配损失景观拓扑，严格证明其保持等距知识嵌入（定理~\ref{thm:isometric}）；其次，通过曲率爆破分析推导出显式相变阈值与临界学习率（定理~\ref{thm:critical}），实现基于几何手术的自动奇点修复（引理~\ref{lem:surgery}）；第三，建立神经网络与共形场论间的类AdS/CFT全息对偶（定理~\ref{thm:ads}），为正则化设计提供纠缠熵上界。实验表明，该方法实现2.1倍收敛加速与63%拓扑简化，同时维持\mathcal{O}(N\log N)复杂度，在少样本准确率上比黎曼基线提升15.2%。理论上，通过结合佩雷尔曼熵与Wasserstein梯度流的新李雅普诺夫函数，证明了指数稳定性（定理~\ref{thm:converge}），显著推动几何深度学习发展。
原文摘要 · Abstract (English)
This paper establishes a unified framework integrating geometric flows with deep learning through three fundamental innovations. First, we propose a thermodynamically coupled Ricci flow that dynamically adapts parameter space geometry to loss landscape topology, formally proved to preserve isometric knowledge embedding (Theorem~\ref{thm:isometric}). Second, we derive explicit phase transition thresholds and critical learning rates (Theorem~\ref{thm:critical}) through curvature blowup analysis, enabling automated singularity resolution via geometric surgery (Lemma~\ref{lem:surgery}). Third, we establish an AdS/CFT-type holographic duality (Theorem~\ref{thm:ads}) between neural networks and conformal field theories, providing entanglement entropy bounds for regularization design. Experiments demonstrate 2.1$\times$ convergence acceleration and 63\% topological simplification while maintaining $\mathcal{O}(N\log N)$ complexity, outperforming Riemannian baselines by 15.2\% in few-shot accuracy. Theoretically, we prove exponential stability (Theorem~\ref{thm:converge}) through a new Lyapunov function combining Perelman entropy with Wasserstein gradient flows, fundamentally advancing geometric deep learning.
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