用分形编码让神经网络突破混沌系统预测极限
The Fractal Neural Operator: Overcoming Spectral Bias in Chaotic Attractors via Prime-Harmonic Weierstrass Encodings
- 用素数谐波编码替代传统几何编码，实现无限频谱分辨率
- 在洛伦兹-63系统上将预测时长远超基准方法2.3倍
- 为混沌系统建模提供新思路，适合动态系统研究者
深度学习模型，尤其是Transformer和神经算子，存在显著的“谱偏见”，本质上像低通滤波器，会平滑高频信息。在流体动力学中影响不大，但在混沌动力系统中却导致灾难性后果，因为其奇异吸引子具有分形几何和无限谱密度特征。本文提出分形神经算子（FNO），采用非共振素数基来逼近连续动力系统。与传统几何编码（如$2^k$）相比，其谐波魏尔斯特拉斯编码能注入无限频谱分辨率。实验表明，FNO将洛伦兹-63系统的有效预测时长远超现有基于储层计算的基准方法，达到347个李雅普诺夫时间，提升幅度达2.3倍。结果表明，'混沌'并非神经网络无法预测，而是需要非可微的分形嵌入流形。
原文摘要 · Abstract (English)
Deep learning models, particularly Transformers and Neural Operators, exhibit a well-documented "spectral bias," effectively acting as low-pass filters that smooth out high-frequency information. While benign in fluid dynamics, this bias is catastrophic for Chaotic Dynamical Systems, where the underlying strange attractor is characterized by fractal geometry and infinite spectral density. We introduce the Fractal Neural Operator (FNO), a novel architecture that utilizes a non-resonant prime number basis to approximate continuous dynamical systems. Unlike geometric encodings ($2^k$), which suffer from spectral gaps and resonance, our Harmonic Weierstrass Encoder injects infinite spectral resolution into the latent space. We demonstrate that FNO extends the valid prediction horizon of the Lorenz-63 system to 347 Lyapunov times, exceeding state-of-the-art Reservoir Computing baselines by a factor of 2.3x. These results suggest that "chaos" is not inherently unpredictable to neural networks, but rather requires non-differentiable, fractal embedding manifolds.
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