# Neural optimal controller for stochastic systems via pathwise HJB operator

**Authors:** Zhe Jiao, Xiaoyan Luo, Xinlei Yi  
**arXiv:** [2402.15592](https://arxiv.org/abs/2402.15592)  
**Date:** 2024-02-23  
**Summary page:** https://www.summarizepaper.com/en/paper/2402.15592v1/

## Key points

- The paper focuses on developing deep learning-based algorithms for high-dimensional stochastic control problems.
- Utilizes physics-informed learning and dynamic programming techniques.
- Introduces a pathwise operator associated with the Hamilton-Jacobi-Bellman (HJB) equation, offering a new perspective on tackling stochastic control problems.
- Presents two numerical methods for solving the physics-informed learning problem based on the explicit representation of optimal control.
- Provides an error analysis to assess the impact of truncation, approximation, and optimization errors on method accuracy.
- Showcases numerical results across various applications to demonstrate algorithm effectiveness and performance.
- Advances deep learning approaches in addressing complex stochastic control problems by incorporating principles of physics-informed learning and dynamic programming concepts.

## Summary

The paper titled "Neural Optimal Controller for Stochastic Systems via Pathwise HJB Operator" by Zhe Jiao, Xiaoyan Luo, and Xinlei Yi focuses on the development of deep learning-based algorithms for high-dimensional stochastic control problems. The approach taken in this work involves utilizing physics-informed learning and dynamic programming techniques. Unlike traditional deep learning methods that rely on a probabilistic representation of the solution to the Hamilton--Jacobi--Bellman (HJB) equation, the authors introduce a pathwise operator associated with the HJB equation. This allows them to formulate a problem of physics-informed learning and offers a novel perspective on tackling stochastic control problems. The paper presents two numerical methods designed to solve the physics-informed learning problem based on whether the optimal control has an explicit representation. An error analysis is provided to assess how truncation, approximation, and optimization errors impact the accuracy of these methods. The authors also showcase numerical results across various applications to demonstrate the effectiveness and performance of the proposed algorithms. Overall, this research contributes to advancing deep learning approaches in addressing complex stochastic control problems by incorporating principles of physics-informed learning and dynamic programming concepts. The innovative use of pathwise operators adds a new dimension to optimizing neural controllers for stochastic systems, offering promising avenues for further exploration in this field.