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An Efficient Particle-Field Algorithm with Neural Interpolation based on a Parabolic-Hyperbolic Chemotaxis System in 3D

Jongwon David Kim, Jack Xin

TL;DR

The paper tackles the computational challenge of simulating 3D chemotaxis/angiogenesis models based on the parabolic-hyperbolic Keller-Segel system. It introduces Neural SIPF (NSIPF), a mesh-free particle-field method where density is an empirical measure of particles and the chemoattractant field is evolved with a CNN interpolator trained on synthetic radial PHKS data; a mollified McKean-Vlasov/propagation-of-chaos framework provides theoretical backing and motivates the neural operator $b_\epsilon$. Key contributions include mass conservation and non-negativity, 3D scalability, and competitive accuracy with FD and SIPF, along with substantial speedups. The results demonstrate diffusion and aggregation behaviors in 3D and highlight potential for physics-informed AI in scalable tumor-vasculature modeling.

Abstract

Tumor angiogenesis involves a collection of tumor cells moving towards blood vessels for nutrients to grow. Angiogenesis, and in general chemotaxis systems have been modeled using partial differential equations (PDEs) and as such require numerical methods to approximate their solutions in 3 space dimensions (3D). This is an expensive computation when solutions develop large gradients at unknown locations, and so efficient algorithms to capture the main dynamical behavior are valuable. Here as a case study, we consider a parabolic-hyperbolic Keller-Segel (PHKS) system in the angiogenesis literature, and develop a mesh-free particle-based neural network algorithm that scales better to 3D than traditional mesh based solvers. From a regularized approximation of PHKS, we derive a neural stochastic interacting particle-field (NSIPF) algorithm where the bacterial density is represented as empirical measures of particles and the field variable (concentration of chemo-attractant) by a convolutional neural network (CNN) trained on low cost synthetic data. As a new model, NSIPF preserves total mass and nonnegativity of the density, and captures the dynamics of 3D multi-bump solutions at much faster speeds compared with classical finite difference (FD) and SIPF methods.

An Efficient Particle-Field Algorithm with Neural Interpolation based on a Parabolic-Hyperbolic Chemotaxis System in 3D

TL;DR

The paper tackles the computational challenge of simulating 3D chemotaxis/angiogenesis models based on the parabolic-hyperbolic Keller-Segel system. It introduces Neural SIPF (NSIPF), a mesh-free particle-field method where density is an empirical measure of particles and the chemoattractant field is evolved with a CNN interpolator trained on synthetic radial PHKS data; a mollified McKean-Vlasov/propagation-of-chaos framework provides theoretical backing and motivates the neural operator . Key contributions include mass conservation and non-negativity, 3D scalability, and competitive accuracy with FD and SIPF, along with substantial speedups. The results demonstrate diffusion and aggregation behaviors in 3D and highlight potential for physics-informed AI in scalable tumor-vasculature modeling.

Abstract

Tumor angiogenesis involves a collection of tumor cells moving towards blood vessels for nutrients to grow. Angiogenesis, and in general chemotaxis systems have been modeled using partial differential equations (PDEs) and as such require numerical methods to approximate their solutions in 3 space dimensions (3D). This is an expensive computation when solutions develop large gradients at unknown locations, and so efficient algorithms to capture the main dynamical behavior are valuable. Here as a case study, we consider a parabolic-hyperbolic Keller-Segel (PHKS) system in the angiogenesis literature, and develop a mesh-free particle-based neural network algorithm that scales better to 3D than traditional mesh based solvers. From a regularized approximation of PHKS, we derive a neural stochastic interacting particle-field (NSIPF) algorithm where the bacterial density is represented as empirical measures of particles and the field variable (concentration of chemo-attractant) by a convolutional neural network (CNN) trained on low cost synthetic data. As a new model, NSIPF preserves total mass and nonnegativity of the density, and captures the dynamics of 3D multi-bump solutions at much faster speeds compared with classical finite difference (FD) and SIPF methods.
Paper Structure (15 sections, 1 theorem, 29 equations, 8 figures, 2 tables, 1 algorithm)

This paper contains 15 sections, 1 theorem, 29 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Parabolic-Hyperbolic KS (PHKS)
  3. Propagation of chaos for the PHKS system
  4. Towards a Neural approach
  5. Computational methods for PHKS
  6. Neural SIPF Algorithm
  7. Neural Interpolator Training
  8. Numerical Experiments and Discussion
  9. Diffusive behavior with one blob
  10. Aggregation behavior with two blobs
  11. Aggregation behavior along annuli boundary
  12. Comparison of numerical methods
  13. Numerical Convergence
  14. Conclusion and Future Research
  15. Acknowledgments

Key Result

Theorem 1

(Propagation of chaos with moderate interaction) Let the interacting particle system and its mollification for PHKS system be where $b_\epsilon$ is defined by (11). Then, by synchronous coupling with moderate interaction, the propagation of chaos holds for the PHKS system. As $N \to \infty$, the empirical measure of the particle system (14) converges weakly to the measure with density $\rho$ sati

Figures (8)

  • Figure 1: Diagram of the CNN architecture.
  • Figure 2: CNN training loss vs. epochs.
  • Figure 3: One bump solution particles at $t = 0$ and $t = 40$ produced by NSIPF with $P = 20000$.
  • Figure 4: Target data points (red) and their neural interpolation at the $y = 50$ slice.
  • Figure 5: Two-blob solution particles aggregating toward the center at t = 0.0, 10.0, 40.0 produced by NSIPF with $P = 20000$
  • ...and 3 more figures

Theorems & Definitions (3)

  • Theorem 1
  • proof : Proof Sketch
  • Remark 1