# A brief review of the Deep BSDE method for solving high-dimensional partial differential equations

arXiv: https://arxiv.org/abs/2505.17032 · HTML: https://mlsanity.com/paper/2505.17032/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 81 references that are in the map.
- First posted on arXiv 7 May 2025.

## Builds on (most cited first, 30 of 46)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space](https://mlsanity.com/paper/2005.05409/) (2020)
- [Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning](https://mlsanity.com/paper/2008.13333/) (2020)
- [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)
- [Learning in Mean Field Games: A Survey](https://mlsanity.com/paper/2205.12944/) (2022)
- [Deep Learning Approximation for Stochastic Control Problems](https://mlsanity.com/paper/1611.07422/) (2016)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (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)
- [Adaptive Deep Learning for High-Dimensional Hamilton-Jacobi-Bellman Equations](https://mlsanity.com/paper/1907.05317/) (2019)
- [Stochastic Optimal Control Matching](https://mlsanity.com/paper/2312.02027/) (2023)
- [Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach](https://mlsanity.com/paper/2002.02600/) (2020)
- [Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: II -- The Finite Horizon Case](https://mlsanity.com/paper/1908.01613/) (2019)
- [Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality](https://mlsanity.com/paper/2106.08900/) (2021)
- [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)
- [Actor-Critic Method for High Dimensional Static Hamilton--Jacobi--Bellman Partial Differential Equations based on Neural Networks](https://mlsanity.com/paper/2102.11379/) (2021)
- [Neural networks-based algorithms for stochastic control and PDEs in finance](https://mlsanity.com/paper/2101.08068/) (2021)
- [Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions](https://mlsanity.com/paper/2007.05384/) (2020)
- [On the Representation of Solutions to Elliptic PDEs in Barron Spaces](https://mlsanity.com/paper/2106.07539/) (2021)
- [Neural networks-based backward scheme for fully nonlinear PDEs](https://mlsanity.com/paper/1908.00412/) (2019)
- [Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks](https://mlsanity.com/paper/2312.14499/) (2023)

## Built on by (most cited first, 5 of 5)

- [Adjoint Matching through the Lens of the Stochastic Maximum Principle in Optimal Control](https://mlsanity.com/paper/2604.08580/) (2026)
- [Finite-Agent Stochastic Differential Games on Large Graphs: II. Graph-Based Architectures](https://mlsanity.com/paper/2509.12484/) (2025)
- [Stochastic Control Methods for Optimization](https://mlsanity.com/paper/2601.01248/) (2026)
- [High-dimensional Bayesian filtering through deep density approximation](https://mlsanity.com/paper/2511.07261/) (2025)
- [Unbiased and Second-Order-Free Training for High-Dimensional PDEs](https://mlsanity.com/paper/2605.14643/) (2026)

## Abstract (arXiv)

High-dimensional partial differential equations (PDEs) pose significant challenges for numerical computation due to the curse of dimensionality, which limits the applicability of traditional mesh-based methods. Since 2017, the Deep BSDE method has introduced deep learning techniques that enable the effective solution of nonlinear PDEs in very high dimensions. This innovation has sparked considerable interest in using neural networks for high-dimensional PDEs, making it an active area of research. In this short review, we briefly sketch the Deep BSDE method, its subsequent developments, and future directions for the field.