# Learning High-Dimensional McKean-Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependence arXiv: https://arxiv.org/abs/2204.11924 ยท HTML: https://mlsanity.com/paper/2204.11924/ ## Key facts - Cited by 13 papers in the map (data through 14 Oct 2026). - Builds on 21 of its 52 references that are in the map. - First posted on arXiv 25 Apr 2022. ## Builds on (most cited first, 21 of 21) - [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017) - [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017) - [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019) - [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017) - [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020) - [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019) - [Convergence of the Deep BSDE Method for Coupled FBSDEs](https://mlsanity.com/paper/1811.01165/) (2018) - [Mean Field Games Flock! The Reinforcement Learning Way](https://mlsanity.com/paper/2105.07933/) (2021) - [Mean Field Games and Applications: Numerical Aspects](https://mlsanity.com/paper/2003.04444/) (2020) - [Numerical resolution of McKean-Vlasov FBSDEs using neural networks](https://mlsanity.com/paper/1909.12678/) (2019) - [Deep Fictitious Play for Finding Markovian Nash Equilibrium in Multi-Agent Games](https://mlsanity.com/paper/1912.01809/) (2019) - [Deep Fictitious Play for Stochastic Differential Games](https://mlsanity.com/paper/1903.09376/) (2019) - [Signatured Deep Fictitious Play for Mean Field Games with Common Noise](https://mlsanity.com/paper/2106.03272/) (2021) - [DeepHAM: A Global Solution Method for Heterogeneous Agent Models with Aggregate Shocks](https://mlsanity.com/paper/2112.14377/) (2021) - [Convergence of Deep Fictitious Play for Stochastic Differential Games](https://mlsanity.com/paper/2008.05519/) (2020) - [A fast iterative PDE-based algorithm for feedback controls of nonsmooth mean-field control problems](https://mlsanity.com/paper/2108.06740/) (2021) - [Provable Fictitious Play for General Mean-Field Games](https://mlsanity.com/paper/2010.04211/) (2020) - [Optimal Policies for a Pandemic: A Stochastic Game Approach and a Deep Learning Algorithm](https://mlsanity.com/paper/2012.06745/) (2020) - [A Class of Dimension-free Metrics for the Convergence of Empirical Measures](https://mlsanity.com/paper/2104.12036/) (2021) - [Convergence of some Mean Field Games systems to aggregation and flocking models](https://mlsanity.com/paper/2004.04403/) (2020) - [A Cucker-Smale inspired deterministic mean field game with velocity interactions](https://mlsanity.com/paper/2009.07519/) (2020) ## Built on by (most cited first, 13 of 13) - [Actor-Critic learning for mean-field control in continuous time](https://mlsanity.com/paper/2303.06993/) (2023) - [Modeling Cell Dynamics and Interactions with Unbalanced Mean Field Schr"odinger Bridge](https://mlsanity.com/paper/2505.11197/) (2025) - [Actor critic learning algorithms for mean-field control with moment neural networks](https://mlsanity.com/paper/2309.04317/) (2023) - [A brief review of the Deep BSDE method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2505.17032/) (2025) - [Neural McKean-Vlasov Processes: Distributional Dependence in Diffusion Processes](https://mlsanity.com/paper/2404.09402/) (2024) - [Operator Learning for Families of Finite-State Mean-Field Games](https://mlsanity.com/paper/2602.13169/) (2026) - [Deep Learning for Mean Field Optimal Transport](https://mlsanity.com/paper/2302.14739/) (2023) - [Nonlocal Mean Field Schr"odinger Bridge with Learned Interactions](https://mlsanity.com/paper/2606.04265/) (2026) - [Deep Generative Modeling with Backward Stochastic Differential Equations](https://mlsanity.com/paper/2304.04049/) (2023) - [A Deep Learning Method for Optimal Investment Under Relative Performance Criteria Among Heterogeneous Agents](https://mlsanity.com/paper/2402.07365/) (2024) - [Mean-field Chaos Diffusion Models](https://mlsanity.com/paper/2406.05396/) (2024) - [Trading Devil Final: Backdoor attack via Stock market and Bayesian Optimization](https://mlsanity.com/paper/2407.14573/) (2024) - [Deep Learning and Elicitability for McKean-Vlasov FBSDEs With Common Noise](https://mlsanity.com/paper/2512.14967/) (2025) ## Abstract (arXiv) One of the core problems in mean-field control and mean-field games is to solve the corresponding McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs). Most existing methods are tailored to special cases in which the mean-field interaction only depends on expectation or other moments and thus inadequate to solve problems when the mean-field interaction has full distribution dependence. In this paper, we propose a novel deep learning method for computing MV-FBSDEs with a general form of mean-field interactions. Specifically, built on fictitious play, we recast the problem into repeatedly solving standard FBSDEs with explicit coefficient functions. These coefficient functions are used to approximate the MV-FBSDEs' model coefficients with full distribution dependence, and are updated by solving another supervising learning problem using training data simulated from the last iteration's FBSDE solutions. We use deep neural networks to solve standard BSDEs and approximate coefficient functions in order to solve high-dimensional MV-FBSDEs. Under proper assumptions on the learned functions, we prove that the convergence of the proposed method is free of the curse of dimensionality (CoD) by using a class of integral probability metrics previously developed in [Han, Hu and Long, arXiv:2104.12036]. The proved theorem shows the advantage of the method in high dimensions. We present the numerical performance in high-dimensional MV-FBSDE problems, including a mean-field game example of the well-known Cucker-Smale model whose cost depends on the full distribution of the forward process.