Deep Learning Method for Stationary Distribution of Reflected Brownian Motion

2026-07-08

AI Summary

This work addresses the long-standing challenge of characterizing the stationary distribution of high-dimensional reflected Brownian motion (RBM), for which closed-form solutions are generally unavailable and key performance metrics such as tail probabilities are difficult to compute accurately. The authors propose a novel approach that integrates deep learning with the Basic Adjoint Relationship (BAR) to directly learn the Laplace transform of the RBM’s stationary distribution. By designing a tailored loss function, an efficient sampling strategy, and a specialized neural network architecture, the method achieves highly accurate predictions of tail probabilities even in settings lacking analytical solutions. In benchmark cases where exact solutions are known, the approach attains near-perfect accuracy, substantially expanding the frontier of performance analysis for complex stochastic systems.

Abstract

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.

Problem

• reflected Brownian motion
• stationary distribution
• tail probabilities
• high-dimensional stochastic systems
• Laplace transform

Innovation

• reflected Brownian motion
• deep learning
• Laplace transform
• basic adjoint relationship
• tail probabilities