[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-127867-en":3,"doc-seo-127867-105":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":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":27,"seo_description":14,"update_tm":28,"read_time":29},127867,2336474466712,"Maeve","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","The Finite-Time Turnpike Property in Machine Learning","Finite-time turnpike property characterizes optimal control trajectories reaching the target state early and remaining there until the terminal time. This work studies a machine learning formulation based on a neural ordinary differential equation viewed as a homogenization of a deep ResNet. With suitable scaling of the quadratic control cost and the non-smooth tracking term, the optimal control problem satisfies the finite-time turnpike property. The hitting time t0 becomes an additional design parameter, balancing network depth against system parameter size.","machines   \nArticle  \nThe Finite-Time Turnpike Property in Machine Learning Martin Gugat   \nDepartment Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Cauerstr. 11,  \n91058 Erlangen, Germany; [martin.gugat@fau.de](martin.gugat@fau.de)  \nAbstract: The finite-time turnpike property describes the situation in an optimal control problem where an optimal trajectory reaches the desired state before the end of the time interval and remains there. We consider a machine learning problem with a neural ordinary differential equation that can be seen as a homogenization of a deep ResNet. We show that with the appropriate scaling of the quadratic control cost and the non-smooth tracking term, the optimal control problem has the finite-time turnpike property; that is, the desired state is reached within the time interval and the optimal state remains there until the terminal time T. The time t 0 where the optimal trajectories reach the desired state can serve as an additional design parameter. Since ResNets can be viewed as discretizations of neural odes, the choice of t 0 corresponds to the choice of the number of layers; that is, the depth of the neural network. The choice of t 0 allows us to achieve a compromise between the depth of the network and the size of the optimal system parameters, which we hope will be useful to determine the optimal depths for neural network architectures in the future.  \nKeywords: ResNet; neural ODE; finite-time turnpike property; turnpike phenomenon; non-smooth tracking term; machine learning; optimal control  \nCitation: Gugat, M. The Finite-Time Turnpike Property in Machine Learning. Machines 2024, 12, 705 . [https://doi.org/10.3390/](https://doi.org/10.3390/)[ ](https://doi.org/10.3390/)machines12100705  \nAcademic Editor: Jan Awrejcewicz  \nReceived: 14 August 2024  \nRevised: 23 September 2024  \nAccepted: 2 October 2024  \nPublished: 4 October 2024  \nCopyright: © 2024 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([https://](https://)[ ](https://)[creativecommons.org/licenses/by/](creativecommons.org/licenses/by/)[ ](creativecommons.org/licenses/by/)[4.0/](4.0/)) .  \n1. Introduction  \nWe study a system that is governed by a neural ODE that can be considered a continuous-time ResNet. Before we can outline the system, some notation is necessary.  \nThe activation function σ is assumed to be continuously differentiable and Lipschitz continuous with a Lipschitz constant that is less than or equal to 1, for example  \nσ (z) = tanh(z),  \nor σ (z) = ~~1~~+ex~~1~~p(−z) where for z ∈ Rd the function σ acts component-wise; that is, σ(z) ∈ Rd with the i-th component, e.g., (σ(z))i = tanh (zi), (i ∈ {1, . . . , d}) .  \nLet a real number T > 0 and natural numbers d and p in {1, 2, 3, . . . } be given. For i ∈ {1, . . . , p} and t ∈ [0, T] almost everywhere, let wi (t) ∈ Rd and ai (t) ∈ Rd be given. The wi (t) are the columns of the matrix W (t) ∈ Rd×p and the ai (t) are the columns of the matrix A (t) ∈ Rd×p . For t ∈ [0, T], almost everywhere, let the bias vector b(t) in Rp with the components bi(t) (i ∈ {1, . . . , p}) be given. In order to state the required regularity assumptions, we introduce the space  \nX (T) = {measurable functions (W(t), A (t), b (t))defined on (0, T)  \nsuch that R0T ∥W(t)∥2 + ∥A(t)∥2 + ∥b(t)∥2 dt \u003C ∞} .  \nFor parameters (W, A, b) ∈ X (T), the system S is defined as follows:  \n􀀸 x(0) = x0 ∈ Rd,  \nS 􀀼􀀺 x′(t) = i σ (ai (t)⊤ x(t) + bi (t)) wi (t) (1)  \nMachines 2024, 12, 705. [https://doi.org/10.3390/machines12100705](https://doi.org/10.3390/machines12100705) [https://www.mdpi.com/journal/machines](https://www.mdpi.com/journal/machines)  \n(see for example [1,2]) .  \nThe motivation to study (1) is that a time-discrete version can be considered as a residual neural network (ResNet) that has been very useful in many applications; see [3] for identi","cbCaidBoyg7HqyJl","https://ap.wps.com/l/cbCaidBoyg7HqyJl","pdf",315718,1,14,"English","en",105,"# Introduction\n## Neural ODE and continuous-time ResNet formulation\n## Training loss with non-smooth tracking term\n## Control cost regularization\n## Exact controllability and motivation","[{\"question\":\"What does the finite-time turnpike property mean in this study?\",\"answer\":\"It means that an optimal trajectory reaches the desired state within the time interval and then stays at that state until the terminal time T.\"},{\"question\":\"How is the machine learning model formulated?\",\"answer\":\"The model uses a neural ordinary differential equation, which can be interpreted as a continuous-time version of a deep ResNet.\"},{\"question\":\"Why is the non-smooth tracking term and its scaling important?\",\"answer\":\"With appropriate scaling of the quadratic control cost and the non-smooth tracking term, the paper proves that the optimal control problem exhibits the finite-time turnpike property. The inclusion of x′ in the loss is essential for bounding the deviation after the hitting time.\"}]","The Finite-Time Turnpike Property in Machine Learning | PDF",1785942426,35,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"the-finite-time-turnpike-property-in-machine-learning","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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/the-finite-time-turnpike-property-in-machine-learning/127867/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"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","2026-08-23","2026-08-05",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 does the finite-time turnpike property mean in this study?","Question",{"text":76,"@type":77},"It means that an optimal trajectory reaches the desired state within the time interval and then stays at that state until the terminal time T.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How is the machine learning model formulated?",{"text":81,"@type":77},"The model uses a neural ordinary differential equation, which can be interpreted as a continuous-time version of a deep ResNet.",{"name":83,"@type":74,"acceptedAnswer":84},"Why is the non-smooth tracking term and its scaling important?",{"text":85,"@type":77},"With appropriate scaling of the quadratic control cost and the non-smooth tracking term, the paper proves that the optimal control problem exhibits the finite-time turnpike property. 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