Unified Learning of the Profile Function in Discrete Keller-Segel Models

TL;DR

This work tackles the problem of identifying the nonparametric profile function in discrete Keller–Segel models by proposing a unified variational learning framework that operates on data from both deterministic particle methods and stochastic particle (SDE) representations. The core idea is to represent the regularized profile $φ^{reg}$ as a linear combination of basis functions and learn its coefficients by solving a variational inverse problem grounded in particle trajectories, using an adaptive knot refinement strategy to capture singular behavior near the origin. The method is validated across 1D–4D deterministic KS dynamics and 2D stochastic KS dynamics, showing accurate trajectory reconstruction and improved kernel learning with adaptive knots, though profile error grows with kernel singularity and stochastic noise. Overall, the approach offers a data-driven, energetically consistent way to infer nonlocal interaction kernels in high-dimensional chemotaxis models, with potential impact on parameter-free modeling and inference in multi-species, multi-physics settings.

Abstract

We propose a unified learning framework for identifying the profile function in discrete Keller-Segel equations, which are widely used mathematical models for understanding chemotaxis. Training data are obtained via either a rigorously developed particle method designed for stable simulation of high-dimensional Keller-Segel systems, or stochastic differential equations approximating the continuous Keller-Segel PDE. Our approach addresses key challenges, including data instability in dimensions higher than two and the accurate capture of singular behavior in the profile function. Additionally, we introduce an adaptive learning strategy to enhance performance. Extensive numerical experiments are presented to validate the effectiveness of our method.

Figures (8)

• Figure 1: \newlabelfig:1D_kernel_uniform_knot0 Comparison of the learned profile functions with the regularized profile functions in the one-dimensional modified Keller–Segel model, using 30 uniform knot points and a truncation parameter of $r_c = 0.01$. The top, middle, and bottom subfigures correspond to the parameters $\chi = 0.35$, $0.55$, and $0.75$, respectively. The corresponding particle trajectories—learned (left) and true (right)—are shown for each $\chi$ value.
• Figure 2: Comparison of the learned profile functions with the regularized profile functions in the one-dimensional modified Keller–Segel model, using 25 adaptive (left) and uniform (right) knot points, with $r_c = 0.01$ and $\chi = 0.55$. Insets highlight the profile behavior close to the origin.
• Figure 3: Comparison of the learned and regularized profile functions in the two-dimensional Keller–Segel model, obtained using 20 uniform knot points and a truncation parameter of $r_c = 0.05$. The left, middle, and bottom subfigures correspond to initial weight parameters $\chi = 1.0$, $2.0$, and $4.0$, respectively. For each $\chi$ value, the corresponding particle trajectories—learned (left) and true (right)—are also shown.
• Figure 4: Comparison of the learned profile functions with the regularized profile functions in the two-dimensional Keller–Segel model using 22 adaptive (left) and uniform (right) knot points, with $r_c = 0.01$ and $\omega = 2.0$. Insets provide a detailed view of the profile near the origin.
• Figure 5: Comparison of the learned profile functions with the regularized profile functions in the three-dimensional modified Keller–Segel model, using 25 uniform knot points and a truncation parameter of $r_c = 0.05$. The left, middle, and bottom subfigures correspond to $\chi = 1.0$, $2.0$, and $4.0$, respectively. The corresponding three-dimensional particle trajectories—learned (left) and true (right)—are also shown, using the same parameter settings. The colorbar represents time.