mlsanityData to 7 Oct 2026

Convergence of a robust deep FBSDE method for stochastic control

arXiv 2201.06854 ↗First posted 18 Jan 2022math.OC cs.NA math.NAcited by 11

Not on the map. Why?

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.

Key facts. Cited by 11 papers in the map (data through 7 Oct 2026). Builds on 19 of its 68 references that are in the map. First posted on arXiv 18 Jan 2022.

Builds on

19 of 68 references.

References · 19

  1. 2017Solving high-dimensional partial differential equations using deep learningcited by 549
  2. 2018Activation Functions: Comparison of trends in Practice and Research for Deep Learningcited by 130
  3. 2020An overview on deep learning-based approximation methods for partial differential equationscited by 65
  4. 2017Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equationscited by 62
  5. 2018Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equationscited by 54
  6. 2019Deep splitting method for parabolic PDEscited by 53
  7. 2016Deep Learning Approximation for Stochastic Control Problemscited by 51
  8. 2018Analysis 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 Equationscited by 44
  9. 2018Convergence of the Deep BSDE Method for Coupled FBSDEscited by 42
  10. 2018Solving the Kolmogorov PDE by means of deep learningcited by 41
  11. 2018A 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 coefficientscited by 31
  12. 2018Machine Learning for semi linear PDEscited by 28
7 more
  1. 2020Approximation error analysis of some deep backward schemes for nonlinear PDEscited by 20
  2. 2019Three algorithms for solving high-dimensional fully-coupled FBSDEs through deep learningcited by 15
  3. 2019Learning Deep Stochastic Optimal Control Policies using Forward-Backward SDEscited by 13
  4. 2019Deep backward schemes for high-dimensional nonlinear PDEscited by 7
  5. 2021Convergence of the Deep BSDE method for FBSDEs with non-Lipschitz coefficientscited by 7
  6. 2022Deep BSDE-ML Learning and Its Application to Model-Free Optimal Controlcited by 5
  7. 2021Learning Locomotion Controllers for Walking Using Deep FBSDEcited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 2; 2023: 0; 2024: 4; 2025: 3; 2026: 2.

11 papers.

  1. 2022Convergence of the Backward Deep BSDE Method with Applications to Optimal Stopping Problemscited by 9
  2. 2022Mean-field neural networks-based algorithms for McKean-Vlasov control problems *cited by 9
  3. 2024A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equationscited by 4
  4. 2024FBSJNN: A Theoretically Interpretable and Efficiently Deep Learning method for Solving Partial Integro-Differential Equationscited by 4
  5. 2025Integration Matters for Learning PDEs with Backward SDEscited by 3
  6. 2024A convergent scheme for the Bayesian filtering problem based on the Fokker--Planck equation and deep splittingcited by 2
  7. 2025A deep solver for backward stochastic Volterra integral equationscited by 1
  8. 2024A forward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equationscited by 0
  9. 2025Error analysis for the deep Kolmogorov methodcited by 0
  10. 2026A Schr"odinger Eigenfunction Method for Long-Horizon Stochastic Optimal Controlcited by 0
  11. 2026Deep Learning for Reflected BSDEs: Regularization and Error Analysiscited by 0