从无组织的数据到新兴的动态模型:问卷到部分微分方程.
David W Sroczynski1,2, Felix P Kemeth2, Anastasia S Georgiou2
1Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08544, USA.
PNAS nexus
|February 3, 2025
概括
本研究提出了一种数据驱动的方法,从无组织的观测中导出参数依赖的部分微分方程 (PDE) 模型. 它使复杂系统能够出现新的空间,时间和参数.
科学领域:
- 复杂系统动力学 复杂系统动力学
- 计算科学 计算科学
- 数学建模的数学建模
背景情况:
- 来自具有不同参数的系统的无组织,空间和时间变化的数据对传统建模构成了挑战.
- 从如此复杂,高维的数据集中推导治理方程需要新的数据驱动方法.
研究的目的:
- 开发一个可概括的,数据驱动的框架,从无组织的观测中推导依赖参数的进化局部微分方程 (PDE) 模型.
- 在这些衍生模型中展示空间,时间和参数的新兴性质.
- 将框架应用于各种系统,包括生物发展和网络动态.
主要方法:
- 使用基于扩散地图的问卷方法来创建新出现的空间,时间和参数维度的平滑参数化.
- 通过观察其在不同轴上的行为来代地组织张量数据.
- 采用机器学习,特别是神经网络,来近似控制新出现的进化方程的运算符.
主要成果:
- 成功地从各种复杂系统的无组织数据中获得了依赖参数的PDE模型.
- 展示了空间,时间和参数的新兴特性,这些参数直接从数据中确定.
- 验证了对向-扩散的方法,Drosophila发展,神经网络和合振荡器模型.
结论:
- 开发的数据驱动方法有效地从复杂,无组织的观测数据中重建控制进化的PDEs.
- 衍生模型的新兴特性,包括对称性破坏和转换不变性,为系统动态提供了洞察力.
- 该框架为理解和建模基础方程未知或难以确定的复杂系统提供了强大的工具.
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