[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85568-en":3,"doc-seo-85568-105":30,"detail-sidebar-cat-0-en-105":91},{"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},85568,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Rare Event Analysis via Stochastic Optimal Control","Rare events govern the behavior of many physical systems, yet unbiased simulation rarely observes them. Transition Path Theory offers a statistical framework by focusing on reactive trajectories between two metastable states and on the committor function, which compactly encodes kinetic and thermodynamic information. A committor-estimation framework is proposed by recasting it as a stochastic optimal control problem, using feedback proportional to the gradient of the log-committor to steer sampling. Two objectives—backpropagation and off-policy Value Matching with optimality guarantees—address hitting-time control. Metastability is handled via an alternative sampling process that preserves reactive current while lowering effective barriers. Results show improved committor, rates, and transition path samples on reversible and non-reversible dynamics.","arXiv :2604 . 13213v3 [ stat .ML] 12 Jul 2026  \nRare Event Analysis via Stochastic Optimal Control  \nYuanqi Du∗1,2, Jiajun He3 , Dinghuai Zhang 1 , Eric Vanden-Eijnden4 , and Carles  \nDomingo-Enrich†1  \n1 Microsoft Research New England  \n2 Cornell University  \n3 University of Cambridge  \n4 Courant Institute of Mathematical Sciences, NYU  \nAbstract  \nRare events, from biomolecular conformational changes, phase transitions, to chemical reactions, are central to the behavior of many physical systems, yet they are extremely difficult to study computationally because unbiased simulations seldom produce them. Transition Path Theory (TPT) provides a rigorous statistical framework for analyzing such events: it characterizes the ensemble of reactive trajectories between two designated metastable states (reactant and product), and its central object—the committor function, which gives the probability that the system will next reach the product rather than the reactant—encodes all essential kinetic and thermodynamic information. We introduce a framework that casts committor estimation as a stochastic optimal control (SOC) problem. In this formulation the committor defines a feedback control—proportional to the gradient of its logarithm—that actively steers trajectories toward the reactive region, thereby enabling efficient sampling of reactive paths. To solve the resulting hitting-time control problem we develop two complementary objectives: a direct backpropagation loss and a principled off-policy Value Matching loss, for which we establish first-order optimality guarantees. We further address metastability, which can trap controlled trajectories in intermediate basins, by introducing an alternative sampling process that preserves the reactive current while lowering effective energy barriers. On both reversible and non-reversible dynamics, the framework yields markedly more accurate committor estimates, reaction rates, and transition path samples than existing methods.  \nKeywords: Stochastic optimal control · Rare event analysis · Transition path theory  \n1 Introduction  \nRare events in physical systems—biomolecular conformational changes, phase transitions, chemical reactions—are infrequent by nature yet key to how these systems evolve [Onsager, 1944 , Levenspiel, 1998 , Bolhuis et al. , 2002 , Seifert, 2008] . Their scarcity makes them notoriously difficult to observe in direct numerical simulations. Quantifying such transitions requires both thermodynamic information ([e.g. free-energy differences](e.g. free-energy differences)) and kinetic information (e.g. reaction rates) . While classical theories such as the Arrhenius equation and transition state theory relate rates to activation barriers [Arrhenius, 1889 , Eyring, 1935], modern rare-event analysis is largely organized around the committor function, a central object of Transition Path Theory (TPT) [Vanden-Eijnden and E, 2006 , 2010] that encodes the full ensemble of reactive trajectories.  \n∗ This work was Y.D.’s summer internship project at Microsoft Research New England, mentored by C. D.-E.  \nCorrespondence: [yuanqidu@microsoft.com](yuanqidu@microsoft.com), [carlesd@microsoft.com](carlesd@microsoft.com)  \n†Corresponding author.  \nSetup and dynamics. Let Xt ∈ X ⊆ Rd denote the state of the system evolving according to the stochastic differential equation  \ndXt = b (Xt)dt + σ dWt , (1)  \nwhere b : X → Rd is the drift, σ ∈ Rd ×d is a constant volatility coefficient (with D = ~~1~~2σσ ⊤ the diffusion tensor), and Wt is a standard d-dimensional Wiener process. Under appropriate conditions on b and σ, the process (1) is ergodic with respect to a unique stationary density ρ satisfying the Fokker–Planck equation ∇ · (bρ − D∇ρ) = 0 .  \nAn important special case is the reversible (gradient) setting, in which the dynamics follow the overdamped Langevin equation dXt = −∇U(Xt)dt +p 2β−1 dWt for a potential U : X → Rand inverse temperature β = 1/(kBT), and the stationary density is the Bol","cbCaisKXZ5EjEEuc","https://ap.wps.com/l/cbCaisKXZ5EjEEuc","pdf",16042245,2,1,104,"English","en",105,"# Introduction\n# Setup and dynamics\n## Setup and dynamics\n# Committor function","[{\"question\":\"What is the role of the committor function in Transition Path Theory?\",\"answer\":\"The committor function gives the probability that the system will next reach the product set before the reactant set. It therefore encodes key kinetic and thermodynamic information about rare transitions.\"},{\"question\":\"How does the proposed method cast committor estimation into stochastic optimal control?\",\"answer\":\"Committor estimation is reformulated as a stochastic optimal control (SOC) problem where the committor induces a feedback control proportional to the gradient of its logarithm. This feedback steers trajectories toward the reactive region to enable efficient sampling.\"},{\"question\":\"How does the framework address metastability during controlled sampling?\",\"answer\":\"Metastability can trap controlled trajectories in intermediate basins. The method introduces an alternative sampling process that preserves the reactive current while lowering effective energy barriers, improving committor and transition path estimates.\"}]",1784204656,262,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"rare-event-analysis-via-stochastic-optimal-control","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/rare-event-analysis-via-stochastic-optimal-control/85568/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the role of the committor function in Transition Path Theory?","Question",{"text":75,"@type":76},"The committor function gives the probability that the system will next reach the product set before the reactant set. It therefore encodes key kinetic and thermodynamic information about rare transitions.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method cast committor estimation into stochastic optimal control?",{"text":80,"@type":76},"Committor estimation is reformulated as a stochastic optimal control (SOC) problem where the committor induces a feedback control proportional to the gradient of its logarithm. This feedback steers trajectories toward the reactive region to enable efficient sampling.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the framework address metastability during controlled sampling?",{"text":84,"@type":76},"Metastability can trap controlled trajectories in intermediate basins. The method introduces an alternative sampling process that preserves the reactive current while lowering effective energy barriers, improving committor and transition path estimates.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]