The Fractal Neural Operator: Overcoming Spectral Bias in Chaotic Attractors via Prime-Harmonic Weierstrass Encodings
2026-06-22 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study how deep learning models typically struggle with chaotic systems because these systems have very complex, fractal patterns that usual models smooth out too much. They propose a new model called the Fractal Neural Operator (FNO) that uses special math based on prime numbers to better capture infinite detail in these chaotic patterns. Their approach allowed the FNO to predict the behavior of a famous chaotic system, Lorenz-63, much longer and more accurately than existing methods. This shows that chaos is not impossible for neural networks to predict, but it needs special ways to represent fractal structures.
spectral biastransformersneural operatorschaotic dynamical systemsfractal geometrystrange attractorLorenz-63 systemLyapunov timeharmonic Weierstrass encoderreservoir computing
Authors
Kanishk Awadhiya
Abstract
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.