# Approximation error analysis of some deep backward schemes for nonlinear PDEs

arXiv: https://arxiv.org/abs/2006.01496 · HTML: https://mlsanity.com/paper/2006.01496/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jun 2020.

## Built on by (most cited first, 21 of 21)

- [NSGA-PINN: A Multi-Objective Optimization Method for Physics-Informed Neural Network Training](https://mlsanity.com/paper/2303.02219/) (2023)
- [Convergence of a robust deep FBSDE method for stochastic control](https://mlsanity.com/paper/2201.06854/) (2022)
- [Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEs](https://mlsanity.com/paper/2405.05192/) (2024)
- [Mean-field neural networks-based algorithms for McKean-Vlasov control problems *](https://mlsanity.com/paper/2212.11518/) (2022)
- [From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEs](https://mlsanity.com/paper/2307.15496/) (2023)
- [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)
- [One model to solve them all: 2BSDE families via neural operators](https://mlsanity.com/paper/2511.01125/) (2025)
- [Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in Lp-sense](https://mlsanity.com/paper/2409.20431/) (2024)
- [Error Analysis of Option Pricing via Deep PDE Solvers: Empirical Study](https://mlsanity.com/paper/2311.07231/) (2023)
- [Uncertainty quantification for deep learning-based schemes for solving high-dimensional backward stochastic differential equations](https://mlsanity.com/paper/2310.03393/) (2023)
- [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)
- [A hierarchical resource-efficient deep policy gradient method for continuous-time optimal control problems](https://mlsanity.com/paper/2502.14141/) (2025)
- [BLADE: Bayesian Langevin Active Discovery with Replica Exchange for Identification of Complex Systems](https://mlsanity.com/paper/2503.02983/) (2025)
- [A Reverse-BSDE Diffusion Sampler](https://mlsanity.com/paper/2505.06800/) (2025)
- [Error analysis for the deep Kolmogorov method](https://mlsanity.com/paper/2508.17167/) (2025)
- [Robust Exploratory Stopping under Ambiguity in Reinforcement Learning](https://mlsanity.com/paper/2510.10260/) (2025)
- [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)
- [Deep Learning for Reflected BSDEs: Regularization and Error Analysis](https://mlsanity.com/paper/2609.05434/) (2026)

## Abstract (arXiv)

Recently proposed numerical algorithms for solving high-dimensional nonlinear partial differential equations (PDEs) based on neural networks have shown their remarkable performance. We review some of them and study their convergence properties. The methods rely on probabilistic representation of PDEs by backward stochastic differential equations (BSDEs) and their iterated time discretization. Our proposed algorithm, called deep backward multistep scheme (MDBDP), is a machine learning version of the LSMDP scheme of Gobet, Turkedjiev (Math. Comp. 85, 2016). It estimates simultaneously by backward induction the solution and its gradient by neural networks through sequential minimizations of suitable quadratic loss functions that are performed by stochastic gradient descent. Our main theoretical contribution is to provide an approximation error analysis of the MDBDP scheme as well as the deep splitting (DS) scheme for semilinear PDEs designed in Beck, Becker, Cheridito, Jentzen, Neufeld (2019). We also supplement the error analysis of the DBDP scheme of Hur'e, Pham, Warin (Math. Comp. 89, 2020). This yields notably convergence rate in terms of the number of neurons for a class of deep Lipschitz continuous GroupSort neural networks when the PDE is linear in the gradient of the solution for the MDBDP scheme, and in the semilinear case for the DBDP scheme. We illustrate our results with some numerical tests that are compared with some other machine learning algorithms in the literature.