# A neural network approach for solving the Monge-Amp`ere equation with transport boundary condition

arXiv: https://arxiv.org/abs/2410.19496 · HTML: https://mlsanity.com/paper/2410.19496/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 33 references that are in the map.
- First posted on arXiv 25 Oct 2024.

## Builds on (most cited first, 10 of 10)

- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [Learning to Optimize](https://mlsanity.com/paper/1606.01885/) (2016)
- [A Survey on Evolutionary Neural Architecture Search](https://mlsanity.com/paper/2008.10937/) (2020)
- [Beyond Periodicity: Towards a Unifying Framework for Activations in Coordinate-MLPs](https://mlsanity.com/paper/2111.15135/) (2021)
- [Where are we now? A large benchmark study of recent symbolic regression methods](https://mlsanity.com/paper/1804.09331/) (2018)
- [A Linearly-Convergent Stochastic L-BFGS Algorithm](https://mlsanity.com/paper/1508.02087/) (2015)
- [Solving the Dirichlet problem for the Monge-Amp`ere equation using neural networks](https://mlsanity.com/paper/2110.03310/) (2021)
- [Learning to Optimize Quasi-Newton Methods](https://mlsanity.com/paper/2210.06171/) (2022)

## Built on by (most cited first, 1 of 1)

- [Direct Optimization of a 3D Finite-Source Reflector via Neural-Network Parameterization](https://mlsanity.com/paper/2609.00899/) (2026)

## Abstract (arXiv)

This paper introduces a novel neural network-based approach to solving the Monge-Amp`ere 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.