[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82929-en":3,"doc-seo-82929-105":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},82929,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Advances in Neural Controlled Differential Equations","Many real-world systems evolve continuously, yet most machine learning models treat time series as discrete sequences. Continuous-time methods instead view a time series as samples from an underlying input path, enabling flexible handling of irregularly sampled or oversampled data. Neural Controlled Differential Equations (NCDEs) model dynamics by evolving hidden states via a dynamical system driven by the input path, using neural parametrizations. This thesis improves NCDE training and scalability by introducing Log-NCDEs, Linear NCDEs, and Structured Linear NCDEs, reducing per-step training time by up to three orders of magnitude while achieving state-of-the-art results on diverse benchmarks.","Advances in Neural Controlled Differential Equations  \nBenjamin Walker  \nBalliol College  \nUniversity of Oxford  \nA thesis submitted for the degree of Doctor of Philosophy  \nNovember 2025  \nTo Mum and Dad, for everything.  \nAcknowledgements  \nTerry, it has been a privilege spending an hour (and often much longer) each week discussing mathematics, machine learning, finance, politics, philosophy, and every other topic we wandered into. Although you “cannot discuss mathematics with someone who doesn’t understand differential geometry,” you certainly did your best with me. I would not be the researcher I am today without you.  \nMum and Dad, I will never be able to express the gratitude I have foryour continuous love, encouragement, and guidance. From Mum spending hours revising with me to Dad patiently reminding me for the hundredth time that there are two solutions to x2 = 4, I owe every step of this journey to you both.  \nSophie, Naomi, and all of my friends, thank you. From adventures Down Under, to European hiking trips and weekend getaways in London, Selsey, Wales, and Milton Keynes, from late-night board games (and those who had to put up with them) to some truly wonderful Christmas meals. I cannot imagine having done this without your support and friendship.  \nThank you to my collaborators and colleagues, Lingyi Yang, Nicola Mu¸ca Cirone, Cris Salvi, Christian Bayer, Andrew McLeod, Felix Krones, Adam Mahdi, Tiexin Qin, Haoliang Li, Cora Cartis, Kate Zhu, Ammar Naseer, Torben Berndt, Alexandre Bloch, Sam Morley, and Elena Gal. This thesis, and I personally, owe much to the time, effort, and enthusiasm you have so generously given.  \nThank you also to my examiners, Marc Deisenroth and Stephen Roberts, for your time, care, and thoughtful feedback on this thesis.  \nFinally, Nathalie, the best part of this journey has been sharing it with you. I cannot wait for our next adventure.  \nThere are many human behaviours which unfold over time. It would be folly to try to understand those behaviours without taking into account their temporal nature.  \n—Jeffrey Elman, Finding Structure in Time (1990)  \nAbstract  \nMany real-world systems evolve continuously, yet most machine learning models interpret time series as discrete sequences. Continuous-time approaches instead treat time series as samples from an underlying input path, a formulation that naturally accommodates irregularly sampled or oversampled data. Among these, Neural Controlled Differential Equations (NCDEs) are a maximally expressive class of models that parametrise a vector field using a neural network and evolve their hidden state by solving a dynamical system driven by the input path. NCDEs typically use a non-linear vector field, so their expressive power and continuous-time flexibility come at the cost of a forward pass that is both computationally expensive and inherently sequential, limiting their scalability and practical applicability.  \nThis thesis advances the training and scalability of NCDEs through three complementary contributions. First, building on neural rough differential equations, Log-NCDEs apply the Log-ODE method to efficiently approximate an NCDE’s solution during training, improving both computational speed and empirical performance. Second, Linear NCDEs replace thenon-linear vector field with a linear one, enabling closed-form solutionsand parallel-in-time computation without sacrificing theoretical expressivity. Third, Structured Linear NCDEs use structured linear vector fields to further enhance efficiency while maintaining theoretical expressiveness and empirical performance.  \nCollectively, these methods reduce the time per training step for an NCDE by up to three orders of magnitude while achieving state-of-the-art performance across diverse time series benchmarks.  \nContents  \n1 Introduction 1  \n1.1 Paths .................................... 1  \n1.1.1 The PhysioNet Challenge 2022 .................. 1  \n1.1.2 From Paths to Signatures ..........","cbCais2fQ3Isfv2g","https://ap.wps.com/l/cbCais2fQ3Isfv2g","pdf",7578849,5,1,188,"English","en",105,"# Introduction\n## Paths\n## Thesis Outline\n# Controlled Differential Equations\n## Tensor Algebra\n## The Signature\n## Controlled Differential Equations\n## The Log-ODE Method\n# Lip(γ) Functions\n## Differentiable Functions\n## Lip(γ) Functions\n## Composition of Lip(γ) Functions","[{\"question\":\"Why do continuous-time models matter for real-world time series?\",\"answer\":\"They treat time series as samples from an underlying input path, which naturally supports irregularly sampled or oversampled data rather than forcing a discrete-sequence interpretation.\"},{\"question\":\"What is the main computational bottleneck of standard NCDEs?\",\"answer\":\"Their non-linear vector field yields strong expressivity, but it requires an expensive forward pass and is inherently sequential, limiting scalability.\"},{\"question\":\"How does the thesis improve NCDE training and scalability?\",\"answer\":\"It proposes three complementary approaches: Log-NCDEs using Log-ODE approximation to speed training, Linear NCDEs enabling closed-form and parallel-in-time computation, and Structured Linear NCDEs that further improve efficiency while keeping expressiveness.\"}]",1784184046,474,{"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},"advances-in-neural-controlled-differential-equations","",{"@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/advances-in-neural-controlled-differential-equations/82929/",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","2026-07-23","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},"Why do continuous-time models matter for real-world time series?","Question",{"text":76,"@type":77},"They treat time series as samples from an underlying input path, which naturally supports irregularly sampled or oversampled data rather than forcing a discrete-sequence interpretation.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What is the main computational bottleneck of standard NCDEs?",{"text":81,"@type":77},"Their non-linear vector field yields strong expressivity, but it requires an expensive forward pass and is inherently sequential, limiting scalability.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the thesis improve NCDE training and scalability?",{"text":85,"@type":77},"It proposes three complementary approaches: Log-NCDEs using Log-ODE approximation to speed training, Linear NCDEs enabling closed-form and parallel-in-time computation, and Structured Linear NCDEs that further improve efficiency while keeping 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