Neural Field Turing Machine: A Differentiable Spatial Computer

We introduce the Neural Field Turing Machine (NFTM), a differentiable architecture unifying symbolic computation, physical simulation, and perceptual inference within continuous spatial fields. NFTM combines a neural controller, continuous memory field, and movable read/write heads performing local updates. At each timestep, the controller reads local patches, computes updates via learned rules, and writes them back while updating head positions. This achieves linear O(N) scaling through fixed-radius neighborhoods while maintaining Turing completeness under bounded error. We demonstrate NFTM on cellular automata (Rule 110), physics-informed PDE solvers (heat equation), and image inpainting (CIFAR-10). These instantiations learn local rules that compose into global dynamics, exhibit stable rollouts, and generalize beyond training horizons, providing a unified substrate bridging discrete algorithms and continuous field dynamics.