Neural Network Field Theories: Non-Gaussianity, Actions, and Locality

The Big Picture

Imagine describing a turbulent ocean two ways: write down the fundamental equations governing every swirl and eddy, or simulate it as an enormous ensemble of tiny interacting particles and let collective statistics do the work. These sound completely different, yet under the right conditions, they describe exactly the same physics.

Researchers at IAIFI have been exploring this situation for quantum fields and neural networks. Both systems are, at their core, machines for generating random functions. A neural network with randomized parameters at initialization produces a random function mapping inputs to outputs. A quantum field theory (physics’ framework for describing particles and forces) uses a sum over all possible field configurations to produce a distribution over the shapes a field can take. The question is simple enough to state: when are these two descriptions the same thing?

The team of Mehmet Demirtas, James Halverson, Anindita Maiti, Matthew Schwartz, and Keegan Stoner has built a systematic mathematical bridge between neural network architecture and the action functionals of field theory. Their construction comes with new Feynman diagram rules and a concrete realization of φ⁴ theory using an infinite-width neural network.

Key Insight: By treating neural networks as probability distributions over functions, the researchers show you can read off a field theory’s governing equations directly from measurements of how network outputs relate to each other. Run the logic backward and you can engineer a network architecture that is a given field theory.

How It Works

The Central Limit Theorem (CLT) is the starting point. Sums of many independent random quantities converge to a Gaussian distribution. In neural networks, the infinite-width limit (where neuron count N → ∞) is exactly the CLT regime: each neuron contributes a tiny random piece, and their sum is Gaussian. In field theory language, a Gaussian distribution over functions is a free field theory, one with no interactions. This is the Neural Network / Gaussian Process (NNGP) correspondence. Infinite-width networks are free field theories.

Free field theories are boring. The real world has interactions. The paper explores two routes to introduce them:

• The 1/N expansion: Work at finite width. The CLT breaks down in a controlled way, and corrections at each order in 1/N introduce connected correlators, statistics that capture genuine interdependencies between network outputs beyond what a Gaussian predicts. These correspond to interaction vertices in field theory.
• Independence breaking: Keep N large but allow network parameters to develop correlations, violating the statistical independence the CLT requires. This subtler deformation can outperform the 1/N expansion with respect to the universal approximation theorem, meaning the resulting field theories can represent a wider class of target functions.

Once non-Gaussianities are generated by either method, the challenge is: given the correlators, what is the action? The answer uses the Edgeworth expansion, a classical statistics tool that describes near-Gaussian distributions as a series of corrections built from Hermite polynomials. Each term captures a specific type of deviation from a bell curve. Each term maps onto a contribution to the action.

The new Feynman diagram prescription is the payoff. Normally, Feynman diagrams have vertices defined by coupling constants in the action. Here the authors flip the logic: the vertices are the connected correlators themselves. You measure how the field’s statistics at different points are wired together (the n-point functions), draw the corresponding diagrams, and reconstruct the action order by order. Feynman rules built from network statistics rather than a Lagrangian written from first principles.

As a demonstration, they work through non-local φ⁴ theory, the quartic interaction ubiquitous in particle physics and statistical mechanics, but with position-dependent couplings. The non-locality arises naturally from the neural network construction, where the Gaussian process kernel introduces spatial structure.

Why It Matters

The ability to engineer network architectures that realize specific field theories goes beyond mathematical curiosity. φ⁴ theory appears in the Higgs mechanism, in models of phase transitions, and as a testbed for renormalization group methods. A neural network that is φ⁴ theory, not an approximation, lets you study field theory phenomena using ML tools like optimization, gradient descent, and generative modeling.

The correspondence also gives physicists a new vocabulary for understanding what neural networks actually compute. The locality question is particularly telling: physical field theories are local (interactions happen at the same spacetime point), but neural network field theories generically produce non-local actions. The paper analyzes when locality emerges, connecting this to the cluster decomposition principle, the requirement that distant experiments be statistically independent. Understanding when a neural network satisfies cluster decomposition tells you whether ML-based approaches to physics are capturing genuine physical structure.

There is also a constructive angle. Rather than discovering field theories by guessing Lagrangians, one might design network architectures encoding desired symmetries and interactions, then train them. This paper provides the dictionary for that translation.

Bottom Line: Neural networks and quantum field theories speak the same mathematical language. This paper provides the translation guide, complete with Feynman rules built from network statistics and a proof-of-concept realization of φ⁴ theory as an infinite-width neural network.