# Variational conditional normalizing flows for computing second-order mean field control problems

arXiv: https://arxiv.org/abs/2503.19580 · HTML: https://mlsanity.com/paper/2503.19580/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Mar 2025.

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

- [Neural Hamilton--Jacobi Characteristic Flows for Optimal Transport](https://mlsanity.com/paper/2510.01153/) (2025)
- [Adaptive Probability Flow Residual Minimization for High-Dimensional Fokker-Planck Equations](https://mlsanity.com/paper/2512.19196/) (2025)
- [Self-supervised In-context Operator Learning for Stochastic Mean-Field Control](https://mlsanity.com/paper/2608.18282/) (2026)

## Abstract (arXiv)

Mean field control (MFC) problems have vast applications in artificial intelligence, engineering, and economics, while solving MFC problems accurately and efficiently in high-dimensional spaces remains challenging. This work introduces variational conditional normalizing flow (VCNF), a neural network-based variational algorithm for solving general MFC problems based on flow maps. Formulating MFC problems as optimal control of Fokker-Planck (FP) equations with suitable constraints and cost functionals, we use VCNF to model the Lagrangian formulation of the MFC problems. In particular, VCNF builds upon conditional normalizing flows and neural spline flows, allowing efficient calculations of the inverse push-forward maps and score functions in MFC problems. We demonstrate the effectiveness of VCNF through extensive numerical examples, including optimal transport, regularized Wasserstein proximal operators, and flow matching problems for FP equations.