Journal ArticleNeural network field theories: non-Gaussianity, actions, and localityDemirtas, Mehmet; Halverson, James; Maiti, Anindita; Schwartz, Matthew D; Stoner, Keegan<title>Abstract</title> <p>Both the path integral measure in field theory (FT) and ensembles of neural networks (NN) describe distributions over functions. When the central limit theorem can be applied in the infinite-width (infinite-<italic>N</italic>) limit, the ensemble of networks corresponds to a free FT. Although an expansion in<inline-formula><tex-math><CDATA/></tex-math><math overflow='scroll'><mn>1</mn><mrow><mo>/</mo></mrow><mi>N</mi></math><inline-graphic href='mlstad17d3ieqn1.gif' type='simple'/></inline-formula>corresponds to interactions in the FT, others, such as in a small breaking of the statistical independence of network parameters, can also lead to interacting theories. These other expansions can be advantageous over the<inline-formula><tex-math><CDATA/></tex-math><math overflow='scroll'><mn>1</mn><mrow><mo>/</mo></mrow><mi>N</mi></math><inline-graphic href='mlstad17d3ieqn2.gif' type='simple'/></inline-formula>-expansion, for example by improved behavior with respect to the universal approximation theorem. Given the connected correlators of a FT, one can systematically reconstruct the action order-by-order in the expansion parameter, using a new Feynman diagram prescription whose vertices are the connected correlators. This method is motivated by the Edgeworth expansion and allows one to derive actions for NN FT. Conversely, the correspondence allows one to engineer architectures realizing a given FT by representing action deformations as deformations of NN parameter densities. As an example,<italic>φ</italic>⁴theory is realized as an infinite-<italic>N</italic>NN FT.</p>IOP Press2024-01-0910519823Machine Learning: Science and Technology510150022632-2153https://doi.org/10.1088/2632-2153/ad17d31848089National Science Foundation