Discovering Partially Known Ordinary Differential Equations: a Case Study on the Chemical Kinetics of Cellulose Degradation

TL;DR

The study tackles modeling cellulose insulation aging in power transformers under data scarcity by leveraging Physics-Informed Neural Networks (PINNs) and symbolic regression to identify partially known ODEs. It first uses PINNs to infer Arrhenius parameters $A$ and $E$ in the Ekenstam equation via scaled formulations, then extends the framework to discover an unknown function in Emsley's ODE and an accompanying parameter $k_2$ through a secondary NN and PySR symbolic regression. The results show recovered parameter values within literature ranges for synthetic and measured data, and accurate reconstruction of DP trajectories with limited observations, while highlighting sensitivity to noise and the need for uncertainty quantification. These findings offer a data-efficient pathway to model transformer insulation aging and to identify novel functional forms in coupled degradation systems.

Abstract

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.

Figures (14)

• Figure 1: Schematic representation of PINNs for the discovery of the scaled unknown parameters $ln(A)$ and $\frac{E}{RT}$ of the Ekenstam ODE.
• Figure 2: Schematic representation of PINNs for the discovery of the unknown function $h(\text{DP},k_1,t)$ and parameter $k_2$ of the Emsley system of ODEs.
• Figure 3: Box plot for the estimated $ln(A)$ values for Dataset 1, its three cases with added noise, Dataset 2 and Dataset 3.
• Figure 4: Box plot for the estimated $\frac{E}{RT}$ values for Dataset 1, its three cases with added noise, Dataset 2 and Dataset 3.
• Figure 5: PINNs prediction (orange-dotted line) of Dataset 1 (blue dots) and the corresponding exact solution (black line). The red-dotted line represents the end-of-life of the power transformer.