Where Does Randomness Matter in Neural Cellular Automata?

Fei Zuo; Jiaqi Shi; Yujing Liu

cs.LGSep 29, 2026

Where Does Randomness Matter in Neural Cellular Automata?

Authors: Fei Zuo, Jiaqi Shi, Yujing Liu

Organizations: Fudan University · Independent Researcher · Shanghai Pupuda Culture Communication Co., Ltd.

Abstract

Stochastic cell updates are often used throughout the life of a neural cellular automaton (NCA), from backpropagation through time to final rollout. This leaves two questions entangled: does update randomness help learn a useful rule, and must that randomness remain at execution? We separate training and evaluation update modes in controlled Growing NCA experiments, then vary the states shown during training. Under the standard constant-rate persist recipe, asynchronous training passes the short-horizon quality test in 10/10 runs, compared with 3/10 synchronous runs. All ten asynchronous models also retain the target for 4,096 steps under deterministic evaluation. For a scalar translation-invariant lattice, we derive an exact mean-square criterion: random masking can damp mean modes, but it also injects variance, and a mean-only test misclassifies four non-marginal settings. Finally, among 30 models that all pass the same reconstruction test, eight of ten grow-trained models become off-target at 4,096 steps, while all persist and regenerate models retain the target; damage recovery separates persist from regenerate. The results distinguish optimization reliability, execution mode, and task-specific behavior instead of treating them as one stability property.

Figures & tables

Training	Target	Quality pass	Retained (det.)	Swap difference $\|\Delta\mathrm{MSE}\|$
Synchronous ( $\alpha=1$ )	Flower	1/5	1/5	$2.3\times 10^{-3}$
	Heart	2/5	3/5	$2.8\times 10^{-3}$
Asynchronous ( $\alpha=0.5$ )	Flower	5/5	5/5	$2.5\times 10^{-3}$
	Heart	5/5	5/5	$1.0\times 10^{-3}$

Table 1: Training and evaluation update modes. Quality is tested at

T=96

in the training mode. Retention is the bounded, on-target class after

4{,}096

deterministic steps. Swap cost is the median absolute MSE change at

T=1024

among quality-passing models. Each target has five models per training mode.

Recipe	Target	MSE $T=96$	$\lambda_{\mathrm{pert}}$ 95% CI	MSE $T=4096$	Recovery (%)	Retained
Grow	Flower	$6.6\times 10^{-5}$	$+0.017$ $[+0.011,+0.021]$	$0.26$	$-46$	1/5
	Heart	$1.9\times 10^{-5}$	$+0.023$ $[+0.017,+0.028]$	$0.34$	$-46$	1/5
Persist	Flower	$1.1\times 10^{-4}$	$-0.024$ $[-0.031,-0.015]$	$6.5\times 10^{-4}$	$-7$	5/5
	Heart	$6.1\times 10^{-5}$	$-0.033$ $[-0.040,-0.027]$	$8.5\times 10^{-5}$	$+4$	5/5
Regenerate	Flower	$4.3\times 10^{-4}$	$-0.008$ $[-0.017,+0.006]$	$7.0\times 10^{-4}$	$+85$	5/5
	Heart	$5.7\times 10^{-4}$	$+0.002$ $[-0.015,+0.025]$	$1.3\times 10^{-3}$	$+99$	5/5

Table 2: Models with similar reconstruction quality learn different later-use behavior. MSE columns are medians over five models per target. Perturbation growth is the model mean, with bootstrap 95% intervals below. Recovery is the mean percentage under the frozen damage protocol. Retention uses the long-horizon threshold, not the stricter

T=96

quality threshold.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Comparison	Original criterion	Observed quantities
Mode-swap asymmetry	Asynchronous-to-synchronous swap ratio above 3, the same direction on both targets, and ratio CI excluding 1.	CI $[0.22,1.24]$ ; directions differ between targets.
Spectral placement	At least 6 asynchronous models with $\rho>1.02$ and all violating multipliers within $\|\mu+1\|<2.05$ ; at most 2 qualifying synchronous survivors.	$5$ asynchronous models and $1$ synchronous survivor qualify.
Temporal oscillation	Synchronous amplitude above twice the asynchronous amplitude on both targets.	The ordering reverses on flower.
Perturbation contrasts	Grow minus persist and grow minus regenerate intervals strictly above zero on both targets.	$3/4$ intervals exclude zero; heart grow minus regenerate is $[-0.002,+0.037]$ .
Long-horizon ordering	Grow median MSE at $T=4096$ above persist and regenerate on both targets.	Grow $0.26/0.34$ ; other groups at most $1.3\times 10^{-3}$ .
Spatial error ratio	Median per-model error ratio from $m=1$ to $m=4$ below 1.2 for both recipes.	Persist $1.45$ ; regenerate $1.197$ .

Appendix

Table 3: Original comparison criteria and observed quantities. Synchronous and asynchronous refer to training. The complete lattice comparison and subsequent non-marginal diagnostic are listed separately within the same row.

$c$	$\alpha=0.25$				$\alpha=0.5$				$\alpha=1.0$
	Mean	MS	Sim.	$S$	Mean	MS	Sim.	$S$	Mean	MS	Sim.	$S$
$0.20$	$\checkmark$	$\checkmark$	$\checkmark$	$0.34$	$\checkmark$	$\checkmark$	$\checkmark$	$0.27$	$\checkmark$	$\checkmark$	$\checkmark$	$0.00$
$0.25$	$\checkmark$	$\checkmark$	$\checkmark$	$0.45$	$\checkmark$	$\checkmark$	$\checkmark$	$0.37$	$\times$	$\times$	$\checkmark^{\dagger}$	$\infty$
$0.30$	$\checkmark$	$\checkmark$	$\checkmark$	$0.56$	$\checkmark$	$\checkmark$	$\checkmark$	$0.50$	$\times$	$\times$	$\times$	$\infty$
$0.35$	$\checkmark$	$\checkmark$	$\checkmark$	$0.68$	$\checkmark$	$\checkmark$	$\checkmark$	$0.67$	$\times$	$\times$	$\times$	$\infty$
$0.40$	$\checkmark$	$\checkmark$	$\checkmark$	$0.81$	$\checkmark$	$\checkmark$	$\checkmark$	$0.92$	$\times$	$\times$	$\times$	$\infty$

Appendix

Table 4: All 27 lattice configurations arranged by update probability. Each block reports the mean-only prediction, second-moment prediction, simulated label, and

S

. A check denotes decay and a cross denotes no decay. A dagger marks an exactly marginal mean multiplier. Bold mean-only predictions identify the four non-marginal disagreements with simulation. An infinite

S

denotes failure of the prerequisite

d_{\alpha}<1

in the implementation.

Recipe	Target	Outcomes	$\max\|x\|$	Outcomes	$\max\|x\|$	Outcomes	$\max\|x\|$
		Masks active		Removed at $T=96$		Absent from seed
Grow	Flower	1/4/0	$[1.02,3.97]$	1/4/0	$[1.02,16.54]$	1/4/0	$[1.03,24.35]$
	Heart	1/4/0	$[1.07,8.55]$	1/4/0	$[1.10,9.62]$	1/4/0	$[1.12,10.17]$
Persist	Flower	5/0/0	$[1.01,1.09]$	5/0/0	$[1.01,1.09]$	5/0/0	$[1.01,1.02]$
	Heart	5/0/0	$[0.99,1.02]$	5/0/0	$[0.99,1.05]$	5/0/0	$[0.99,1.04]$
Regenerate	Flower	5/0/0	$[1.02,1.39]$	5/0/0	$[1.02,1.31]$	5/0/0	$[1.02,1.14]$

Appendix

Table 5: Living-mask ablation with matched random update tapes. Each group contains five models. Outcome counts give retained, off-target, and diverged or dead runs in that order. State magnitudes are ranges across the five models at

T=4{,}096

Recipe	Target	Seed	MSE@96	$\lambda_{\mathrm{pert}}$	$\rho$	MSE@4096	Outcome
Grow	Flower	0	$6.6\times 10^{-5}$	$+0.0198$	$1.134$	$0.39$	Off-target
		1	$1.4\times 10^{-5}$	$+0.0247$	$1.262$	$0.05$	Retained
		2	$2.0\times 10^{-3}$	$+0.0182$	$1.117$	$0.24$	Off-target
		3	$6.8\times 10^{-5}$	$+0.0135$	$1.078$	$0.26$	Off-target
		4	$2.3\times 10^{-5}$	$+0.0080$	$1.204$	$0.35$	Off-target
	Heart	0	$1.9\times 10^{-5}$	$+0.0239$	$1.204$	$0.34$	Off-target

Appendix

Table 6: The thirty recipe models. Reconstruction errors, finite-time perturbation growth, largest returned frozen multiplier, and stochastic long-horizon outcome are listed for every training seed. Retained is the archived bounded outcome class. Outcome labels use unrounded measurements.

Training	Target	Seed	Quality MSE	Pass	Det. outcome	Signed swap $\Delta\mathrm{MSE}$
sync	Flower	0	Dead †	No	Dead	n/a
		1	$2.1\times 10^{-3}$	Yes	Retained	$2.3\times 10^{-3}$
		2	Dead †	No	Dead	n/a
		3	Dead †	No	Dead	n/a
		4	Dead †	No	Dead	n/a
	Heart	0	$1.0\times 10^{-3}$	Yes	Retained	$1.9\times 10^{-3}$

Appendix

Table 7: The twenty factorial entries. The last column is the signed mode-swap difference, so a negative value denotes lower error in the unmatched mode. A dagger denotes an empty model for which the frozen quality evaluator records a sentinel rather than an ordinary error value. Its true empty-canvas image loss is identified separately in the training traces and final-state probe.

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Neural Cellular Automata Brain-Inspired Synergistic Framework Visual Reasoning

Open paper AskPDF arXiv

Modern AI architectures used to solve visual reasoning tasks typically rely heavily on global connectivity and synchronization. As biological systems demonstrate, though, sophisticated computation can be performed in a more decentralized fashion. In this work, we test the reasoning capabilities of Neural Cellular Automata (NCAs), networks of recurrent cells that use strictly local connectivity and asynchronous updates. NCAs have been extensively studied in artificial life experiments, but it is unclear whether they can perform complex multi-step reasoning. We show that NCAs produce spatio-temporal dynamics capable of solving challenging visual reasoning tasks, including large mazes, Sudoku, and ARC-AGI-1. Furthermore, we provide evidence that NCAs generalize out-of-distribution when running with larger grids, longer rollouts, or parallel trials; and that the latter can be made more efficient via pruning of redundant trajectories. We find that these generalization capabilities depend on training with sample replay and stochastic perturbations, and that stochasticity remains beneficial at test time. Finally, we show that NCAs are robust reasoners capable of dynamically modulating compute to recover efficiently from damage, and that they can scale to solve reasoning in raw pixel space.

Abstract

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