Balancing Centralized Learning and Distributed Self-Organization: A Hybrid Model for Embodied Morphogenesis

We investigate how to couple a learnable brain-like’’ controller to a cell-like’’ Gray–Scott substrate to steer pattern formation with minimal effort. A compact convolutional policy is embedded in a differentiable PyTorch reaction–diffusion simulator, producing spatially smooth, bounded modulations of the feed and kill parameters ($\mathrm{\Delta }F$, $\mathrm{\Delta }K$) under a warm–hold–decay gain schedule. Training optimizes Turing-band spectral targets (FFT-based) while penalizing control effort (${\mathrm{\ell }}_1\mathrm{/}{\mathrm{\ell }}_2$) and instability. We compare three regimes: pure reaction–diffusion, NN-dominant, and a hybrid coupling. The hybrid achieves reliable, fast formation of target textures: 100% strict convergence in $\sim 165$ steps, matching cell-only spectral selectivity (0.436 vs.\ 0.434) while using $\sim 15\times$ less ${\mathrm{\ell }}_1$ effort and $>200\times$ less ${\mathrm{\ell }}_2$ power than NN-dominant control. An amplitude sweep reveals a non-monotonic Goldilocks’’ zone ($A\approx 0.03$–$0.045$) that yields 100% quasi convergence in 94–96 steps, whereas weaker or stronger gains fail to converge or degrade selectivity. These results quantify morphological computation: the controller seeds then cedes,‘’ providing brief, sparse nudges that place the system in the correct basin of attraction, after which local physics maintains the pattern. The study offers a practical recipe for building steerable, robust, and energy-efficient embodied systems that exploit an optimal division of labor between centralized learning and distributed self-organization.