# A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations

arXiv: https://arxiv.org/abs/2404.08456 · HTML: https://mlsanity.com/paper/2404.08456/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 69 references that are in the map.
- First posted on arXiv 12 Apr 2024.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [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)
- [Convergence of the Deep BSDE Method for Coupled FBSDEs](https://mlsanity.com/paper/1811.01165/) (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)
- [Deep Neural Network Framework Based on Backward Stochastic Differential Equations for Pricing and Hedging American Options in High Dimensions](https://mlsanity.com/paper/1909.11532/) (2019)
- [Learning Deep Stochastic Optimal Control Policies using Forward-Backward SDEs](https://mlsanity.com/paper/1902.03986/) (2019)
- [Convergence of a robust deep FBSDE method for stochastic control](https://mlsanity.com/paper/2201.06854/) (2022)
- [Differential Machine Learning](https://mlsanity.com/paper/2005.02347/) (2020)
- [A deep solver for BSDEs with jumps](https://mlsanity.com/paper/2211.04349/) (2022)
- [Convergence of the Deep BSDE method for FBSDEs with non-Lipschitz coefficients](https://mlsanity.com/paper/2101.01869/) (2021)
- [Deep learning algorithms for solving high dimensional nonlinear backward stochastic differential equations](https://mlsanity.com/paper/2010.01319/) (2020)
- [A deep neural network algorithm for semilinear elliptic PDEs with applications in insurance mathematics](https://mlsanity.com/paper/2010.15757/) (2020)
- [Multistep schemes for solving backward stochastic differential equations on GPU](https://mlsanity.com/paper/1909.13560/) (2019)
- [A deep learning method for solving stochastic optimal control problems driven by fully-coupled FBSDEs](https://mlsanity.com/paper/2204.05796/) (2022)
- [Generalized convergence of the deep BSDE method: a step towards fully-coupled FBSDEs and applications in stochastic control](https://mlsanity.com/paper/2403.18552/) (2024)
- [A novel control method for solving high-dimensional Hamiltonian systems through deep neural networks](https://mlsanity.com/paper/2111.02636/) (2021)
- [Uncertainty quantification for deep learning-based schemes for solving high-dimensional backward stochastic differential equations](https://mlsanity.com/paper/2310.03393/) (2023)

## Built on by (most cited first, 4 of 4)

- [A deep solver for backward stochastic Volterra integral equations](https://mlsanity.com/paper/2505.18297/) (2025)
- [Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions](https://mlsanity.com/paper/2508.07559/) (2025)
- [Nonlinear filtering based on density approximation and deep BSDE prediction](https://mlsanity.com/paper/2508.10630/) (2025)
- [Deep Neural networks for solving high-dimensional parabolic partial differential equations](https://mlsanity.com/paper/2601.13256/) (2026)

## Abstract (arXiv)

In this work, we propose a novel backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations (BSDEs), where the deep neural network (DNN) models are trained not only on the inputs and labels but also the differentials of the corresponding labels. This is motivated by the fact that differential deep learning can provide an efficient approximation of the labels and their derivatives with respect to inputs. The BSDEs are reformulated as differential deep learning problems by using Malliavin calculus. The Malliavin derivatives of solution to a BSDE satisfy themselves another BSDE, resulting thus in a system of BSDEs. Such formulation requires the estimation of the solution, its gradient, and the Hessian matrix, represented by the triple of processes left(Y, Z, Gammaright). All the integrals within this system are discretized by using the Euler-Maruyama method. Subsequently, DNNs are employed to approximate the triple of these unknown processes. The DNN parameters are backwardly optimized at each time step by minimizing a differential learning type loss function, which is defined as a weighted sum of the dynamics of the discretized BSDE system, with the first term providing the dynamics of the process Y and the other the process Z. An error analysis is carried out to show the convergence of the proposed algorithm. Various numerical experiments up to 50 dimensions are provided to demonstrate the high efficiency. Both theoretically and numerically, it is demonstrated that our proposed scheme is more efficient compared to other contemporary deep learning-based methodologies, especially in the computation of the process Gamma.