[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84309-en":3,"doc-seo-84309-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84309,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","Deep Learning Method for Stationary Distribution of Reflected Brownian Motion","Stationary distributions of reflected Brownian motion (RBM) are central to analyzing high-dimensional stochastic systems, yet closed-form solutions exist only for special cases, making performance metrics like tail probabilities difficult to compute. This paper introduces a deep learning framework that learns the Laplace transform of high-dimensional RBMs using a Laplace-form basic adjoint relationship (BAR). A tailored loss function, training-data sampling scheme, and dimension-stable neural architecture enable accurate and efficient prediction in settings with known ground-truth tails, with code provided.","arXiv :2607 .0809 1v 1 [ cs .LG] 9 Jul 2026  \nDeep Learning Method for Stationary Distribution of Reflected  \nBrownian Motion  \nJim Dai  \nOperations Research and Information Engineering, Cornell University, [jd694@cornell.edu](jd694@cornell.edu)  \nZhanhao Zhang  \nOperations Research and Information Engineering, Cornell University, [zz564@cornell.edu](zz564@cornell.edu)  \nAbstract  \nThe 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](https://github.com/zhangz73/NN4MGF).  \n1 Introduction  \nReflected Brownian motion (RBM) plays an important role in the analysis of multiclass queueing networks, where it often arises as a diffusion approximation under heavy traffic. In such settings, the stationary distribution of an RBM provides useful approximations for the steady-state behavior of the underlying queueing network. Closed-form expressions for stationary distributions are known only for a few special classes of RBMs. This paper develops a scalable deep learning method for computing the Laplace transform of the stationary distribution of a high dimensional RBM. The computed Laplace transforms can be used to estimate tail probabilities that serve as important performance metrics, such as tail latency for queueing networks.  \nWe consider a d-dimensional RBM Z = {Z(t), t ≥ 0} associated with data (Σ,µ, R), which  \nsatisfies the following equations:  \nZ (t) = Z(0) + X(t) + RY(t), t ≥ 0 ,  \nX = {X(t), t ≥ 0} is a d-dimensional Brownian motion with  \ncovariance matrix Σ and drift µ ,  \nY (0) = 0, Y(·) is non-decreasing,  \nZ0 ∞ Zk(t)dYk(t) = 0, k = 1 ,..., d.  \nThe d × d matrix R is known as the reflection matrix. We assume that the RBM is well defined and has a unique stationary distribution π, which is satisfied, e.g., when R is an M-matrix and R −1µ \u003C 0 (Harrison-Williams 1987) . Define the Laplace transform of Z at steady state as  \nφ0 (θ) = Eπ he⟨−θ,Z(0)⟩i ,  \nφk (θ) = Eπ 􀀔Z01 e⟨−θ,Z(t)⟩dYk (t)􀀕 ,  \nθ ∈ R  \nk = 1 , . . . , d,  \nwhere Z(0) follows the stationary distribution π . Lemma 1 of [1] shows that the Laplace transforms φ and φk are uniquely characterized by the Laplace version of basic adjoint relationship (BAR)  \nd  \nγ0 (θ)φ0 (θ) = X γk(θ)φk(θ), (1)  \nk=1  \nwhere γ0 (θ) = − ~~1~~2 ⟨θ,Σθ⟩ + ⟨µ,θ⟩ and γk(θ) = −⟨R(k),θ⟩ . Here, R (k) denotes the k-th column of R. In [1], the authors prove that the Laplace version BAR (1) is equivalent to the PDE version of the BAR that was first advanced in Harrison-Williams (1987) .  \nIn contrast to [2], which develops an efficient approach for estimating steady-state expectations of high-dimensional RBMs, our work targets the Laplace transforms of their stationary distributions. Since the Laplace transform can be numerically inverted to recover tail probabilities and, more broadly, the full stationary distribution [3], it provides a significantly richer characterization of system performance. To the best of our knowledge, this is the first framework for estimating the Laplace transform φk(·) of high-dimensional RBMs. The main methodological contribution is to turn the BA","cbCaii1nQnbzMj5t","https://ap.wps.com/l/cbCaii1nQnbzMj5t","pdf",1046086,5,1,19,"English","en",105,"# Abstract\n# Introduction\n# Problem Setup and Laplace BAR Formulation\n# Related Work and Literature Review","[{\"question\":\"What problem does the paper address in reflected Brownian motion analysis?\",\"answer\":\"It targets computing the Laplace transform of the stationary distribution of high-dimensional RBMs, which enables deriving tail probabilities and other steady-state performance metrics that are otherwise hard to obtain.\"},{\"question\":\"How does the proposed method connect deep learning with RBM theory?\",\"answer\":\"The approach trains a neural network to satisfy the Laplace-version basic adjoint relationship (BAR), turning the BAR characterization into a scalable learning objective.\"},{\"question\":\"Why are Laplace transforms useful for performance evaluation?\",\"answer\":\"Laplace transforms can be numerically inverted to recover tail probabilities and, more broadly, reconstruct the stationary distribution, providing richer characterization of system performance than limited closed-form steady-state results.\"}]",1784194736,48,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"deep-learning-method-for-stationary-distribution-of-reflected-brownian-motion","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/deep-learning-method-for-stationary-distribution-of-reflected-brownian-motion/84309/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-28","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What problem does the paper address in reflected Brownian motion analysis?","Question",{"text":76,"@type":77},"It targets computing the Laplace transform of the stationary distribution of high-dimensional RBMs, which enables deriving tail probabilities and other steady-state performance metrics that are otherwise hard to obtain.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the proposed method connect deep learning with RBM theory?",{"text":81,"@type":77},"The approach trains a neural network to satisfy the Laplace-version basic adjoint relationship (BAR), turning the BAR characterization into a scalable learning objective.",{"name":83,"@type":74,"acceptedAnswer":84},"Why are Laplace transforms useful for performance evaluation?",{"text":85,"@type":77},"Laplace transforms can be numerically inverted to recover tail probabilities and, more broadly, reconstruct the stationary distribution, providing richer characterization of system performance than limited closed-form 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