DiffeoMorph: Learning to Morph 3D Shapes Using Differentiable Agent-Based Simulations

TL;DR

DiffeoMorph is a differentiable framework that learns agent-based protocols to morph 3D shapes into target structures using shape-matching loss and SE(3)-equivariant neural networks.

Contribution

It introduces a novel shape-matching loss based on 3D Zernike polynomials and an end-to-end differentiable system for learning morphogenesis protocols.

Findings

Successfully morphs initial conditions into complex 3D shapes.

Shape-matching loss outperforms standard shape comparison metrics.

Demonstrates versatility in forming various shapes from minimal initial patterns.

Abstract

Biological systems can form complex three-dimensional structures through the collective behavior of agents that share a common update rule and operate 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 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 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 global orientation. To achieve rotation invariance while preserving reflection sensitivity, we include an alignment step that optimally rotates the predicted Zernike spectrum to match the target before computing the loss. We perform 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 complex shapes from minimally patterned initial conditions. DiffeoMorph provides a general framework for learning distributed control strategies for morphogenesis, swarm robotics, and programmable self-assembly.