# Convergence of a robust deep FBSDE method for stochastic control

arXiv: https://arxiv.org/abs/2201.06854 · HTML: https://mlsanity.com/paper/2201.06854/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 68 references that are in the map.
- First posted on arXiv 18 Jan 2022.

## Builds on (most cited first, 19 of 19)

- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Activation Functions: Comparison of trends in Practice and Research for Deep Learning](https://mlsanity.com/paper/1811.03378/) (2018)
- [Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations](https://mlsanity.com/paper/1709.05963/) (2017)
- [An overview on deep learning-based approximation methods for partial differential equations](https://mlsanity.com/paper/2012.12348/) (2020)
- [Deep Learning Approximation for Stochastic Control Problems](https://mlsanity.com/paper/1611.07422/) (2016)
- [Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations](https://mlsanity.com/paper/1804.07010/) (2018)
- [Deep splitting method for parabolic PDEs](https://mlsanity.com/paper/1907.03452/) (2019)
- [Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations](https://mlsanity.com/paper/1809.03062/) (2018)
- [Convergence of the Deep BSDE Method for Coupled FBSDEs](https://mlsanity.com/paper/1811.01165/) (2018)
- [Solving the Kolmogorov PDE by means of deep learning](https://mlsanity.com/paper/1806.00421/) (2018)
- [A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients](https://mlsanity.com/paper/1809.07321/) (2018)
- [Machine Learning for semi linear PDEs](https://mlsanity.com/paper/1809.07609/) (2018)
- [Approximation error analysis of some deep backward schemes for nonlinear PDEs](https://mlsanity.com/paper/2006.01496/) (2020)
- [Three algorithms for solving high-dimensional fully-coupled FBSDEs through deep learning](https://mlsanity.com/paper/1907.05327/) (2019)
- [Learning Deep Stochastic Optimal Control Policies using Forward-Backward SDEs](https://mlsanity.com/paper/1902.03986/) (2019)
- [Deep backward schemes for high-dimensional nonlinear PDEs](https://mlsanity.com/paper/1902.01599/) (2019)
- [Convergence of the Deep BSDE method for FBSDEs with non-Lipschitz coefficients](https://mlsanity.com/paper/2101.01869/) (2021)
- [Deep BSDE-ML Learning and Its Application to Model-Free Optimal Control](https://mlsanity.com/paper/2201.01318/) (2022)
- [Learning Locomotion Controllers for Walking Using Deep FBSDE](https://mlsanity.com/paper/2107.07931/) (2021)

## Built on by (most cited first, 12 of 12)

- [Mean-field neural networks-based algorithms for McKean-Vlasov control problems *](https://mlsanity.com/paper/2212.11518/) (2022)
- [Convergence of the Backward Deep BSDE Method with Applications to Optimal Stopping Problems](https://mlsanity.com/paper/2210.04118/) (2022)
- [A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations](https://mlsanity.com/paper/2404.08456/) (2024)
- [FBSJNN: A Theoretically Interpretable and Efficiently Deep Learning method for Solving Partial Integro-Differential Equations](https://mlsanity.com/paper/2412.11010/) (2024)
- [Integration Matters for Learning PDEs with Backward SDEs](https://mlsanity.com/paper/2505.01078/) (2025)
- [A convergent scheme for the Bayesian filtering problem based on the Fokker--Planck equation and deep splitting](https://mlsanity.com/paper/2409.14585/) (2024)
- [A deep solver for backward stochastic Volterra integral equations](https://mlsanity.com/paper/2505.18297/) (2025)
- [A forward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations](https://mlsanity.com/paper/2408.05620/) (2024)
- [An Optimization-Based Framework for Solving Forward-Backward Stochastic Differential Equations: Convergence Analysis and Error Bounds](https://mlsanity.com/paper/2507.15234/) (2025)
- [Error analysis for the deep Kolmogorov method](https://mlsanity.com/paper/2508.17167/) (2025)
- [A Schr"odinger Eigenfunction Method for Long-Horizon Stochastic Optimal Control](https://mlsanity.com/paper/2603.23173/) (2026)
- [Deep Learning for Reflected BSDEs: Regularization and Error Analysis](https://mlsanity.com/paper/2609.05434/) (2026)

## Abstract (arXiv)

In this paper, we propose a deep learning based numerical scheme for strongly coupled FBSDEs, stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs, fails for a simple linear-quadratic control problem, and motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.