Application of partial differential equations to the optimization of electoral district formation

Title	Application of partial differential equations to the optimization of electoral district formation
Authors	A. M. Abrashkin¹ ¹Ulyanovsk State Technical University
Annotation	The article is devoted to the analysis of the possibilities of using mathematical modeling to identify politically motivated formation of the boundaries of electoral districts and their further optimization. The problem of forming electoral districts is presented as a problem of partitioning a weighted graph using methods for solving partial differential equations (PDE). The approach is based on minimizing the “energy of compactness”, ensuring the optimal shape of the districts while respecting the principle of equal representation of the population. The procedure for calculating the gradient flow is implemented through the Merriman-Bence-Osher algorithm and Bertsekas auction dynamics. The method allows you to quantitatively evaluate existing maps for artificial distortions, and generate alternative, geographically based options for dividing the region into districts.
Keywords	Gerrymandering, differential equations, partial derivatives.
Citation	Abrashkin A. M. ''Application of partial differential equations to the optimization of electoral district formation'' [Electronic resource]. Mathematical modeling, numerical methods, and software systems: Collection of materials from the 12th All‑Russian Scientific Youth School‑Seminar named after E. V. Voskresensky (Saransk, July 21–24, 2026). - pp. 14-19. Available at: https://conf.svmo.ru/files/2026/papers/paper64.pdf. - Date of access: 25.09.2026.

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Application of partial differential equations to the optimization of electoral district formation

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