Neural optimization of the most probable paths of three-dimensional active Brownian particles

Bin Zheng; Zhongqiang Xiong; Changhao Li; Zhanglin Hou; Ziluo Zhang; Xinpeng Xu; Li-Shing Lin; Kenta Ishimoto; Kento Yasuda; Shigeyuki Komura

doi:10.1103/ys1b-1lrd

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Neural optimization of the most probable paths of three-dimensional active Brownian particles

Bin Zheng ¹, Zhongqiang Xiong ¹, Changhao Li ^2,3,1, Zhanglin Hou ¹, Ziluo Zhang ⁴, Xinpeng Xu ^5,6, Li-Shing Lin ⁷, Kenta Ishimoto ⁸, Kento Yasuda ⁹ et al.

Shigeyuki Komura ^1,*

¹Zhejiang Key Laboratory of Soft Matter Biomedical Materials, Wenzhou Institute, University of Chinese Academy of Sciences, Wenzhou, Zhejiang 325001, China
²College of Energy Engineering and State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou, Zhejiang 310003, China
³Joint Research Centre on Medicine, Xiangshan Hospital of Wenzhou Medical University, Ningbo, Zhejiang 315700, China
⁴Department of Physics, Xiamen University, Xiamen, Fujian 361005, China
⁵Department of Physics and MATEC Key Lab, Guangdong Technion–Israel Institute of Technology, Shantou, Guangdong 515063, China
⁶Technion - Israel Institute of Technology, Haifa 32000, Israel
⁷Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
⁸Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
⁹Laboratory of Physics, College of Science and Technology, Nihon University, Funabashi, Chiba 274-8501, Japan

^*Contact author: komura@wiucas.ac.cn

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Phys. Rev. E 114, L012103 – Published 7 July, 2026

DOI: https://doi.org/10.1103/ys1b-1lrd

Abstract

We develop a variational neural-network framework to determine the most probable path (MPP) of a 3D active Brownian particle (ABP) by directly minimizing the Onsager-Machlup integral (OMI). To obtain the OMI, we use the Onsager-Machlup variational principle for active systems and construct the Rayleighian of the ABP by including its active power. This approach reveals geometric transitions of the MPP from in-plane I- and U-shaped paths to 3D helical paths as the final time and net displacement are varied. We also demonstrate that the initial and final boundary conditions have a significant impact on the MPPs. Our results show that neural optimization combined with the Onsager-Machlup variational principle provides an efficient and versatile framework for exploring optimal transition pathways in active and nonequilibrium systems.

Physical Review E

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Neural optimization of the most probable paths of three-dimensional active Brownian particles

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Neural optimization of the most probable paths of three-dimensional active Brownian particles

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