一种用于多物种趋化性中亚临界和超临界动力学的混合物理信息神经网络框架
A Hybrid Physics-Informed Neural Network Framework for Subcritical and Supercritical Dynamics in Multi-Species Chemotaxis
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中文总结 AI 辅助
研究两物种趋化系统的亚临界和超临界动力学，提出混合物理信息神经网络框架，亚临界用连续时间PINN，超临界用离散时间PINN并结合多种策略，数值实验验证该框架能准确解析不同状态下的动力学。
中文摘要 AI 辅助
我们研究了一个两物种趋化系统，它表现出两种性质不同的状态：亚临界动力学，其中解保持光滑；超临界动力学，其中强聚集可能导致有限时间爆破。在亚临界状态下，我们使用标准的连续时间物理信息神经网络（PINN）对耦合场进行交替训练，结果表明它能提供准确有效的近似。然而，在超临界状态下，这种公式不够稳健，无法解析与爆破相关的高度局部化结构和陡峭梯度。为解决这一限制，我们引入基于向后欧拉时间步长的离散时间PINN，结合对数变换以稳定大的解值，以及一种将训练点集中在高活动区域附近的残差自适应配置策略。这种混合框架使我们在解光滑时能使用简单且计算成本低的PINN，而在出现奇异动力学时切换到更稳健的离散公式。数值实验证实，连续时间公式能准确解析亚临界动力学，而离散时间公式能可靠地捕捉超临界状态下的奇异聚集和有限时间爆破。
英文摘要
We study a two-species chemotaxis system that exhibits two qualitatively different regimes: subcritical dynamics, in which solutions remain smooth, and supercritical dynamics, in which strong aggregation may lead to finite-time blow-up. In the subcritical regime, we use a standard continuous-time physics-informed neural network (PINN) with alternating training for the coupled fields and show that it provides accurate and efficient approximations. In the supercritical regime, however, this formulation is not sufficiently robust to resolve the highly localized structures and steep gradients associated with blow-up. To address this limitation, we introduce a discrete-time PINN based on backward Euler time stepping, combined with a logarithmic transformation to stabilize large solution values and a residual-adaptive collocation strategy that concentrates training points near regions of high activity. This hybrid framework allows us to use a simple and computationally inexpensive PINN when the solution is smooth, while switching to a more robust discrete formulation when singular dynamics emerge. Numerical experiments confirm that the continuous-time formulation accurately resolves subcritical dynamics, while the discrete-time formulation reliably captures singular aggregation and finite-time blow-up in the supercritical regime.
