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.

翻译：本文提出了一种统一的学习框架，用于识别离散Keller-Segel方程中的剖面函数，该方程是理解趋化现象的广泛应用数学模型。训练数据通过两种方式获取：一是为高维Keller-Segel系统稳定模拟而严格设计的粒子方法，二是逼近连续Keller-Segel偏微分方程的随机微分方程。我们的方法解决了关键挑战，包括二维以上数据的不稳定性以及剖面函数奇异行为的精确捕捉。此外，我们引入了一种自适应学习策略以提升性能。通过大量数值实验验证了该方法的有效性。