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2D Spatial Reasoning with Adaptive Neural Cellular Automata

NeurIPS 2026 · Vision-language and multimodal › Multimodal LLMs · approach: Spatial reasoning in VLMs

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Proposes Adaptive Neural Cellular Automata (aNCA), which replace fixed NCA perception with DCNv2 deformable convolutions, and evaluates them on Sudoku, image-based MNIST Sudoku, 30x30 maze solving, and global color balance.

Problem

Standard neural cellular automata have static, spatially invariant perception, which limits their ability to reason over 2D spatial relations on grid data such as Sudoku and maze shortest-path tasks.

Approach

Replace the fixed k×k perception of NCAs with input-dependent deformable convolutions (DCNv2 with learned offsets and modulation scalars), so each cell adaptively samples from spatially variant, step-dependent neighborhoods; the learned offsets align with Sudoku row, column and box constraints.

Key results

• On hard Sudoku (array), aNCA reaches 87.78% ASR vs 78.23% for the 3x3 NCA baseline, with 165k parameters.
• On 30x30 mazes, aNCA achieves 98.56% Valid Path and 94.18% Valid Optimal, surpassing HRM (74.5%), ItrSA++ (78.6%) and TRM-Att (85.3%) on Valid Optimal.
• On MNIST Sudoku Hard, aNCA reaches 82.10% ASR versus 51.6% for SRM (118M parameters), using a frozen codec plus a 165k-parameter NCA.

Surprising findings

• Enlarging the fixed convolution kernel to 7x7 or 9x9 does not improve Sudoku performance; the 9x9 variant is unstable across seeds (±37.15 on Hard).
• On global color balance, the 5x5 NCA achieves the lowest mean count error (0.07) while aNCA has higher exact-ratio accuracy (96.8% vs 95.3%), with the gap within run-to-run variability.

Limitations

• Adaptive perception incurs higher per-step compute: aNCA inference takes about 833 ms per puzzle vs about 160 ms for the 3x3 NCA.
• Current architecture targets regular 2D grids; irregular graphs, 3D scenes and variable resolution are not covered.
• Maze results rely on the Sapient maze dataset and similar-resolution transfer; the paper's maze valid-optimal number is inconsistent between text (95.2%) and table (94.18%).

Experimental setup

Datasets: SRM Sudoku dataset (one million puzzles) MNIST 30x30 Sapient maze dataset Prim-style generated mazes (generalization)

Benchmarks: SRM Sudoku test split (Easy/Medium/Hard) MNIST Sudoku 30x30 maze (Sapient) Global color balance

Metrics: Average Solve Rate Average Cell Precision Valid Path Valid Optimal Imbalance Exact-ratio accuracy

Baselines: SRM HRM TRM-Att ItrSA++ Diffusion Model NCA 3x3 Conv NCA 5x5 Conv NCA dilated conv NCA DCNv1

Builds on: Neural Cellular Automata DCNv2 deformable convolution NCAtorch Sudoku Reasoning Model (SRM) Hierarchical Reasoning Model

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