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

Yani Feng, Michael K. Ng, Zhiwen Zhang

TL;DR

This work introduces DeepLagrangian, a Lagrangian-based, flow-enabled approach for learning and generating aggregation patterns in Keller–Segel chemotaxis systems, including near-singular blow-up behavior in 2D and 3D. By normalizing the KS density to a PDF and reformulating the dynamics into a continuity-ODE framework, the method uses a time-dependent KRnet to learn the PDF under a physics-informed Lagrangian loss, with a time marching strategy to handle rapid temporal changes. A theoretical bound ties the KL divergence between the learned and true PDFs to the Lagrangian loss, ensuring principled accuracy. Numerical experiments demonstrate high fidelity in diverse scenarios, outperforming Eulerian PINN-based approaches, and showcase the method's ability to capture complex aggregation and blow-up patterns under varying initial conditions and advection parameters.

Abstract

The Keller-Segel (KS) chemotaxis system is used to describe the overall behavior of a collection of cells under the influence of chemotaxis. However, solving the KS chemotaxis system and generating its aggregation patterns remain challenging due to the emergence of solutions exhibiting near-singular behavior, such as finite-time blow-up or concentration phenomena. Building on a Lagrangian framework of the KS system, we develop DeepLagrangian, a self-adaptive density estimation method that learns and generates aggregation patterns and near-singular solutions of the KS system in two- and three-dimensional (2D and 3D) space under different physical parameters. The main advantage of the Lagrangian framework is its inherent ability to adapt to near-singular solutions. To develop this framework, we normalize the KS solution into a probability density function (PDF), derive the corresponding normalized KS system, and utilize the property of the continuity equation to rewrite the system into a Lagrangian framework. We then define a physics-informed Lagrangian loss to enforce this framework and incorporate a flow-based generative model, called the time-dependent KRnet, to approximate the PDF by minimizing the loss. Furthermore, we integrate time-marching strategies with the time-dependent KRnet to enhance the accuracy of the PDF approximation. After obtaining the approximate PDF, we recover the original KS solution. We also prove that the Lagrangian loss effectively controls the Kullback-Leibler (KL) divergence between the approximate PDF and the exact PDF. In the numerical experiments, we demonstrate the accuracy of our DeepLagrangian method for the 2D and 3D KS chemotaxis system with/without advection.

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

TL;DR

This work introduces DeepLagrangian, a Lagrangian-based, flow-enabled approach for learning and generating aggregation patterns in Keller–Segel chemotaxis systems, including near-singular blow-up behavior in 2D and 3D. By normalizing the KS density to a PDF and reformulating the dynamics into a continuity-ODE framework, the method uses a time-dependent KRnet to learn the PDF under a physics-informed Lagrangian loss, with a time marching strategy to handle rapid temporal changes. A theoretical bound ties the KL divergence between the learned and true PDFs to the Lagrangian loss, ensuring principled accuracy. Numerical experiments demonstrate high fidelity in diverse scenarios, outperforming Eulerian PINN-based approaches, and showcase the method's ability to capture complex aggregation and blow-up patterns under varying initial conditions and advection parameters.

Abstract

The Keller-Segel (KS) chemotaxis system is used to describe the overall behavior of a collection of cells under the influence of chemotaxis. However, solving the KS chemotaxis system and generating its aggregation patterns remain challenging due to the emergence of solutions exhibiting near-singular behavior, such as finite-time blow-up or concentration phenomena. Building on a Lagrangian framework of the KS system, we develop DeepLagrangian, a self-adaptive density estimation method that learns and generates aggregation patterns and near-singular solutions of the KS system in two- and three-dimensional (2D and 3D) space under different physical parameters. The main advantage of the Lagrangian framework is its inherent ability to adapt to near-singular solutions. To develop this framework, we normalize the KS solution into a probability density function (PDF), derive the corresponding normalized KS system, and utilize the property of the continuity equation to rewrite the system into a Lagrangian framework. We then define a physics-informed Lagrangian loss to enforce this framework and incorporate a flow-based generative model, called the time-dependent KRnet, to approximate the PDF by minimizing the loss. Furthermore, we integrate time-marching strategies with the time-dependent KRnet to enhance the accuracy of the PDF approximation. After obtaining the approximate PDF, we recover the original KS solution. We also prove that the Lagrangian loss effectively controls the Kullback-Leibler (KL) divergence between the approximate PDF and the exact PDF. In the numerical experiments, we demonstrate the accuracy of our DeepLagrangian method for the 2D and 3D KS chemotaxis system with/without advection.
Paper Structure (18 sections, 4 theorems, 58 equations, 9 figures, 1 algorithm)

This paper contains 18 sections, 4 theorems, 58 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Lagrangian framework
  3. Methodology
  4. Physics-informed Lagrangian loss
  5. Time-dependent KRnet
  6. Time marching strategy
  7. Compare with Eulerian methods
  8. Theoretical analysis
  9. Numerical experiments
  10. Test problem 1: 2D KS system without advection
  11. Test problem 2: 2D KS system without advection for different initial conditions
  12. Case 1: Two closely spaced distributions composing initial conditions
  13. Case 2: Two widely spaced distributions composing initial conditions
  14. Test problem 3: 2D KS system with advection
  15. Dependence on the evolution time
  16. ...and 3 more sections

Key Result

Lemma 4.1

Assume that $\bar{\rho}_{\theta}$ and $\bar{\rho}$ are the solutions to fks_density and fks_approx, respectively. The following equation holds,

Figures (9)

  • Figure 1: 2D KS solutions without time marching, test problem 1.
  • Figure 2: 2D KS solutions with time marching, test problem 1.
  • Figure 3: Comparison of 2D KS solutions obtained by our DeepLagrangian method, PINNs and Adaptive-PINNs, test problem 1.
  • Figure 4: Solutions of the 2D KS system without advection for Case 1, test problem 2.
  • Figure 5: Solutions of the 2D KS system without advection for Case 2, test problem 2.
  • ...and 4 more figures

Theorems & Definitions (7)

  • Lemma 4.1
  • Proof 1
  • Lemma 4.2
  • Proof 2
  • Lemma 4.3
  • Theorem 4.4
  • Proof 3