Optimal Architecture and Fundamental Bounds in Neural Network Field Theory

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.