[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82985-en":3,"doc-seo-82985-105":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},82985,687197207639,"Asher","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Learning Stable Controlled Dynamical Systems via Input Contraction Neural Differential Models","Learning continuous-time representations of dynamical systems from observation data is central to data-driven control and scientific machine learning. Existing neural differential equation methods lack strict structural guarantees for time-varying inputs or rely on overly restrictive assumptions such as constant or vanishing excitation. The paper introduces the Input-Contraction Neural Differential Model (ICNDM), which incorporates time-varying control inputs and ensures incremental exponential convergence through input-dependent contraction regularization. Using an input encoder and a parameterized Riemannian contraction metric, the approach learns both nonautonomous neural vector fields and stability properties. Sufficient conditions are proven under bounded excitations and validated on nonlinear oscillators and PMSM experimental data.","Learning Stable Controlled Dynamical Systems via Input-Contraction Neural Differential Models  \nSyed Pouladi  \nCollege of Engineering and Physical Sciences, Khalifa University, Abu Dhabi, United Arab Emirates  \narXiv :2607 .05718v1 [ ee ss . SY] 7 Jul 2026  \nAbstract—Learning continuous-time representations of dynamical systems from observation data has emerged as a cornerstone of data-driven control and scientific machine learning. However, existing neural differential equations either treat external control inputs heuristically without providing strict structural guarantees, or enforce stability properties under the restrictive assumption of constant or vanishing inputs. This paper proposes the Input-Contraction Neural Differential Model (ICNDM), a novel deep learning framework that seamlessly incorporates timevarying control inputs while ensuring incremental exponential convergence via input-dependent contraction regularization. By leveraging an embedded input encoder and a parameterized metric network, the proposed architecture learns both the nonautonomous neural vector fields and a generalized Riemannian contraction metric simultaneously. We derive sufficient conditions for input-dependent contraction and formally establish an inputto-state contraction property under bounded external excitations. Extensive numerical evaluations on highly nonlinear chaotic oscillators and experimental data from a Permanent Magnet Synchronous Motor (PMSM) drive system demonstrate that ICNDM yields substantial reductions in long-horizon rollout errors and exhibits superior structural robustness against input perturbations compared with state-of-the-art neural differential benchmarks.  \nIndex Terms—Neural Ordinary Differential Equations, Contraction Theory, System Identification, Permanent Magnet Synchronous Motor, Physics-Informed Machine Learning.  \nI. INTRODUCTION  \nModeling and predicting the behavior of complex nonlinear dynamical systems from physical observations is an foundational challenge in automation, robotics, and industrial process control [4] . Traditional methods rely heavily on physics-based principles, which frequently necessitate simplifying assumptions, linearization, or laborious parameter calibration [5] . In recent years, data-driven approaches have gained remarkable momentum, driven by the expressive capacity of deep neural networks. In particular, Neural Ordinary Differential Equations (Neural ODEs) [6] have redefined system identification by parameterizing the underlying vector fields as continuous-time neural architectures, providing elegant solutions for handling irregularly sampled time-series data and preserving continuous trajectories.  \nDespite their successful deployment in modeling autonomous behaviors, practical systems in engineering disciplines—such as collaborative robotic arms, autonomous intelligent vehicles, and power electronics networks—inherently evolve under external time-varying control actions. To generalize neural architectures to these controlled configurations, several extensions have emerged, such as  \nControlled Neural Differential Equations (Neural CDEs) [7], Universal Differential Equations (UDEs) [8], and the ICODE framework [1] . The latter focuses heavily on modeling dynamical structures containing extrinsic input information. Concurrently, ControlSynth Neural ODEs [2] established methods to model dynamical representations with strict convergence guarantees. However, a persistent bottleneck in these formulations is the absence of rigorous closed-loop stability guarantees under generic, non-vanishing input sequences. Since data-driven approximations inevitably suffer from subtle interpolation and extrapolation errors, recursively integrating an unconstrained neural vector field over extended horizons often leads to error accumulation, state divergence, and extreme sensitivity to external perturbations [10] .  \nTo address this instability, several researchers have incorporated contractio","cbCaiguY6HEawrcO","https://ap.wps.com/l/cbCaiguY6HEawrcO","pdf",263875,3,1,5,"English","en",105,"# Introduction\n## Motivation: data-driven control and neural ODEs\n## Related work: controlled and contraction-based neural differential methods\n## Problem: lack of closed-loop stability under non-vanishing inputs\n## Proposed method: ICNDM and key contributions","[{\"question\":\"What limitation in existing neural differential equation approaches motivates ICNDM?\",\"answer\":\"Prior neural differential methods either treat control inputs without rigorous structural guarantees or enforce stability only under restrictive assumptions like constant or vanishing inputs, leaving open-loop error accumulation and poor robustness for generic non-vanishing inputs.\"},{\"question\":\"How does ICNDM guarantee stability with time-varying control inputs?\",\"answer\":\"ICNDM ensures incremental exponential convergence by using input-dependent contraction regularization and jointly optimizing a parameterized metric network with an input embedding, yielding an input-to-state contraction property under bounded excitations.\"},{\"question\":\"What evidence supports ICNDM’s effectiveness and robustness?\",\"answer\":\"Numerical experiments on highly nonlinear chaotic oscillators and experimental data from a Permanent Magnet Synchronous Motor (PMSM) drive show reduced long-horizon rollout errors and stronger structural robustness against input perturbations compared with neural differential baselines.\"}]",1784184468,13,{"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},"learning-stable-controlled-dynamical-systems-via-input-contraction-neural-differential-models","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/learning-stable-controlled-dynamical-systems-via-input-contraction-neural-differential-models/82985/",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","2026-07-23","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 limitation in existing neural differential equation approaches motivates ICNDM?","Question",{"text":75,"@type":76},"Prior neural differential methods either treat control inputs without rigorous structural guarantees or enforce stability only under restrictive assumptions like constant or vanishing inputs, leaving open-loop error accumulation and poor robustness for generic non-vanishing inputs.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does ICNDM guarantee stability with time-varying control inputs?",{"text":80,"@type":76},"ICNDM ensures incremental exponential convergence by using input-dependent contraction regularization and jointly optimizing a parameterized metric network with an input embedding, yielding an input-to-state contraction property under bounded excitations.",{"name":82,"@type":73,"acceptedAnswer":83},"What evidence supports ICNDM’s effectiveness and robustness?",{"text":84,"@type":76},"Numerical experiments on highly nonlinear chaotic oscillators and experimental data from a Permanent Magnet Synchronous Motor (PMSM) drive show reduced long-horizon rollout errors and stronger structural robustness against input perturbations compared with neural differential baselines.","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},[93,97,101,105,109,114,119,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & 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