发现部分已知的常微分方程：以纤维素降解的化学动力学为例研究
Discovering Partially Known Ordinary Differential Equations: a Case Study on the Chemical Kinetics of Cellulose Degradation
AI总结：
本研究针对电力变压器纤维素绝缘老化数据稀缺问题，利用PINNs和符号回归发现并恢复Ekenstam与Emsley常微分方程系统中的未知参数与函数形式。
摘要：
聚合度（DP）是基于聚合物绝缘系统（如电力组件中的纤维素绝缘）老化估计的方法之一。聚合物中的主要降解机制是水解、热解和氧化。这些机制共同作用导致DP的降低。然而，这类问题的数据可用性通常很少。本研究使用来自电力变压器的纤维素降解数据分析绝缘老化。电力变压器内部浸入矿物油中的纤维素老化问题用常微分方程（ODEs）进行建模。我们使用物理信息神经网络（PINNs）和符号回归恢复降解系统的控制方程。我们将PINNs应用于发现Ekenstam ODE中描述纤维素污染含量以及与温度相关的材料老化过程的Arrhenius方程的未知参数，针对合成数据和真实DP值。Ekenstam ODE的一种修改由Emsley的ODE系统给出，其中由Arrhenius方程表示的速率常数在新公式中随时间递减。我们使用PINNs和符号回归恢复该系统其中一个ODE的函数形式，并识别一个未知参数。
英文摘要：
The degree of polymerization (DP) is one of the methods for estimating the aging of the polymer based insulation systems, such as cellulose insulation in power components. The main degradation mechanisms in polymers are hydrolysis, pyrolysis, and oxidation. These mechanisms combined cause a reduction of the DP. However, the data availability for these types of problems is usually scarce. This study analyzes insulation aging using cellulose degradation data from power transformers. The aging problem for the cellulose immersed in mineral oil inside power transformers is modeled with ordinary differential equations (ODEs). We recover the governing equations of the degradation system using Physics-Informed Neural Networks (PINNs) and symbolic regression. We apply PINNs to discover the Arrhenius equation's unknown parameters in the Ekenstam ODE describing cellulose contamination content and the material aging process related to temperature for synthetic data and real DP values. A modification of the Ekenstam ODE is given by Emsley's system of ODEs, where the rate constant expressed by the Arrhenius equation decreases in time with the new formulation. We use PINNs and symbolic regression to recover the functional form of one of the ODEs of the system and to identify an unknown parameter.
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