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Physics-Informed Neural Networks for Electrical Circuit Analysis: Applications in Dielectric Material Modeling

Reyhaneh Taj

TL;DR

Pedagogical insights are provided for those looking to use PINNs in both forward and inverse modes, particularly within the DeepXDE framework, in addressing both forward and inverse problems related to dielectric properties.

Abstract

Scientific machine learning (SciML) represents a significant advancement in integrating machine learning (ML) with scientific methodologies. At the forefront of this development are Physics-Informed Neural Networks (PINNs), which offer a promising approach by incorporating physical laws directly into the learning process, thereby reducing the need for extensive datasets. However, when data is limited or the system becomes more complex, PINNs can face challenges, such as instability and difficulty in accurately fitting the training data. In this article, we explore the capabilities and limitations of the DeepXDE framework, a tool specifically designed for implementing PINNs, in addressing both forward and inverse problems related to dielectric properties. Using RC circuit models to represent dielectric materials in HVDC systems, we demonstrate the effectiveness of PINNs in analyzing and improving system performance. Additionally, we show that applying a logarithmic transformation to the current (ln(I)) significantly enhances the stability and accuracy of PINN predictions, especially in challenging scenarios with sparse data or complex models. In inverse mode, however, we faced challenges in estimating key system parameters, such as resistance and capacitance, in more complex scenarios with longer time domains. This highlights the potential for future work in improving PINNs through transformations or other methods to enhance performance in inverse problems. This article provides pedagogical insights for those looking to use PINNs in both forward and inverse modes, particularly within the DeepXDE framework.

Physics-Informed Neural Networks for Electrical Circuit Analysis: Applications in Dielectric Material Modeling

TL;DR

Pedagogical insights are provided for those looking to use PINNs in both forward and inverse modes, particularly within the DeepXDE framework, in addressing both forward and inverse problems related to dielectric properties.

Abstract

Scientific machine learning (SciML) represents a significant advancement in integrating machine learning (ML) with scientific methodologies. At the forefront of this development are Physics-Informed Neural Networks (PINNs), which offer a promising approach by incorporating physical laws directly into the learning process, thereby reducing the need for extensive datasets. However, when data is limited or the system becomes more complex, PINNs can face challenges, such as instability and difficulty in accurately fitting the training data. In this article, we explore the capabilities and limitations of the DeepXDE framework, a tool specifically designed for implementing PINNs, in addressing both forward and inverse problems related to dielectric properties. Using RC circuit models to represent dielectric materials in HVDC systems, we demonstrate the effectiveness of PINNs in analyzing and improving system performance. Additionally, we show that applying a logarithmic transformation to the current (ln(I)) significantly enhances the stability and accuracy of PINN predictions, especially in challenging scenarios with sparse data or complex models. In inverse mode, however, we faced challenges in estimating key system parameters, such as resistance and capacitance, in more complex scenarios with longer time domains. This highlights the potential for future work in improving PINNs through transformations or other methods to enhance performance in inverse problems. This article provides pedagogical insights for those looking to use PINNs in both forward and inverse modes, particularly within the DeepXDE framework.

Paper Structure

This paper contains 17 sections, 10 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Physics-Informed Neural Networks (PINNs)
  3. RC Circuit Models for PINNs
  4. Case 0: Single RC Circuit
  5. Case 1: Two Parallel RC Circuits
  6. Case 2: Three Parallel RC Circuits
  7. Generalization to Case $N$
  8. Logarithmic Approach
  9. Implementation in DeepXDE
  10. Forward Mode
  11. Inverse Mode
  12. Results
  13. Forward Mode: Current Prediction
  14. Logarithmic Transformation for Enhanced Predictive Performance
  15. Hyperparameter Optimization and Time Domain Challenges
  16. ...and 2 more sections

Figures (6)

  • Figure 1: Overview of Physics-Informed Neural Networks (PINNs) architecture.
  • Figure 2: RC circuit configurations. (a) One R-C circuit in Case 0. (b) Three parallel RC branches in Case 2. This setup can be generalized to Case $N$ with additional RC branches in parallel
  • Figure 3: Predicted current for Case 0 as a function of time in forward mode. 'True data' refers to the analytical solution.
  • Figure 4: Predicted $\ln(I)$ with respect to time for Case 1. 'True data' refers to the analytical solution.
  • Figure 5: Predicted $\ln(I)$ with respect to time for Case 1 (a), Case 3 (b). 'True data' refers to the analytical solution. The logarithmic transformation significantly improves prediction accuracy for more complex cases.
  • ...and 1 more figures