# Fractional-Boundary-Regularized Deep Galerkin Method for Variational Inequalities in Mixed Optimal Stopping and Control

arXiv: https://arxiv.org/abs/2505.19309 · HTML: https://mlsanity.com/paper/2505.19309/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 16 references that are in the map.
- First posted on arXiv 25 May 2025.

## Builds on (most cited first, 5 of 5)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Adaptive Deep Learning for High-Dimensional Hamilton-Jacobi-Bellman Equations](https://mlsanity.com/paper/1907.05317/) (2019)
- [Deep learning of free boundary and Stefan problems](https://mlsanity.com/paper/2006.05311/) (2020)
- [Solving high-dimensional optimal stopping problems using deep learning](https://mlsanity.com/paper/1908.01602/) (2019)
- [A neural network approach to learning solutions of a class of elliptic variational inequalities](https://mlsanity.com/paper/2411.18565/) (2024)

## Abstract (arXiv)

Mixed optimal stopping and stochastic control problems define variational inequalities with non-linear Hamilton-Jacobi-Bellman (HJB) operators, whose numerical solution is notoriously difficult and lack of reliable benchmarks. We first use the dual approach to transform it into a linear operator, and then introduce a Fractional-Boundary-Regularized Deep Galerkin Method (FBR-DGM) that augments the classical L2 loss with Sobolev-Slobodeckij norms on the parabolic boundary, enforcing regularity and yielding consistent improvements in the network approximation and its derivatives. The improved accuracy allows the network to be converted back to the original solution using the dual transform. The self-consistency and stability of the network can be tested by checking the primal-dual relationship among optimal value, optimal wealth, and optimal control, offering innovative benchmarks in the absence of analytical solutions.