A DeepParticle method for learning and generating aggregation patterns in multi-dimensional Keller-Segel chemotaxis systems

TL;DR

This paper introduces a DeepParticle method that combines particle simulations and deep neural networks to learn, generate, and analyze aggregation patterns in multi-dimensional Keller-Segel chemotaxis systems, handling complex flow regimes.

Contribution

It develops a novel DeepParticle framework that learns to generate solutions of Keller-Segel systems under parameter variations using Wasserstein distance minimization.

Findings

Successfully learned and generated Keller-Segel dynamics in laminar and chaotic flows.

Demonstrated the method's ability to handle near singular solutions and blowup scenarios.

Reduced computational cost with an iterative divide-and-conquer algorithm.

Abstract

We study a regularized interacting particle method for computing aggregation patterns and near singular solutions of a Keller-Segal (KS) chemotaxis system in two and three space dimensions, then further develop DeepParticle (DP) method to learn and generate solutions under variations of physical parameters. The KS solutions are approximated as empirical measures of particles which self-adapt to the high gradient part of solutions. We utilize the expressiveness of deep neural networks (DNNs) to represent the transform of samples from a given initial (source) distribution to a target distribution at finite time T prior to blowup without assuming invertibility of the transforms. In the training stage, we update the network weights by minimizing a discrete 2-Wasserstein distance between the input and target empirical measures. To reduce computational cost, we develop an iterative divide-and-conquer algorithm to find the optimal transition matrix in the Wasserstein distance. We present numerical results of DP framework for successful learning and generation of KS dynamics in the presence of laminar and chaotic flows. The physical parameter in this work is either the small diffusivity of chemo-attractant or the reciprocal of the flow amplitude in the advection-dominated regime.