  We introduce a modern Hopfield network with continuous states and a
corresponding update rule. The new Hopfield network can store exponentially
(with the dimension of the associative space) many patterns, retrieves the
pattern with one update, and has exponentially small retrieval errors. It has
three types of energy minima (fixed points of the update): (1) global fixed
point averaging over all patterns, (2) metastable states averaging over a
subset of patterns, and (3) fixed points which store a single pattern. The new
update rule is equivalent to the attention mechanism used in transformers. This
equivalence enables a characterization of the heads of transformer models.
These heads perform in the first layers preferably global averaging and in
higher layers partial averaging via metastable states. The new modern Hopfield
network can be integrated into deep learning architectures as layers to allow
the storage of and access to raw input data, intermediate results, or learned
prototypes. These Hopfield layers enable new ways of deep learning, beyond
fully-connected, convolutional, or recurrent networks, and provide pooling,
memory, association, and attention mechanisms. We demonstrate the broad
applicability of the Hopfield layers across various domains. Hopfield layers
improved state-of-the-art on three out of four considered multiple instance
learning problems as well as on immune repertoire classification with several
hundreds of thousands of instances. On the UCI benchmark collections of small
classification tasks, where deep learning methods typically struggle, Hopfield
layers yielded a new state-of-the-art when compared to different machine
learning methods. Finally, Hopfield layers achieved state-of-the-art on two
drug design datasets. The implementation is available at:
https://github.com/ml-jku/hopfield-layers
