[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84557-en":3,"doc-seo-84557-105":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},84557,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","Physics Informed Neural Networks for Nonlinear Delay Differential Equations","A physics-informed neural network framework is proposed for solving general first-order delay differential equations. The method integrates a differentiable history switch, a trial-solution formulation that explicitly enforces history constraints, and a segmented collocation strategy to stabilize gradient propagation across large time horizons. This enables scalable, physics-consistent approximations that maintain continuity across subintervals. Numerical experiments validate the effectiveness of the approach.","Physics Informed Neural Networks for Nonlinear Delay Differential Equations  \nStone Yao ∗ Vipin Kumar ∗∗ Roberto Guglielmi ∗∗∗  \n∗ Department of Computer Science, University of Waterloo, Waterloo, ON, Canada (e-mail: [stone.yao@uwaterloo.ca](stone.yao@uwaterloo.ca)).  \n∗∗ Department of Mathematics & Computing, Dr B. R. Ambedkar National Institute of Technology Jalandhar, Punjab, India (e-mail: [vipinkumar@nitj.ac.in](vipinkumar@nitj.ac.in))  \n∗∗∗ Department of Applied Mathematics, University of Waterloo,  \narXiv :2607 .00380v1 [math .NA] 1 Jul 2026  \nWaterloo, ON, Canada (e-mail: [roberto.guglielmi@uwaterloo.ca](roberto.guglielmi@uwaterloo.ca))  \nAbstract: In this paper we propose a novel physics-informed neural network framework for solving general first-order delay differential equations. Our approach combines a differentiable history switch, a trial-solution formulation that explicitly enforces history constraints, and a segmented collocation strategy to stabilize gradient propagation across large temporal domains. The method enables a scalable and physics-consistent approximation of delay differential equation solutions while maintaining continuity across subintervals. Numerical experiments demonstrate the effectiveness of the proposed method.  \nKeywords: Physics-informed neural network; Delay differential equations; Trial solution  \n1. INTRODUCTION  \nDifferential equations arise in many physical applications of science and engineering, such as population dynamics, mechanical systems, electrical circuits, and biological processes. In many real-world phenomena, the dynamics of a system at a given time depend not only on its current state but also on its past history—introducing delay into the dynamics. Such systems are modelled by delay differential equations (DDEs), which naturally occur in diverse areas including population dynamics, epidemiology, neural networks, control theory, and economics, see Smith (2011) . In contrast to ordinary differential equations (ODEs), the presence of delay introduces infinite-dimensional dynamics, leading to richer but more complex behaviours such as oscillations, bifurcations, and stability switches, see Niculescu (2002); Milano and Anghel (2011); Wu et al.(2004) .  \nAn accurate method for approximating the solutions of DDEs is therefore essential for understanding and predicting the evolution of time-delay systems, and shall take these challenges into account. Indeed, analytical methods for DDEs are typically limited to specific forms or linearized systems, often relying on characteristic equations or Laplace transforms, see Walter (2013) . For most nonlinear and non-autonomous problems, closed-form solutions are unavailable, motivating the development of numerical methods. Classical numerical solvers, such as the method of steps, Runge–Kutta-based schemes, or collocation techniques (e.g., MATLAB’s dde23), approximate the solution by discretizing time and propagating it iteratively using previously computed delay values, see Atkinson et al.(2009); Butcher (2016) . While effective, these methods can become computationally expensive or unstable for  \nlong-time horizons, high-dimensional systems, or when the delay term is state-dependent or distributed.  \nEarly neural approaches for differential equations date back to the 1990s, where feed-forward networks were trained to minimize residual errors of ODEs and PDEs Lagaris et al. (1998); Meade and Fernandez (1994); Dissanayake and Phan-Thien (1994) . Subsequent works have extended these ideas to various equation types, including stiff ODEs Kim et al. (2021), fractional systems Firoozsalari et al. (2023), and delay equations Panghaland Kumar (2022); Fang et al. (2020); Noor and Malik (2025) . For DDEs, some recent studies have used recurrent neural networks or hybrid architectures combining history functions with data-driven delay modelling Vinodbhai and Dubey (2023); Chen et al. (2025) . However, most of these methods are problem-specific, re","cbCaikc3LpnTx8rV","https://ap.wps.com/l/cbCaikc3LpnTx8rV","pdf",2157107,1,6,"English","en",105,"# Introduction\n## Background: delay differential equations and challenges\n## Classical numerical methods and limitations\n## Neural approaches and the PINN framework\n## Motivation and proposed contributions","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper targets numerical approximation of general first-order delay differential equations (DDEs), where the system depends on its past history.\"},{\"question\":\"How does the proposed PINN framework enforce history constraints?\",\"answer\":\"It uses a trial-solution formulation that explicitly enforces history constraints, supported by a differentiable history switch.\"},{\"question\":\"What technique helps stabilize training over large temporal domains?\",\"answer\":\"A segmented collocation strategy is introduced to stabilize gradient propagation across large time horizons while maintaining continuity across subintervals.\"}]",1784196723,15,{"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},"physics-informed-neural-networks-for-nonlinear-delay-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/physics-informed-neural-networks-for-nonlinear-delay-differential-equations/84557/",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 problem does the paper address?","Question",{"text":74,"@type":75},"The paper targets numerical approximation of general first-order delay differential equations (DDEs), where the system depends on its past history.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the proposed PINN framework enforce history constraints?",{"text":79,"@type":75},"It uses a trial-solution formulation that explicitly enforces history constraints, supported by a differentiable history switch.",{"name":81,"@type":72,"acceptedAnswer":82},"What technique helps stabilize training over large temporal domains?",{"text":83,"@type":75},"A segmented collocation strategy is introduced to stabilize gradient propagation across large time horizons while maintaining continuity across subintervals.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,113,118,121,126,129,133],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":21,"doc_module":4,"doc_module_name":45,"category_name":110,"show_sort_weight":111,"slug":112},"Technology",50,"technology",{"id":114,"doc_module":4,"doc_module_name":45,"category_name":115,"show_sort_weight":116,"slug":117},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":119,"slug":120},30,"research-report",{"id":122,"doc_module":4,"doc_module_name":45,"category_name":123,"show_sort_weight":124,"slug":125},9,"Religion & Spirituality",20,"religion-spirituality",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":127,"show_sort_weight":124,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":45,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":45,"category_name":135,"show_sort_weight":105,"slug":136},19,"General","general"]