PI-NCA: Neural Cellular Automata that conserve by construction

Current learned PDE emulators can be computationally heavy to simulate and are not always readily scalable. Neural cellular automata (NCAs) are therefore an appealing template: a small local rule shared by every cell mirrors the locality of a differential operator. Used directly, however, an NCA predicts unconstrained state increments; in our evaluation, its mass drifts by tens of units, and on Navier-Stokes it performs worse than predicting no change at all. We introduce Physics-Informed Neural Cellular Automata (PI-NCA), together with its multi-channel variant, MC-PI-NCA, to address this failure structurally. The Physics-Informed Neural Cellular Automaton (PI-NCA) makes the shared rule output a flux rather than an increment and advances the state by the flux's periodic discrete divergence, so the flux-divergence update enforces exact conservation of the field total, regardless of the learned weights. MC-PI-NCA extends the architecture to coupled systems, predicting one flux per field and conserving each field's total independently. Additionally, MC-PI-NCA can simultaneously predict multiple physical variables, such as temperature, pressure, and density, while applying a parameter-free projection to keep the predicted states within physically valid range. We compare both architectures with the Fourier neural operator (FNO), full and matched-size ResNet and U-Net surrogates, the standard NCA, and an identity floor on ten 2-D phenomena, using five to ten seeds and Holm-corrected paired tests. Ablations isolate the effects of the flux head and the projection and identify the regimes in which the conservation prior should not be used. On the shallow-water equations, PI-NCA and MC-PI-NCA are the best-performing models with the lowest relative L2 error while using $112\times$ and $54\times$ fewer parameters than the FNO.On Cahn-Hilliard, PI-NCA has a relative L2 error $20.63$ lower than the best baseline and $43.50$ lower than the FNO; on heat and advection-diffusion, the proposed models are second only to the FNO while using $60$ to $130\times$ fewer parameters. PI-NCA outperforms NCA on seven out of the ten evaluated phenomena with $0.33\times$ fewer parameters and trains $8.3\times$ faster than the FNO.Overall, PI-NCA demonstrates that conservation can be incorporated into compact NCA as an architectural property rather than a training objective, yielding competitive accuracy with significantly lower parameter and computational costs.