用可微分空间场统一计算、模拟与推理，实现类图灵机的连续计算。
Neural Field Turing Machine: A Differentiable Spatial Computer
- 控制器读取局部区域，通过学习规则更新连续记忆场并移动读写头。
- 在固定半径邻域下实现线性扩展，支持长程稳定推演与超训练时长泛化。
- 适用于细胞自动机、物理方程求解和图像修复等多场景统一建模。
我们提出神经场图灵机（NFTM），一种可微分架构，将符号计算、物理模拟与感知推理统一于连续空间场中。NFTM包含神经控制器、连续记忆场及可移动读写头，每步从局部区域读取，通过学习规则计算更新并回写，同时调整头位置。该设计在固定半径邻域下实现线性O(N)扩展，且在有界误差下保持图灵完备性。我们展示了三种实例：1）规则110细胞自动机模拟，2）基于物理的二维热方程求解器，3）CIFAR-10图像补全的迭代优化。这些实例学习局部更新规则，组合成全局动态，表现出稳定长时推演能力，并能泛化至训练时长之外。NFTM提供了一个统一的可微分计算框架，连接离散算法与连续场动力学。
原文摘要 · Abstract (English)
We introduce the Neural Field Turing Machine (NFTM), a differentiable architecture that unifies symbolic computation, physical simulation, and perceptual inference within continuous spatial fields. NFTM combines a neural controller, continuous memory field, and movable read/write heads that perform local updates. At each timestep, the controller reads local patches, computes updates via learned rules, and writes them back while updating head positions. This design achieves linear O(N) scaling through fixed-radius neighborhoods while maintaining Turing completeness under bounded error. We demonstrate three example instantiations of NFTM: cellular automata simulation (Rule 110), physics-informed PDE solvers (2D heat equation), and iterative image refinement (CIFAR-10 inpainting). These instantiations learn local update rules that compose into global dynamics, exhibit stable long-horizon rollouts, and generalize beyond training horizons. NFTM provides a unified computational substrate bridging discrete algorithms and continuous field dynamics within a single differentiable framework.
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