Differentiable Voronoi Diagrams for Simulation of Cell-Based Mechanical Systems
Logan Numerow, Yue Li, Stelian Coros, Bernhard Thomaszewski
TL;DR
This work introduces differentiable restricted Voronoi diagrams as a cell-centered representation for simulating mechanical cellular systems with dynamic topology. By encoding each cell with a Voronoi site and using power-diagram geometry, the approach implicitly defines interfaces, reduces degrees of freedom, and provides closed-form first and second derivatives to support Newton-type optimization and equilibrium/inverse problems. The framework couples to deformable boundaries and demonstrates applications in tissue growth, foam coarsening, and image-based foam matching, achieving qualitatively similar results to explicit models but with substantially faster runtimes. The method offers a scalable, differentiable pathway for large-scale cell-based simulations and inverse analyses, with potential extensions toward richer cell shapes and boundary interactions.
Abstract
Navigating topological transitions in cellular mechanical systems is a significant challenge for existing simulation methods. While abstract models lack predictive capabilities at the cellular level, explicit network representations struggle with topology changes, and per-cell representations are computationally too demanding for large-scale simulations. To address these challenges, we propose a novel cell-centered approach based on differentiable Voronoi diagrams. Representing each cell with a Voronoi site, our method defines shape and topology of the interface network implicitly. In this way, we substantially reduce the number of problem variables, eliminate the need for explicit contact handling, and ensure continuous geometry changes during topological transitions. Closed-form derivatives of network positions facilitate simulation with Newton-type methods for a wide range of per-cell energies. Finally, we extend our differentiable Voronoi diagrams to enable coupling with arbitrary rigid and deformable boundaries. We apply our approach to a diverse set of examples, highlighting splitting and merging of cells as well as neighborhood changes. We illustrate applications to inverse problems by matching soap foam simulations to real-world images. Comparative analysis with explicit cell models reveals that our method achieves qualitatively comparable results at significantly faster computation times.