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.

In the paper, we present a multi-scale search space which is casted into a differentiable supernet consisting of three modules, i.e., parallel module, transition module, and fusion module. As shown in Figure 1.(a), there are As mentioned in the main body of the paper, the super-network we build for restoration contains three cells and each cell consists of four cascade modules. Namely, there are 12 cascade modules in total. The strided convolution is used to down sample features. The convolutional sequence is arranged in a residual manner for each parallel direction.

Additionally, existing zero-cost proxies fail to generalize across neural architecture design spaces.

KITTI 2012 contains 194 training image pairs and 195 test image pairs. We use a maximum disparity level of 192 in this dataset. Most of the stereo pairs are indoor scenes with handcrafted layouts. This dataset contains many thin objects and large disparity ranges. We provide more qualitative results on the SceneFlow, KITTI 2012, KITTI 2015 and Middlebury datasets in Figure 1 2 3 4, respectively.

The "Repeat" represents the maximum number of repeated blocks in a group. " represents the inverted bottleneck " represents the inverted bottleneck

Recent extensions of Cellular Automata (CA) have incorporated key ideas from modern deep learning, dramatically extending their capabilities and catalyzing a new family of Neural Cellular Automata (NCA) techniques.

Elementary Cellular Automata (ECAs) exhibit diverse behaviours often categorized by Wolfram's qualitative classification. To provide a quantitative basis for understanding these behaviours, we investigate the global dynamics of such automata and we describe a method that allows us to compute the number of all configurations leading to short attractors in a limited number of time steps. This computation yields exact results in the thermodynamic limit (as the CA grid size grows to infinity), and is based on the Transfer Matrix Method (TMM) that we adapt for our purposes. Specifically, given two parameters $(p, c)$ we are able to compute the entropy of all initial configurations converging to an attractor of size $c$ after $p$ time-steps. By calculating such statistics for various ECA rules, we establish a quantitative connection between the entropy and the qualitative Wolfram classification scheme. Class 1 rules rapidly converge to maximal entropy for stationary states ($c=1$) as $p$ increases. Class 2 rules also approach maximal entropy quickly for appropriate cycle lengths $c$, potentially requiring consideration of translations. Class 3 rules exhibit zero or low finite entropy that saturates after a short transient. Class 4 rules show finite positive entropy, similar to some Class 3 rules. This method provides a precise framework for quantifying trajectory statistics, although its exponential computational cost in $p+c$ restricts practical analysis to short trajectories.

Spontaneous self-replication in cellular automata has long been considered rare, with most known examples requiring careful design or artificial initialization. In this paper, we present formal, causal evidence that such replication can emerge unassisted -- and that it can do so in a distributed, multi-component form. Building on prior work identifying complex dynamics in the Outlier rule, we introduce a data-driven framework that reconstructs the full causal ancestry of patterns in a deterministic cellular automaton. This allows us to rigorously identify self-replicating structures via explicit causal lineages. Our results show definitively that self-replicators in the Outlier CA are not only spontaneous and robust, but are also often composed of multiple disjoint clusters working in coordination, raising questions about some conventional notions of individuality and replication in artificial life systems.