# Deep BSDE-ML Learning and Its Application to Model-Free Optimal Control

arXiv: https://arxiv.org/abs/2201.01318 · HTML: https://mlsanity.com/paper/2201.01318/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jan 2022.

## Built on by (most cited first, 6 of 6)

- [Convergence of a robust deep FBSDE method for stochastic control](https://mlsanity.com/paper/2201.06854/) (2022)
- [Deep Signature Algorithm for Multi-dimensional Path-Dependent Options](https://mlsanity.com/paper/2211.11691/) (2022)
- [Integration Matters for Learning PDEs with Backward SDEs](https://mlsanity.com/paper/2505.01078/) (2025)
- [How important are specialized transforms in Neural Operators?](https://mlsanity.com/paper/2308.09293/) (2023)
- [Deep Generative Modeling with Backward Stochastic Differential Equations](https://mlsanity.com/paper/2304.04049/) (2023)
- [An Optimization-Based Framework for Solving Forward-Backward Stochastic Differential Equations: Convergence Analysis and Error Bounds](https://mlsanity.com/paper/2507.15234/) (2025)

## Abstract (arXiv)

A modified Deep BSDE (backward differential equation) learning method with measurability loss, called Deep BSDE-ML method, is introduced in this paper to solve a kind of linear decoupled forward-backward stochastic differential equations (FBSDEs), which is encountered in the policy evaluation of learning the optimal feedback policies of a class of stochastic control problems. The measurability loss is characterized via the measurability of BSDE's state at the forward initial time, which differs from that related to terminal state of the known Deep BSDE method. Though the minima of the two loss functions are shown to be equal, this measurability loss is proved to be equal to the expected mean squared error between the true diffusion term of BSDE and its approximation. This crucial observation extends the application of the Deep BSDE method -- approximating the gradients of the solution of a partial differential equation (PDE) instead of the solution itself. Simultaneously, a learning-based framework is introduced to search an optimal feedback control of a deterministic nonlinear system. Specifically, by introducing Gaussian exploration noise, we are aiming to learn a robust optimal controller under this stochastic case. This reformulation sacrifices the optimality to some extent, but as suggested in reinforcement learning (RL) exploration noise is essential to enable the model-free learning.