Optimal Architecture and Fundamental Bounds in Neural Network Field Theory

TL;DR

This paper explores architectural choices in neural network field theory, identifying an optimal configuration that minimizes errors and establishes fundamental bounds, advancing NNFT as a practical tool for field theory simulations.

Contribution

It uncovers a new architectural parameter in NNFT that affects finite-width errors and demonstrates an optimal setting that minimizes variance and removes IR-sensitive corrections.

Findings

Optimal architecture parameter $eta=0$ minimizes finite-width variance.

Bias can be eliminated through extrapolation to infinite width.

Variance sets a fundamental limit on the signal-to-noise ratio in NNFT.

Abstract

Neural network field theory (NNFT) represents fields as neural networks and samples field configurations by drawing network parameters from a probability distribution. We identify a previously unexplored architectural freedom in NNFT, parameterized by $\alpha$, that leaves the infinite-width theory invariant but dramatically affects finite-width errors in the calculation of correlation functions. For a massive scalar field, we show that $\alpha=0$, corresponding to propagator-weighted neuron momenta and constant neuron amplitudes, is optimal: it minimizes finite-width variance and uniquely removes IR-sensitive corrections in the interacting theory. Even at $\alpha=0$, relative errors from both bias and variance grow exponentially with distance beyond the correlation length. The bias can be removed by extrapolating to infinite width, which we demonstrate numerically, while the variance imposes a fundamental bound on the achievable signal-to-noise ratio as in lattice field theory. These results chart a path toward developing NNFT into a practical tool for the numerical study of field theories.