Sahand Hormoz

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.

Seong Ho Pahng, Sahand Hormoz

Graph Neural Networks (GNNs) traditionally employ a message-passing mechanism that resembles diffusion over undirected graphs, which often leads to homogenization of node features and reduced discriminative power in tasks such as node classification. Our key insight for addressing this limitation is to assign fuzzy edge directions -- that can vary continuously from node $i$ pointing to node $j$ to vice versa -- to the edges of a graph so that features can preferentially flow in one direction between nodes to enable long-range information transmission across the graph. We also introduce a novel complex-valued Laplacian for directed graphs with fuzzy edges where the real and imaginary parts represent information flow in opposite directions. Using this Laplacian, we propose a general framework, called Continuous Edge Direction (CoED) GNN, for learning on graphs with fuzzy edges and prove its expressivity limits using a generalization of the Weisfeiler-Leman (WL) graph isomorphism test for directed graphs with fuzzy edges. Our architecture aggregates neighbor features scaled by the learned edge directions and processes the aggregated messages from in-neighbors and out-neighbors separately alongside the self-features of the nodes. Since continuous edge directions are differentiable, they can be learned jointly with the GNN weights via gradient-based optimization. CoED GNN is particularly well-suited for graph ensemble data where the graph structure remains fixed but multiple realizations of node features are available, such as in gene regulatory networks, web connectivity graphs, and power grids. We demonstrate through extensive experiments on both synthetic and real graph ensemble datasets that learning continuous edge directions significantly improves performance both for undirected and directed graphs compared with existing methods.

Farzan Vafa, Sahand Hormoz

Hidden Markov Models (HMMs) are powerful tools for modeling sequential data, where the underlying states evolve in a stochastic manner and are only indirectly observable. Traditional HMM approaches are well-established for linear sequences, and have been extended to other structures such as trees. In this paper, we extend the framework of HMMs on trees to address scenarios where the tree-like structure of the data includes coupled branches -- a common feature in biological systems where entities within the same lineage exhibit dependent characteristics. We develop a dynamic programming algorithm that efficiently solves the likelihood, decoding, and parameter learning problems for tree-based HMMs with coupled branches. Our approach scales polynomially with the number of states and nodes, making it computationally feasible for a wide range of applications and does not suffer from the underflow problem. We demonstrate our algorithm by applying it to simulated data and propose self-consistency checks for validating the assumptions of the model used for inference. This work not only advances the theoretical understanding of HMMs on trees but also provides a practical tool for analyzing complex biological data where dependencies between branches cannot be ignored.