[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81525-en":3,"doc-seo-81525-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},81525,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782109480056885918",8,"Research & Report","Towards Identifiability of Interventional Stochastic Differential Equations","Investigates identifiability of stochastic differential equations under multiple interventions when only samples from stationary distributions are available. Establishes the first provable bounds enabling unique recovery of SDE parameters from stationary-measure data. Derives tight intervention-count bounds for linear SDEs with shift interventions and provides upper bounds for nonlinear SDEs in the small-noise regime. Validates recovery on synthetic data and uses gene regulatory dynamics to show benefits of parameterizations with learnable activation functions.","Towards Identifiability of Interventional Stochastic Differential Equations  \nAaron Zweig 1,2,7 Zaikang Lin 1,3,4,5 Elham Azizi2,6,7 David A. Knowles 1,2  \n1New York Genome Center, New York, U.S.  \n2Department of Computer Science, Columbia University, New York, U.S.  \n3Department of Applied Mathematics and Applied Physics, Columbia University, New York, U.S.  \n4Institute of Computational Biology, Helmholtz Munich, Munich, Germany  \n5Department of Mathematics, Technische Universität München, Munich, Germany  \n6Department of Biomedical Engineering, Columbia University, New York, U.S  \n7Irving Institute for Cancer Dynamics, Columbia University  \narXiv :2505 . 15987v 5 [ cs .LG] 10 Jul 2026  \nAbstract  \nWe study identifiability of stochastic differential equations (SDE) under multiple interventions. Our results give the first provable bounds for unique recovery of SDE parameters given samples from their stationary distributions. We give tight boundson the number of necessary interventions for linear SDEs, and upper bounds for nonlinear SDEs in the small noise regime. We experimentally validate the recovery of true parameters in synthetic data, and motivated by our theoretical results, demonstrate the advantage of parameterizations with learnable activation functions in application to gene regulatory dynamics.  \n1 INTRODUCTION  \nStochastic dynamical systems are ubiquitous as models for natural data. They are perfectly suited for application to time-series data, and therefore also a good candidate to characterize systems that reach a steady state in the limit. If a system is governed by some stochastic differential equation (SDE) and the same system is observed under different interventions, ideally one would learn the underlying parameters governing the dynamics, and guarantee accurate prediction under new interventions.  \nHowever, in many natural settings, data is modeled as following an SDE even if one does not have access to explicit trajectories. Studies of ecological systems focus on the long-term survival of multiple species modeled by the quasi-stationary state of SDEs with environmental factors as perturbations [Hening and Li, 2021] . The application of  \nflow cytometry to protein signaling networks under perturbation [Sachs et al., 2005] is destructive and yields protein quantification at one time point, modeled using the stationary distributions of linear SDEs in Varando and Hansen [2020] .  \nOne highly motivating application is single-cell genomic sequencing with high-throughput CRISPR perturbations. Biologists are often interested in inferring the gene regulatory network (GRN) that characterizes the dynamics of gene expression, informing which genes should be targeted for treatment [Dixit et al., 2016] . But the destructive nature of sequencing makes it impossible to observe the trajectory of a single cell at multiple time-points, and in general it is difficult to obtain any time-series genomics data due to the high expense. Therefore, practitioners often only collect data at the end of an experiment, i.e., from the stationary distribution of the system.  \nUnderstanding the dynamics is essential for extrapolating to unseen settings, but noise and latent confounding makes it non-trivial to determine the true dynamics. Causal disentanglement aims to learn causal factors in spite of these confounders, mainly focusing on directed acyclic graph (DAG) based methods. To demonstrate these methods are well-founded, there is considerable effort devoted to understanding which models have identifiability guarantees [Lachapelle et al., 2022] . However, these models suffer from inherent weakness, in particular 1) being unable to represent cycles or 2) approximate continuous-time dynamical models.  \nThere has been renewed interest in modeling with stochastic differential equations (SDE) directly [Peters et al., 2022] . In the genomic context, there is precedent for this typeof modeling to represent the so-called “Waddington landscape”","cbCailxcmE4LuD4V","https://ap.wps.com/l/cbCailxcmE4LuD4V","pdf",478330,1,26,"English","en",105,"# Abstract\n# Introduction\n# Setup\n## Notation","[{\"question\":\"What does the document study regarding stochastic differential equations?\",\"answer\":\"It studies how to identify the parameters of stochastic differential equations when data comes only from stationary distributions under multiple interventions.\"},{\"question\":\"What are the main theoretical results about identifiability?\",\"answer\":\"The work provides provable bounds for unique recovery of SDE parameters from stationary-distribution samples, including tight bounds for linear SDEs and upper bounds for nonlinear SDEs in the small-noise regime.\"},{\"question\":\"How is the theory validated and where is it applied?\",\"answer\":\"The paper validates parameter recovery on synthetic data and applies the ideas to gene regulatory dynamics, demonstrating improved performance using parameterizations with learnable activation functions.\"}]",1784174017,66,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"towards-identifiability-of-interventional-stochastic-differential-equations","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/towards-identifiability-of-interventional-stochastic-differential-equations/81525/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What does the document study regarding stochastic differential equations?","Question",{"text":74,"@type":75},"It studies how to identify the parameters of stochastic differential equations when data comes only from stationary distributions under multiple interventions.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What are the main theoretical results about identifiability?",{"text":79,"@type":75},"The work provides provable bounds for unique recovery of SDE parameters from stationary-distribution samples, including tight bounds for linear SDEs and upper bounds for nonlinear SDEs in the small-noise regime.",{"name":81,"@type":72,"acceptedAnswer":82},"How is the theory validated and where is it applied?",{"text":83,"@type":75},"The paper validates parameter recovery on synthetic data and applies the ideas to gene regulatory dynamics, demonstrating improved performance using parameterizations with learnable activation 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