Variational neural transport for aggregation-diffusion equations
Abstract
We develop a neural transport framework for nonlinear, nonlocal aggregationdiffusion equations arising in chemotaxis, polymer dynamics, and related applications. Based on the Onsager variational principle, we reformulate the governing equation as a constrained optimal control problem, taking the velocity field as the control and evolving the density via the associated continuity equation. The velocity field is parameterized by a neural network trained on the variational objective evaluated along particle trajectories. By coupling the density and generalized score equations into a single system solved via the Euler method, the score function propagates efficiently for loss evaluation. Furthermore, we establish a rigorous error analysis that decomposes the total error into approximation, optimization, Monte Carlo sampling, and time-discretization components, and prove corresponding error bounds. Numerical experiments across several representative models demonstrate the accuracy and effectiveness of the proposed approach.
Related
- Aggregation--Diffusion Equations with Regular or Repulsive Interactions: Gaussian and Self-Similar Asymptotics
- A diffusion-free multi-layer neural-network method for multidimensional nonlinear hyperbolic equations
- Shape exclusion for diffusion-limited aggregation
- Quantitative Stability Of Aggregation-Diffusion Under Local Attraction
- Sharp Quantitative Convergence of Compactly Supported Aggregation-Diffusion
BibTeX
@misc{li2026variational,
title = {{Variational neural transport for aggregation-diffusion equations}},
author = {Qianyi Li and Hailiang Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12745},
eprint = {2610.12745},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12745},
}
RIS
TY - UNPB
TI - Variational neural transport for aggregation-diffusion equations
AU - Qianyi Li
AU - Hailiang Liu
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12745
UR - https://arxiv.org/abs/2610.12745
AB - We develop a neural transport framework for nonlinear, nonlocal aggregationdiffusion equations arising in chemotaxis, polymer dynamics, and related applications. Based on the Onsager variational principle, we reformulate the governing equation as a constrained optimal control problem, taking the velocity field as the control and evolving the density via the associated continuity equation. The velocity field is parameterized by a neural network trained on the variational objective evaluated along particle trajectories. By coupling the density and generalized score equations into a single system solved via the Euler method, the score function propagates efficiently for loss evaluation. Furthermore, we establish a rigorous error analysis that decomposes the total error into approximation, optimization, Monte Carlo sampling, and time-discretization components, and prove corresponding error bounds. Numerical experiments across several representative models demonstrate the accuracy and effectiveness of the proposed approach.
ER -