Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From Obstacle Problems to Neural Insights: Feed Forward Neural Network Modeling of Ice Thickness

Kapil Chawla, William Holmes, Roger Temam

TL;DR

This work addresses modeling ice thickness under an obstacle problem formulation by marrying a variational-inequality framework with deep neural networks. The authors formulate an energy-minimization problem, introduce a composite loss that enforces PDE residuals, obstacle constraints, and boundary conditions, and approximate the solution with fully connected networks trained via Adam. They validate the approach through 1D and 2D MMS tests and apply it to Greenland data from NSIDC-0092, employing bedrock pretraining to stabilize learning; results show accurate, data-consistent thickness estimates and robust performance across $(p\ge2)$ cases. The study highlights the potential of combining classical mathematical modeling with modern neural methods for reliable ice-thickness estimation and broader geophysical applications.

Abstract

In this study, we integrate the established obstacle problem formulation from ice sheet modeling with cutting-edge deep learning methodologies to enhance ice thickness predictions, specifically targeting the Greenland ice sheet. By harmonizing the mathematical structure with an energy minimization framework tailored for neural network approximations, our method's efficacy is confirmed through both 1D and 2D numerical simulations. Utilizing the NSIDC-0092 dataset for Greenland and incorporating bedrock topography for model pre-training, we register notable advances in prediction accuracy. Our research underscores the potent combination of traditional mathematical models and advanced computational techniques in delivering precise ice thickness estimations.

From Obstacle Problems to Neural Insights: Feed Forward Neural Network Modeling of Ice Thickness

TL;DR

This work addresses modeling ice thickness under an obstacle problem formulation by marrying a variational-inequality framework with deep neural networks. The authors formulate an energy-minimization problem, introduce a composite loss that enforces PDE residuals, obstacle constraints, and boundary conditions, and approximate the solution with fully connected networks trained via Adam. They validate the approach through 1D and 2D MMS tests and apply it to Greenland data from NSIDC-0092, employing bedrock pretraining to stabilize learning; results show accurate, data-consistent thickness estimates and robust performance across cases. The study highlights the potential of combining classical mathematical modeling with modern neural methods for reliable ice-thickness estimation and broader geophysical applications.

Abstract

In this study, we integrate the established obstacle problem formulation from ice sheet modeling with cutting-edge deep learning methodologies to enhance ice thickness predictions, specifically targeting the Greenland ice sheet. By harmonizing the mathematical structure with an energy minimization framework tailored for neural network approximations, our method's efficacy is confirmed through both 1D and 2D numerical simulations. Utilizing the NSIDC-0092 dataset for Greenland and incorporating bedrock topography for model pre-training, we register notable advances in prediction accuracy. Our research underscores the potent combination of traditional mathematical models and advanced computational techniques in delivering precise ice thickness estimations.
Paper Structure (14 sections, 35 equations, 16 figures, 1 table)

This paper contains 14 sections, 35 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical Formulation of the Model
  3. Energy Minimization Formulation
  4. Solution Approximation Using Deep Neural Networks
  5. Numerical Illustrations: 1D and 2D Examples
  6. 1D Example: Case with p=2
  7. Analysis of Relative Error in the Context of Sample Variability
  8. 2D Problem: General Case with Any p >= 2
  9. Application to Greenland Data
  10. Neural Network Initialization and Training
  11. Analysis of Training Losses
  12. Loss Analysis for Different p Values
  13. Summary
  14. Acknowledgements

Figures (16)

  • Figure 1: Cross-sectional view of an ice sheet with the respective notation, based on Jouvet et al. (2012)
  • Figure 2: Comparison of the 1D solution, obstacle function (denoted b), and the exact solution.
  • Figure 3: Evolution of different losses during training, plotted against the number of iterations on a logarithmic scale (with base 10). (a) Total Loss, represented as "Log Loss Value" on the y-axis and computed from Equation \ref{['total_loss']}, symbolizes the amalgamation of all individual loss terms. (b) Loss 1, denoted as "Log Loss Value" on the y-axis from Equation \ref{['loss_1']}, illustrates the residual loss term. (c) Loss 2, weighted by coefficient $\alpha$ and articulated as "Log Loss Value" on the y-axis in Equation \ref{['loss_2']}, encapsulates the obstacle loss term. (d) Loss 3, accentuated by coefficient $\beta$ and reflected as "Log Loss Value" on the y-axis from Equation \ref{['loss_3']}, highlights the boundary loss inherent in the problem.
  • Figure 4: Evolution of the $L^1$ error between the approximated solution $u^*$ and the exact solution $u_{\text{exact}}$ over training iterations. The plot showcases the difference in magnitudes as training progresses, shedding light on the convergence and accuracy of the neural network's predictions.
  • Figure 5: Graphical representation of relative error against varying sample sizes. The blue dashed line represents the regression line with a slope of -0.83, which lies between the theoretical slopes of -1/2 (red) and -1 (gray).
  • ...and 11 more figures