# Learning Optimal Control via Forward and Backward Stochastic Differential Equations

arXiv: https://arxiv.org/abs/1509.02195 · HTML: https://mlsanity.com/paper/1509.02195/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Sep 2015.

## Built on by (most cited first, 11 of 11)

- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [Deep Generalized Schr"odinger Bridge](https://mlsanity.com/paper/2209.09893/) (2022)
- [Deep Momentum Multi-Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2303.01751/) (2023)
- [Decentralized Safe Multi-agent Stochastic Optimal Control using Deep FBSDEs and ADMM](https://mlsanity.com/paper/2202.10658/) (2022)
- [Base Models for Parabolic Partial Differential Equations](https://mlsanity.com/paper/2407.12234/) (2024)
- [Deep Graphic FBSDEs for Opinion Dynamics Stochastic Control](https://mlsanity.com/paper/2204.02506/) (2022)
- [Solving a class of stochastic optimal control problems by physics-informed neural networks](https://mlsanity.com/paper/2402.15592/) (2024)
- [Deep L1 Stochastic Optimal Control Policies for Planetary Soft-landing](https://mlsanity.com/paper/2109.00183/) (2021)
- [Learning Control Policies of Hodgkin-Huxley Neuronal Dynamics](https://mlsanity.com/paper/2311.07563/) (2023)
- [Optimal Control of Agent-Based Dynamics under Deep Galerkin Feedback Laws](https://mlsanity.com/paper/2406.09141/) (2024)
- [Deep Generalized Schr"odinger Bridges: From Image Generation to Solving Mean-Field Games](https://mlsanity.com/paper/2412.20279/) (2024)

## Abstract (arXiv)

In this paper we present a novel sampling-based numerical scheme designed to solve a certain class of stochastic optimal control problems, utilizing forward and backward stochastic differential equations (FBSDEs). By means of a nonlinear version of the Feynman-Kac lemma, we obtain a probabilistic representation of the solution to the nonlinear Hamilton-Jacobi-Bellman equation, expressed in the form of a decoupled system of FBSDEs. This system of FBSDEs can then be simulated by employing linear regression techniques. To enhance the efficiency of the proposed scheme when treating more complex nonlinear systems, we then derive an iterative modification based on Girsanov's theorem on the change of measure, which features importance sampling. The modified scheme is capable of learning the optimal control without requiring an initial guess. We present simulations that validate the algorithm and demonstrate its efficiency in treating nonlinear dynamics.