# Algorithm for Hamilton-Jacobi equations in density space via a generalized Hopf formula

arXiv: https://arxiv.org/abs/1805.01636 · HTML: https://mlsanity.com/paper/1805.01636/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 May 2018.

## Built on by (most cited first, 2 of 2)

- [Deep Learning for Mean Field Games with non-separable Hamiltonians](https://mlsanity.com/paper/2301.02877/) (2023)
- [Mean Field Game GAN](https://mlsanity.com/paper/2103.07855/) (2021)

## Abstract (arXiv)

We design fast numerical methods for Hamilton-Jacobi equations in density space (HJD), which arises in optimal transport and mean field games. We overcome the curse-of-infinite-dimensionality nature of HJD by proposing a generalized Hopf formula in density space. The formula transfers optimal control problems in density space, which are constrained minimizations supported on both spatial and time variables, to optimization problems over only spatial variables. This transformation allows us to compute HJD efficiently via multi-level approaches and coordinate descent methods.