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

Zhongjian Wang, Jack Xin, Zhiwen Zhang

TL;DR

This work addresses learning and generating multi-dimensional Keller–Segel chemotaxis patterns, including near-singular aggregation, by combining a regularized interacting particle method with a DeepParticle framework. DP learns a transport map from an initial particle distribution to the finite-time distribution before blow-up by minimizing a discrete $2$-Wasserstein distance, while an iterative LP-based procedure computes the optimal transition matrix on subproblems to reduce cost. The approach is demonstrated in 2D and 3D KS systems with and without advection, including laminar and chaotic Kolmogorov flows, showing accurate generation and extrapolation to unseen parameter values. The results enable fast parametric studies and could be extended to other pattern-forming PDEs and KS-like models in biology and fluid dynamics.

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.

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

TL;DR

This work addresses learning and generating multi-dimensional Keller–Segel chemotaxis patterns, including near-singular aggregation, by combining a regularized interacting particle method with a DeepParticle framework. DP learns a transport map from an initial particle distribution to the finite-time distribution before blow-up by minimizing a discrete -Wasserstein distance, while an iterative LP-based procedure computes the optimal transition matrix on subproblems to reduce cost. The approach is demonstrated in 2D and 3D KS systems with and without advection, including laminar and chaotic Kolmogorov flows, showing accurate generation and extrapolation to unseen parameter values. The results enable fast parametric studies and could be extended to other pattern-forming PDEs and KS-like models in biology and fluid dynamics.

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.
Paper Structure (18 sections, 21 equations, 10 figures, 2 tables, 1 algorithm)

This paper contains 18 sections, 21 equations, 10 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Regularized interacting particle method of KS model
  3. Blow up behavior in KS model
  4. Regularized interacting particle methods
  5. Deep particle method
  6. Discrete Wasserstein distance
  7. Training data and neural network configuration
  8. Iterative method in finding transition matrix $\gamma$
  9. Full Training Algorithm
  10. Numerical Examples
  11. 2D KS Simulation and Generation in the Absence of Advection
  12. KS Simulation and Generation in the Presence of 2D Laminar Flows
  13. Learning the dependence on $A$
  14. Learning the dependence on evolution time
  15. KS Simulation and Generation in the Presence of 3D flows
  16. ...and 3 more sections

Figures (10)

  • Figure 1: Performance of the IPM (reference) and DeepParticle (network) algorithms.
  • Figure 2: Comparison of particle distributions obtained by the DeepParticle method (network output) and IPM (reference) solver at different times and $A=0$.
  • Figure 3: Learning particle aggregation at different $A$ values with $t=0.02$ fixed. Both the interpolation and extrapolation performances of the network are tested.
  • Figure 4: Learning particle aggregation at different times with fixed flow amplitude $A=100$. The extrapolation performance of the network is also tested.
  • Figure 5: Three cross sections of generated distributions at different $A$ values in a 3D laminar flow (\ref{['Laminar']}).
  • ...and 5 more figures

Theorems & Definitions (1)

  • Remark 3.1