Seong Ho Pahng, Guoye Guan, Benjamin Fefferman et al.
Biological systems can form complex three-dimensional structures through the collective behavior of identical agents -- cells that follow the same internal rules and communicate without central control. How such distributed control gives rise to precise global patterns remains a central question not only in developmental biology but also in distributed robotics, programmable matter, and multi-agent learning. Here, we introduce DiffeoMorph, an end-to-end differentiable framework for learning a morphogenesis protocol that guides a population of agents to morph into a target 3D shape. Each agent updates its position and internal state using an attention-based SE(3)-equivariant graph neural network, based on its own internal state and signals received from other agents. To train this system, we introduce a new shape-matching loss based on the 3D Zernike polynomials, which compares the predicted and target shapes as continuous spatial distributions, not as discrete point clouds, and is invariant to agent ordering, number of agents, and rigid-body transformations. To enforce full SO(3) invariance -- invariant to rotations yet sensitive to reflections, we include an alignment step that optimally rotates the predicted Zernike spectrum to match the target before computing the loss. This results in a bilevel problem, with the inner loop optimizing a unit quaternion for the best alignment and the outer loop updating the agent model. We compute gradients through the alignment step using implicit differentiation. We perform systematic benchmarking to establish the advantages of our shape-matching loss over other standard distance metrics for shape comparison tasks. We then demonstrate that DiffeoMorph can form a range of shapes -- from simple ellipsoids to complex morphologies -- using only minimal spatial cues.