# Solving the Dirichlet problem for the Monge-Amp`ere equation using neural networks

arXiv: https://arxiv.org/abs/2110.03310 · HTML: https://mlsanity.com/paper/2110.03310/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 60 references that are in the map.
- First posted on arXiv 7 Oct 2021.

## Builds on (most cited first, 6 of 6)

- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't](https://mlsanity.com/paper/2009.10713/) (2020)
- [On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics](https://mlsanity.com/paper/2007.15623/) (2020)
- [Machine Learning and Computational Mathematics](https://mlsanity.com/paper/2009.14596/) (2020)
- [Calibrating Multivariate L'evy Processes with Neural Networks](https://mlsanity.com/paper/1812.08883/) (2018)

## Built on by (most cited first, 3 of 3)

- [A neural network approach for solving the Monge-Amp`ere equation with transport boundary condition](https://mlsanity.com/paper/2410.19496/) (2024)
- [Convex Physics Informed Neural Networks for the Monge-Amp`ere Optimal Transport Problem](https://mlsanity.com/paper/2501.10162/) (2025)
- [Port-Hamiltonian Neural Networks for Systems with Multiple Asymptotically Stable Equilibria](https://mlsanity.com/paper/2610.01356/) (2026)

## Abstract (arXiv)

The Monge-Amp`ere equation is a fully nonlinear partial differential equation (PDE) of fundamental importance in analysis, geometry and in the applied sciences. In this paper we solve the Dirichlet problem associated with the Monge-Amp`ere equation using neural networks and we show that an ansatz using deep input convex neural networks can be used to find the unique convex solution. As part of our analysis we study the effect of singularities, discontinuities and noise in the source function, we consider nontrivial domains, and we investigate how the method performs in higher dimensions. We investigate the convergence numerically and present error estimates based on a stability result. We also compare this method to an alternative approach in which standard feed-forward networks are used together with a loss function which penalizes lack of convexity.