# Deep Learning Method for Stationary Distribution of Reflected Brownian Motion

arXiv: https://arxiv.org/abs/2607.08091 · HTML: https://mlsanity.com/paper/2607.08091/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 23 references that are in the map.
- First posted on arXiv 9 Jul 2026.

## Builds on (most cited first, 11 of 11)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning](https://mlsanity.com/paper/2008.13333/) (2020)
- [Singular Control of (Reflected) Brownian Motion: A Computational Method Suitable for Queueing Applications](https://mlsanity.com/paper/2312.11823/) (2023)
- [Deep Learning for Computing Convergence Rates of Markov Chains](https://mlsanity.com/paper/2405.20435/) (2024)
- [Deep Learning for Markov Chains: Lyapunov Functions, Poisson's Equation, and Stationary Distributions](https://mlsanity.com/paper/2508.16737/) (2025)
- [A Computational Method for Solving the Stochastic Joint Replenishment Problem in High Dimensions](https://mlsanity.com/paper/2511.11830/) (2025)
- [Dynamic Scheduling of a Multiclass Queue in the Halfin-Whitt Regime: A Computational Approach for High-Dimensional Problems](https://mlsanity.com/paper/2311.18128/) (2023)
- [Dynamic control of stochastic matching systems in heavy traffic: An effective computational method for high-dimensional problems](https://mlsanity.com/paper/2509.00809/) (2025)

## Abstract (arXiv)

The stationary distribution of reflected Brownian motion (RBM) plays an important role in the analysis of high-dimensional stochastic systems, yet closed-form solutions are known only for a few special cases. Computing important performance metrics, such as tail probabilities, is even more intractable, despite their practical relevance. In this paper, we develop a deep learning approach that accurately and efficiently learns the Laplace transform of high-dimensional RBMs based on the basic adjoint relationship (BAR). Our framework combines a careful design of the loss function, training data sampling procedure, and neural network architecture. We evaluate the proposed method on RBM instances with known ground-truth tail probabilities and demonstrate near-perfect prediction in high-dimensional settings, highlighting its potential as a general tool for analyzing stochastic systems beyond analytically tractable regimes. Our code can be found at https://github.com/zhangz73/NN4MGF.