A Martingale Neural Network for Solving High-Dimensional Hamilton-Jacobi-Bellman Equations without Explicit Minimization of the Hamiltonian

Core Concepts

The proposed SOC-MartNet is an efficient numerical method for solving high-dimensional Hamilton-Jacobi-Bellman (HJB) equations and stochastic optimal control problems without the need for explicit minimization of the Hamiltonian.

Abstract

The paper proposes a novel numerical method called SOC-MartNet for solving high-dimensional Hamilton-Jacobi-Bellman (HJB) equations and stochastic optimal control problems. The key ideas are:

Reformulate the HJB equation into a stochastic neural network learning process, where a control network and a value network are trained such that the associated Hamiltonian process is minimized and the cost process becomes a martingale.

Employ an adversarial network to enforce the martingale property of the cost process, based on the projection property of conditional expectations. This avoids the need for explicit minimization of the Hamiltonian over the control space at each time-space point.

The training algorithm enjoys parallel efficiency, as it is free of time-direction iterations during gradient computation, in contrast to existing deep learning methods for PDEs.

The proposed SOC-MartNet is shown to be effective and efficient for solving HJB-type equations and stochastic optimal control problems with dimensions up to 500.

Stats

The paper does not provide specific numerical data or statistics. The focus is on the methodology and algorithm development.

Quotes

The paper does not contain any direct quotes that are important to highlight.

Key Insights Distilled From

SOC-MartNet: A Martingale Neural Network for the Hamilton-Jacobi-Bellman Equation without Explicit inf H in Stochastic Optimal Controls

by Wei Cai,Shui... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.03169.pdf

Deeper Inquiries

How can the SOC-MartNet be extended to handle HJB equations with non-smooth or non-convex Hamiltonians

To extend the SOC-MartNet to handle HJB equations with non-smooth or non-convex Hamiltonians, we can introduce regularization terms in the loss function that penalize non-smoothness or non-convexity. By incorporating penalty terms that encourage smoothness or convexity in the Hamiltonian function, the neural network can learn to approximate these types of functions more effectively. Additionally, the adversarial network can be trained to generate test functions that specifically target the non-smooth or non-convex regions of the Hamiltonian, aiding in the optimization process. By adapting the loss function and training strategy to account for these characteristics, the SOC-MartNet can successfully handle HJB equations with non-smooth or non-convex Hamiltonians.

What are the theoretical guarantees on the convergence and optimality of the solutions obtained by the SOC-MartNet

Theoretical guarantees on the convergence and optimality of the solutions obtained by the SOC-MartNet can be established through the framework of stochastic optimization and martingale theory. Convergence guarantees can be derived by analyzing the optimization process of the neural networks and adversarial network within the SOC-MartNet algorithm. By ensuring that the loss function decreases over iterations and that the martingale condition is satisfied, convergence to a local minimum can be theoretically proven.
Optimality guarantees can be established by considering the relationship between the learned control and value functions and the Hamiltonian in the HJB equation. If the SOC-MartNet successfully minimizes the Hamiltonian process and enforces the martingale property for the cost process, the optimality of the solutions can be inferred. Theoretical analysis of the algorithm's performance in terms of approximation accuracy, convergence speed, and solution quality can provide insights into the optimality of the solutions obtained by the SOC-MartNet.

Can the SOC-MartNet framework be applied to other types of high-dimensional partial differential equations beyond HJB equations

The SOC-MartNet framework can be applied to a wide range of high-dimensional partial differential equations beyond HJB equations. By adapting the loss function, network architecture, and training strategy, the SOC-MartNet can be tailored to solve various types of PDEs, including parabolic, elliptic, and hyperbolic equations. The key lies in formulating the PDE problem in a way that aligns with the martingale learning process and adversarial training approach of the SOC-MartNet.
For parabolic equations, similar to the HJB equations, the SOC-MartNet can be used to learn the value function and control strategies that minimize the Hamiltonian and enforce the martingale property for the cost process. By adjusting the network structures and loss functions to suit the specific characteristics of different types of PDEs, the SOC-MartNet can effectively tackle a broad spectrum of high-dimensional PDE problems.