Deep Learning Method for Stationary Distribution of Reflected Brownian Motion

2026-07-11 · Source: cs.AI updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Expert, extended

Summary

A new deep learning method accurately and efficiently learns the Laplace transform of high-dimensional Reflected Brownian Motions (RBMs). This framework addresses the intractability of computing stationary distributions and tail probabilities in complex stochastic systems, where closed-form solutions are rare. The approach leverages the Basic Adjoint Relationship (BAR) and integrates a carefully designed loss function, a two-stage training data sampling procedure, and a scalable neural network architecture. The loss function includes normalized BAR error, pairwise consistency, monotonicity, Cauchy-Riemann, and zero-anchoring penalties, using log-parameterization for stability. The architecture employs shared coordinate encoders and additive aggregation, ensuring its size does not scale with the RBM's dimension. Evaluated on 2-dimensional, 20-dimensional, and 30-dimensional RBMs, the method demonstrated near-perfect prediction of tail probabilities (up to 1%) via numerical Laplace inversion, showcasing its potential for analyzing large-scale stochastic systems.

Key takeaway

For research scientists analyzing high-dimensional stochastic systems, this deep learning framework offers a robust alternative to intractable analytical solutions. You can utilize its scalable architecture and specialized loss function to accurately compute Laplace transforms and derive critical performance metrics like tail probabilities for systems such as multiclass queueing networks. Consider integrating this method to characterize system behavior beyond analytically tractable regimes, especially for RBMs up to 30 dimensions.

Key insights

Deep learning can accurately approximate high-dimensional Reflected Brownian Motion Laplace transforms for tail probability estimation.

Principles

• Enforce structural properties via loss functions.
• Design network architecture for scalability.
• Targeted sampling improves high-dimensional coverage.

Method

The method parameterizes Laplace transform logarithms using a scalable neural network. It minimizes a multi-component loss function based on the BAR, structural properties, and uses a two-stage conditional sampling strategy.

In practice

• Estimate tail probabilities for queueing networks.
• Analyze high-dimensional stochastic systems.
• Compute moments from learned Laplace transforms.

Topics

• Reflected Brownian Motion
• Deep Learning
• Laplace Transform
• Stationary Distribution
• Tail Probabilities
• Stochastic Systems

Code references

• zhangz73/NN4MGF

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by cs.AI updates on arXiv.org.