NeurIPS 2026 (accepted; arXiv preprint) · 2026

2D Spatial Reasoning with Adaptive Neural Cellular Automata

Martin Spitznagel, Janis Keuper

Why this publication matters

Solving a Sudoku or tracing a path through a maze requires local decisions to respect relationships across an entire grid. This paper gives neural cellular automata adaptive perception, allowing each cell to learn which other locations matter at each iteration. The resulting compact models tackle several controlled spatial reasoning benchmarks, while visualizing their learned sampling locations helps reveal the relationships they use. These experiments provide task-specific evidence about spatial reasoning on 2D grids.

Abstract

Many modern learning approaches are still struggling with spatial reasoning tasks, i.e. they lack the ability to utilize geometric information of perceived entities and their spatial relation to each other to solve problems. We introduce a novel Adaptive Neural Cellular Automata (aNCA) architecture which replaces the static and spatially invariant perception of standard NCAs by learnable, spatially variant and data-adaptive perceptive fields. We show that this conceptual change enables NCAs to iteratively reason over 2D spatial relations on grid-like data structures (e.g. images). Empirical results on public benchmarks show state of the art comprehensible results with high generalization abilities for solving image based puzzles like Sudoku or finding the shortest path in a maze. The implementation and all experimental setups are openly accessible as an extension to the NCAtorch framework at https://github.com/mspitzna/NCAtorch.

Abstract source ↗

Figures

Input and target examples for array Sudoku, handwritten-digit Sudoku, shortest-path maze solving, and global color balance. — **Figure 3.** The four 2D spatial reasoning tasks evaluated in this work. Each column shows the input (top) and target (bottom). View in source ↗

Fixed and adaptive perception patterns alongside an aNCA update step using learned deformable-convolution offsets and modulation. — **Figure 5.** Comparison of standard 3×3, dilated, constraint-aligned, and deformable perception patterns, followed by one aNCA update step. The perception module predicts offsets and modulations from the current state, guiding feature aggregation at adaptive grid locations; the update module then produces the next state. View in source ↗

Cite this paper

Download .bib

@misc{spitznagel20262dspatialreasoningadaptive,
  title = {2D Spatial Reasoning with Adaptive Neural Cellular Automata},
  author = {Martin Spitznagel and Janis Keuper},
  year = {2026},
  eprint = {2610.08518},
  archivePrefix = {arXiv},
  primaryClass = {cs.CV},
  url = {https://arxiv.org/abs/2610.08518}
}

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