5 papers
An SO(3) Gauge Theory of Turbulence with Spontaneous Symmetry Breaking
Ahmed Farooq
Fully developed isotropic turbulence exhibits a dual nature: a continuous, scale-invariant energy cascade coexists with discrete, intense vortex filaments. We show that this dualit…
A Statistical Field Theory for Isotropic Turbulence
Ahmed Farooq
This article establishes a first-principles statistical field theory of fully developed isotropic turbulence. Applying an exact Helmholtz decomposition to the local angular momentu…
A Hybrid Multi-Well Hopfield-CNN with Feature Extraction and K-Means for MNIST Classification
Ahmed Farooq
This study presents a hybrid model for classifying handwritten digits in the MNIST dataset, combining convolutional neural networks (CNNs) with a multi-well Hopfield network. The a…
Long-Sequence Memory with Temporal Kernels and Dense Hopfield Functionals
Ahmed Farooq
In this study we introduce a novel energy functional for long-sequence memory, building upon the framework of dense Hopfield networks which achieves exponential storage capacity th…
A Framework for Non-Linear Attention via Modern Hopfield Networks
Ahmed Farooq
In this work we propose an energy functional along the lines of Modern Hopfield Networks (MNH), the stationary points of which correspond to the attention due to Vaswani et al. [12…