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A neural network approach for solving the Monge-Ampère equation with transport boundary condition

Roel Hacking, Lisa Kusch, Koondanibha Mitra, Martijn Anthonissen, Wilbert IJzerman

TL;DR

A novel neural network-based approach to solving the Monge-Amp\`ere equation with the transport boundary condition, specifically targeted towards optical design applications by leveraging multilayer perceptron networks to learn approximate solutions by minimizing a loss function that encompasses the equation's residual, boundary conditions, and convexity constraints.

Abstract

This paper introduces a novel neural network-based approach to solving the Monge-Ampère equation with the transport boundary condition, specifically targeted towards optical design applications. We leverage multilayer perceptron networks to learn approximate solutions by minimizing a loss function that encompasses the equation's residual, boundary conditions, and convexity constraints. Our main results demonstrate the efficacy of this method, optimized using L-BFGS, through a series of test cases encompassing symmetric and asymmetric circle-to-circle, square-to-circle, and circle-to-flower reflector mapping problems. Comparative analysis with a conventional least-squares finite-difference solver reveals the competitive, and often superior, performance of our neural network approach on the test cases examined here. A comprehensive hyperparameter study further illuminates the impact of factors such as sampling density, network architecture, and optimization algorithm. While promising, further investigation is needed to verify the method's robustness for more complicated problems and to ensure consistent convergence. Nonetheless, the simplicity and adaptability of this neural network-based approach position it as a compelling alternative to specialized partial differential equation solvers.

A neural network approach for solving the Monge-Ampère equation with transport boundary condition

TL;DR

A novel neural network-based approach to solving the Monge-Amp\`ere equation with the transport boundary condition, specifically targeted towards optical design applications by leveraging multilayer perceptron networks to learn approximate solutions by minimizing a loss function that encompasses the equation's residual, boundary conditions, and convexity constraints.

Abstract

This paper introduces a novel neural network-based approach to solving the Monge-Ampère equation with the transport boundary condition, specifically targeted towards optical design applications. We leverage multilayer perceptron networks to learn approximate solutions by minimizing a loss function that encompasses the equation's residual, boundary conditions, and convexity constraints. Our main results demonstrate the efficacy of this method, optimized using L-BFGS, through a series of test cases encompassing symmetric and asymmetric circle-to-circle, square-to-circle, and circle-to-flower reflector mapping problems. Comparative analysis with a conventional least-squares finite-difference solver reveals the competitive, and often superior, performance of our neural network approach on the test cases examined here. A comprehensive hyperparameter study further illuminates the impact of factors such as sampling density, network architecture, and optimization algorithm. While promising, further investigation is needed to verify the method's robustness for more complicated problems and to ensure consistent convergence. Nonetheless, the simplicity and adaptability of this neural network-based approach position it as a compelling alternative to specialized partial differential equation solvers.

Paper Structure

This paper contains 17 sections, 25 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Parallel-to-far-field reflector problem
  4. Artificial neural networks
  5. Least-squares solver
  6. Methods
  7. Modified loss function
  8. Optimization
  9. Numerical examples
  10. Problem A: Symmetrical circle-to-circle
  11. Problem B: Asymmetrical circle-to-circle 1
  12. Problem C: Asymmetrical circle-to-circle 2
  13. Problem D: Square-to-circle
  14. Problem E: Circle-to-flower
  15. Hyperparameter study
  16. ...and 2 more sections

Figures (17)

  • Figure 1: Visualization of the parallel-to-far-field reflector problem
  • Figure 2: Stereographic projection of the reflected ray.
  • Figure 3: Example of a fully-connected feedforward neural network. Adapted from tikzNeuralNetworks.
  • Figure 4: The $2500$ interior points used for optimization for the unit circle and square. The unit circle is used for Problems \ref{['A']}, \ref{['B']}, \ref{['C']}, and \ref{['E']}. The unit square is used only for Problem \ref{['D']}.
  • Figure 5: [ Problem \ref{['A']}] In order, $u(x_1, x_2)$, $\det(D^2 u(x_1, x_2))$, $f(x_1, x_2)$, and $g(y_1, y_2)$.
  • ...and 12 more figures