Neural Field Turing Machine: A Differentiable Spatial Computer

Akash Malhotra, Nacera seghouani

Abstract:

We introduce the Neural Field Turing Machine (NFTM), a theoretical framework
for computation on spatial fields through local, iterative updates. NFTM defines
a computational model where a stateless neural controller performs read-write
operations on a neurally continuous spatial memory field via movable heads with
bounded support regions. Unlike Neural Turing Machines, which access discrete
memory locations globally, or Neural Cellular Automata, which operate on dis-
crete grids with fixed update rules, NFTM formalizes computation on neurally
continuous domains, fields with finite parameterization, continuous evaluation, and
differentiable dynamics. The framework supports both sequential and parallel up-
date schedules, with Turing completeness under bounded error established through
Rule 110 emulation. We demonstrate NFTM’s practical realizability through three
minimal instantiations using standard neural architectures: cellular automata (Rule
110), physics-informed PDE solvers (heat equation), and image inpainting (CIFAR-
10). These examples, operating on discrete grids as practical approximations, show
that the framework captures real computational patterns across symbolic, physi-
cal, and perceptual domains. NFTM provides a compelling perspective on local
iterative computation over spatial fields.