Diffeomorph Achieves 3D Shape Morphing Via Differentiable Agent-Based Simulations

Understanding how collections of simple agents create complex three-dimensional structures is a fundamental challenge in biology and increasingly relevant to robotics and artificial intelligence, and Seong Ho Pahng from Harvard University, Guoye Guan and Benjamin Fefferman from Harvard Medical School, with Sahand Hormoz also at Harvard Medical School, have now developed a new framework to address this problem. Their research introduces DiffeoMorph, a system that learns how to guide a group of agents to form a desired three-dimensional shape through a process mimicking biological morphogenesis. This innovative approach uses a differentiable simulation, allowing the system to learn from its mistakes and refine its control strategy, and crucially, incorporates a novel shape-matching loss function based on 3D Zernike polynomials. By comparing shapes as continuous distributions rather than discrete points, and ensuring invariance to rotations and agent arrangement, DiffeoMorph achieves a significant advance in the ability to model and control complex shape formation, demonstrating the creation of diverse morphologies from simple spatial cues.

Biological systems can form complex three-dimensional structures through the collective behaviour 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. This work introduces DiffeoMorph, an end-to-end differentiable framework for learning a morphogenesis protocol that guides a population of agents to.

Benchmarking Metrics and Mathematical Formulations

This is an incredibly thorough and detailed supplementary document. It provides a wealth of information, mathematical formulations, and explanations that would be invaluable to anyone reading the associated paper. The document’s strengths are numerous. It is highly complete, covering a wide range of material, from detailed explanations of benchmarking metrics to precise mathematical expressions and even the underlying principles such as Clebsch–Gordan coefficients. Despite the technical complexity, the writing remains generally clear and well organized, with standard and appropriate use of mathematical notation. The mathematical rigor is strong, with accurate and complete formulations, clear definitions of symbols, and helpful explanations of underlying ideas, such as the connection between Fourier transforms and spatial correlations. Structurally, the document is well arranged, with clear headings and subheadings and a logical flow that is easy to follow. Importantly, the explanations go beyond formulas by providing context and meaning, for example explaining how the power spectrum relates to the two-point correlation function. The benchmarking metrics are treated with exceptional thoroughness, with detailed explanations and mathematical definitions that make the evaluation process easy to understand. Overall, it reads like a very well-written appendix to a research paper, offering everything needed for reproducibility and deeper understanding.

There are a few minor, mostly stylistic suggestions. Notation consistency should be double-checked to ensure that the same symbols always represent the same quantities. Adding equation numbers for key formulas would make referencing easier in the main text or other analyses. Including a few simple visual aids could help readers visualize certain concepts, particularly those related to angular momentum coupling. Cross-referencing between this document and the main paper would also improve navigation. For readability, additional whitespace between sections and slightly larger font sizing could be beneficial. Finally, defining the sets “IB” and “IT” earlier, perhaps when first introducing the bispectrum, would improve the flow. Overall, this is an outstanding supplementary document that demonstrates deep understanding and a strong commitment to clarity and completeness, making it a valuable resource for researchers in the field.

3D Morphogenesis via Zernike Polynomial Matching

Scientists have developed DiffeoMorph, a new framework for controlling the collective behavior of agents to form complex three-dimensional shapes, mirroring processes seen in biological systems like cell morphogenesis. The core of this achievement lies in a novel shape-matching loss function based on 3D Zernike polynomials, which represents shapes as continuous spatial distributions rather than discrete point clouds, overcoming limitations of existing shape comparison methods. This new loss function is invariant to agent ordering and number, and crucially, remains sensitive to reflections while being fully invariant to rotations, a key feature for accurately capturing complex morphologies., The team’s shape-matching loss overcomes limitations of existing methods like Chamfer distance, Earth Mover’s distance, and Gromov, Wasserstein distances by simultaneously satisfying permutation invariance, point-count robustness, rotation invariance, and chirality sensitivity, while also capturing high-frequency shape details. Unlike methods that eliminate directional information, DiffeoMorph retains full spectral structure and explicitly aligns the predicted shape’s spectrum to the target by optimizing a unit quaternion, maximizing spectral overlap.

This alignment process is integrated into a bilevel optimization structure, where gradients are computed through implicit differentiation, enabling efficient end-to-end training., Benchmarking experiments demonstrate the superiority of this new loss function, and the framework successfully guides agents to form a range of shapes, from simple ellipsoids to complex morphologies, using only minimal spatial cues. The system achieves this by combining the spectral shape-matching loss with an SE(3)-equivariant force model, which updates each agent’s position and internal state based on its configuration and signals from other agents. This work provides a principled objective for learning 3D morphogenesis and has implications for engineering artificial systems, including self-assembly, swarm robotics, and organoid development.