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A Method for Auto-Differentiation of the Voronoi Tessellation (2312.16192v3)

Published 22 Dec 2023 in cs.CG and cs.LG

Abstract: Voronoi tessellation, also known as Voronoi diagram, is an important computational geometry technique that has applications in various scientific disciplines. It involves dividing a given space into regions based on the proximity to a set of points. Autodifferentiation is a powerful tool for solving optimization tasks. Autodifferentiation assumes constructing a computational graph that allows to compute gradients using backpropagation algorithm. However, often the Voronoi tessellation remains the only non-differentiable part of a pipeline, prohibiting end-to-end differentiation. We present the method for autodifferentiation of the 2D Voronoi tessellation. The method allows one to construct the Voronoi tessellation and pass gradients, making the construction end-to-end differentiable. We provide the implementation details and present several important applications. To the best of our knowledge this is the first autodifferentiable realization of the Voronoi tessellation providing full set of Voronoi geometrical parameters in a differentiable way.

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References (24)
  1. Abdelkhalek, M. Sutherland–hodgman. https://github.com/mhdadk/sutherland-hodgman, 2022.
  2. Voronoi tessellation based statistical volume element characterization for use in fracture modeling. Computer Methods in Applied Mechanics and Engineering, 336:135–155, 2018.
  3. Differentiable biology: using deep learning for biophysics-based and data-driven modeling of molecular mechanisms. Nature methods, 18(10):1169–1180, 2021.
  4. Differentiable quantum computational chemistry with pennylane. arXiv preprint arXiv:2111.09967, 2021.
  5. VORONOI DIAGRAMS AND DELAUNAY TRIANGULATIONS. World Scientific Publishing Co. Pte. Ltd., 1st edition, 2013.
  6. Braden, B. The surveyor’s area formula. The College Mathematics Journal, 17(4):326–337, 1986.
  7. Semi-discrete normalizing flows through differentiable tessellation. Advances in Neural Information Processing Systems, 35:14878–14889, 2022.
  8. The formation of voronoi diagrams in chemical and physical systems: experimental findings and theoretical models. International journal of bifurcation and chaos, 14(07):2187–2210, 2004.
  9. Edge detecting new physics the voronoi way. Europhysics letters, 114(4):41001, 2016.
  10. Differentiable graph-structured models for inverse design of lattice materials. arXiv preprint arXiv:2304.05422, 2023.
  11. Clustering biological data using voronoi diagram. In Advanced Computing, Networking and Security: International Conference, ADCONS 2011, Surathkal, India, December 16-18, 2011, Revised Selected Papers, pp.  188–197. Springer, 2012.
  12. Cellular topology optimization on differentiable voronoi diagrams. International Journal for Numerical Methods in Engineering, 124(1):282–304, 2023.
  13. Voronoi tessellations and their application to climate and global modeling. 1 2011. URL https://www.osti.gov/biblio/1090872.
  14. Space: The re-visioning frontier of biological image analysis with graph theory, computational geometry, and spatial statistics. Mathematics, 9(21), 2021. ISSN 2227-7390. doi: 10.3390/math9212726. URL https://www.mdpi.com/2227-7390/9/21/2726.
  15. Dqc: A python program package for differentiable quantum chemistry. The Journal of chemical physics, 156(8), 2022.
  16. Crystal voronoi diagram and its applications. Future Generation Computer Systems, 18(5):681–692, 2002.
  17. Differentiable physics simulation. In ICLR 2020 Workshop on Integration of Deep Neural Models and Differential Equations, 2020.
  18. An intelligent modeling framework to optimize the spatial layout of ocean moored buoy observing networks. Frontiers in Marine Science, 10:1134418, 2023.
  19. Methods for the study of cellular sociology: Voronoi diagrams and parametrization of the spatial relationships. Journal of theoretical biology, 154(3):359–369, 1992.
  20. Naumann, U. The art of differentiating computer programs : an introduction to algorithmic differentiation. SIAM, 1st edition, 2012. ISBN ISBN 978-1-611972-06-1.
  21. Nowak, A. Application of voronoi diagrams in contemporary architecture and town planning. Challenges of Modern Technology, 6(2):30–34, 2015.
  22. Automatic differentiation in pytorch. 2017.
  23. Microscale modelling of the deformation of a martensitic steel using the voronoi tessellation method. Journal of the Mechanics and Physics of Solids, 113:35–55, 2018.
  24. End-to-end differentiable construction of molecular mechanics force fields. Chemical Science, 13(41):12016–12033, 2022.
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