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Physics informed neural network for forward and inverse radiation heat transfer in graded-index medium

K. Murari, S. Sundar

TL;DR

The paper tackles the challenge of solving the radiative transfer equation in graded-index media, where spatially varying refractive index induces curved light paths and high-dimensional complexity. It introduces a physics-informed neural network (PINN) framework that solves both forward and inverse problems in this setting, with theoretical generalization bounds and an ensemble-training strategy to select hyperparameters. The authors validate the approach through multiple 1D and 2D test cases, including Gaussian and discontinuous sources, demonstrating accurate, stable results and improved robustness over traditional mesh-based methods. The work offers a mesh-free, scalable, and fast solver for complex radiative-transfer problems in graded-index media, with potential for high-dimensional and data-assimilation applications.

Abstract

Radiation heat transfer in a graded-index medium often suffers accuracy problems due to the gradual changes in the refractive index. The finite element method, meshfree, and other numerical methods often struggle to maintain accuracy when applied to this medium. To address this issue, we apply physics-informed neural networks (PINNs)-based machine learning algorithms to simulate forward and inverse problems for this medium. We also provide the theoretical upper bounds. This theoretical framework is validated through numerical experiments of predefined and newly developed models that demonstrate the accuracy and robustness of the algorithms in solving radiation transport problems in the medium. The simulations show that the novel algorithm goes on with numerical stability and effectively mitigates oscillatory errors, even in cases with more pronounced variations in the refractive index.

Physics informed neural network for forward and inverse radiation heat transfer in graded-index medium

TL;DR

The paper tackles the challenge of solving the radiative transfer equation in graded-index media, where spatially varying refractive index induces curved light paths and high-dimensional complexity. It introduces a physics-informed neural network (PINN) framework that solves both forward and inverse problems in this setting, with theoretical generalization bounds and an ensemble-training strategy to select hyperparameters. The authors validate the approach through multiple 1D and 2D test cases, including Gaussian and discontinuous sources, demonstrating accurate, stable results and improved robustness over traditional mesh-based methods. The work offers a mesh-free, scalable, and fast solver for complex radiative-transfer problems in graded-index media, with potential for high-dimensional and data-assimilation applications.

Abstract

Radiation heat transfer in a graded-index medium often suffers accuracy problems due to the gradual changes in the refractive index. The finite element method, meshfree, and other numerical methods often struggle to maintain accuracy when applied to this medium. To address this issue, we apply physics-informed neural networks (PINNs)-based machine learning algorithms to simulate forward and inverse problems for this medium. We also provide the theoretical upper bounds. This theoretical framework is validated through numerical experiments of predefined and newly developed models that demonstrate the accuracy and robustness of the algorithms in solving radiation transport problems in the medium. The simulations show that the novel algorithm goes on with numerical stability and effectively mitigates oscillatory errors, even in cases with more pronounced variations in the refractive index.

Paper Structure

This paper contains 27 sections, 79 equations, 28 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement and PINN approximation
  3. The model
  4. The underlying model and domain
  5. The underlying model for forward problem
  6. The underlying model for inverse problem
  7. Quadrature rules
  8. Training points
  9. Interior training points
  10. Temporal boundary training points
  11. Spatial boundary training points
  12. Data training points
  13. Neural networks
  14. Residuals
  15. Loss functions and optimization
  16. ...and 12 more sections

Figures (28)

  • Figure 1: An illustration of a Snell's law.
  • Figure 2: Training points (forward problem): A depiction of the training set $\boldsymbol{S}$ with randomly selected training points. Yellow dots represent interior points, while green and blue dots indicate the temporal and spatial boundary points.
  • Figure 3: Training points (inverse problem): A representation of the training set $\boldsymbol{S}$ with randomly selected training points. Yellow dots indicate interior points, and grey dots represent Sobol points.
  • Figure 4: In this diagram, input layer neurons are depicted in red, hidden layer neurons in yellow, and output layer neurons in green.
  • Figure 5: Source term of Eq.\ref{['eq:1d']}.
  • ...and 23 more figures

Theorems & Definitions (4)

  • Remark 2.3
  • Remark 2.4
  • proof
  • proof