Simulate, Don't Interpolate: Recovering Continuous Spatial Dynamics from Discrete, Unpaired Snapshots

Spatiotemporal processes are central to many scientific questions, from how structure emerges in physical systems to how organisms form, or diseases develop. However, measuring such systems often relies on destructive experiments that preclude repeated time-series measurements, especially in biology and medicine. The alternative, unpaired snapshots, generally cannot fully determine the underlying dynamical processes. Recovering them requires constraining the solution space; implementing this at the level of the rules governing the system remains an open problem. We introduce REGIS (REcovering Generative dynamics from Independent Snapshots), a framework that recovers spatial dynamics from time-labelled images by learning a unified update rule that is spatially local, Markovian, and time-invariant, and that is iterated to reach every observed time. We implement these constraints with neural cellular automata, trained adversarially by matching generated rollouts to the observed distributions. On synthetic data, REGIS outperforms state-of-the-art baselines in trajectory quality within and beyond the training horizon, data efficiency, and recovery of the underlying dynamics. Applied to a real biological system, a spatial atlas of zebrafish heart regeneration, REGIS recovers a rule that simulates regenerative dynamics. When used in an in silico functional screen, REGIS identifies critical cell types and hierarchical dependencies consistent with known biology, giving the learned rule a mechanistic interpretation. The result is a general recipe for building simulators from unpaired spatial snapshot data by sharing weights across space and time under a locality constraint. The resulting simulators are mechanistic and perturbable, offering a practical route to digital twins for biological systems.