[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84684-en":3,"doc-seo-84684-105":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},84684,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","CSympNet-ID Conformal Symplectic Map Learning for Linearly Damped Hamiltonian Systems","Learning dissipative dynamics from discrete observations is crucial for reliable long-horizon prediction and physically meaningful parameter identification. For linearly damped Hamiltonian systems, trajectories are conformally symplectic, contracting the canonical symplectic form by a scalar dissipation factor. CSympNet-ID is a discrete-time map-learning framework that learns one-step flow maps from snapshot pairs while enforcing exact discrete conformal symplecticity by construction, avoiding penalty terms or projections. The method uses an exact symplectic neural core plus diagonal scaling layers with positive, exponentially parameterized damping-rate control.","arXiv :2607 .03339v 1 [ cs .LG] 3 Jul 2026  \nCSympNet-ID: conformal-symplectic map learning for linearly damped Hamiltonian systems  \nJiale Gong 1 Pengzhan Jin2 Dongyang Kuang 1 Lu Li 1 ∗ Yifa Tang3  \n1 School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, China  \n2 National Engineering Laboratory for Big Data Analysis and Applications, Peking University, Beijing 100871, China  \n3 State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China  \nAbstract  \nLearning dissipative dynamics from discrete observations is essential for reliable longhorizon prediction and physically meaningful parameter identification. For linearly damped Hamiltonian systems, the exact flow is generally not symplectic but conformally symplectic, contracting the canonical symplectic form by a scalar factor that reflects the net dissipation. We propose Conformal Symplectic Networks with damping identification (CSympNet-ID) , a discrete-time map-learning framework that learns the one-step flow map directly from snapshot pairs while enforcing exact discrete conformal symplecticity by construction, without penalty terms or projection. The architecture composes an exact symplectic neural core with explicit diagonal scaling layers whose factors are parameterized exponentially by a scalar damping-rate parameter, thereby guaranteeing positivity and interpretability of the learned dissipation factor. We establish a scaling-conjugacy factorization for conformal symplectic maps and derive a pointwise-in-step density result for CSympNet-ID. We evaluate an irregularstep damped oscillator, a damped spring-mass chain, a damped nonlinear cubic oscillator, and additional high-dimensional extensions. CSympNet-ID gives the most favorable overall results among the compared models in the reported experiments, particularly in data-scarce regimes, target contraction-law recovery, and high-dimensional tests where unstructured baselines degrade rapidly.  \nKeywords: damping identification; conformal symplecticity; symplectic networks; map learning; long-horizon prediction.  \n1 Introduction  \nDissipation plays a central role in many real-world dynamical systems, affecting stability, longhorizon predictability, and the identification of physically meaningful parameters. A widely used idealization is a linearly damped Hamiltonian system with a scalar damping rate. Although more general models may involve anisotropic, nonlinear, or state-dependent dissipation, a single effective damping coefficient arises naturally in reduced-order, modal, and homogenized descriptions of dissipative physics. In mechanical and structural dynamics, proportional or modal damping is routinely used to model vibration decay, and each lightly damped mode can often be represented as an oscillator with velocity-proportional damping [1] . A direct example from molecular simulation and sampling is underdamped Langevin dynamics, whose deterministic drift contains the linear friction term, where the scalar coefficient controls relaxation, decorrelation, and sampling efficiency [2] . In dynamical-system learning, a broad line of work aims to model unknown dynamical laws or their evolution maps from observation data [3] . For dissipative systems, however, matching short-time trajectories alone is often insufficient: a model that  \n∗ Corresponding author. Email: [lilu86@mail.sysu.edu.cn](lilu86@mail.sysu.edu.cn)  \nreproduces an incorrect dissipation law may yield wrong decay envelopes, distorted phase portraits, or unstable rollouts when extrapolated beyond the observation window.  \nA principled route to robust extrapolation is to encode physics and geometry as inductive bias, as emphasized in physics-informed and structure-preserving learning frameworks [4,5] . The structure-aware methods discussed below follow two complementary routes: continuous-time models that learn vector fields or variational structures, and discr","cbCaijBBKmybYtm0","https://ap.wps.com/l/cbCaijBBKmybYtm0","pdf",787475,5,1,24,"English","en",105,"# Introduction\n## Dissipation in dynamical systems\n## Structure-preserving and physics-informed learning approaches\n## Continuous-time structure-aware models\n## Discrete-time map-learning and symplectic networks","[{\"question\":\"Why is learning dissipative dynamics from discrete observations important?\",\"answer\":\"It supports reliable long-horizon prediction and helps identify parameters with physical meaning from data snapshots rather than requiring exact time derivatives.\"},{\"question\":\"What does conformal symplecticity mean for linearly damped Hamiltonian systems?\",\"answer\":\"The exact flow is not symplectic but conformally symplectic: it contracts the canonical symplectic form by a scalar factor tied to net dissipation.\"},{\"question\":\"How does CSympNet-ID enforce conformal symplecticity during training?\",\"answer\":\"It learns the one-step flow map directly from snapshot pairs and guarantees exact discrete conformal symplecticity by construction using an exact symplectic neural core combined with parameterized diagonal scaling layers, without penalty terms or projections.\"}]",1784197648,60,{"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},"csympnet-id-conformal-symplectic-map-learning-for-linearly-damped-hamiltonian-systems","",{"@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/csympnet-id-conformal-symplectic-map-learning-for-linearly-damped-hamiltonian-systems/84684/",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","2026-07-29","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 is learning dissipative dynamics from discrete observations important?","Question",{"text":76,"@type":77},"It supports reliable long-horizon prediction and helps identify parameters with physical meaning from data snapshots rather than requiring exact time derivatives.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What does conformal symplecticity mean for linearly damped Hamiltonian systems?",{"text":81,"@type":77},"The exact flow is not symplectic but conformally symplectic: it contracts the canonical symplectic form by a scalar factor tied to net dissipation.",{"name":83,"@type":74,"acceptedAnswer":84},"How does CSympNet-ID enforce conformal symplecticity during training?",{"text":85,"@type":77},"It learns the one-step flow map directly from snapshot pairs and guarantees exact discrete conformal symplecticity by construction using an exact symplectic neural core combined with parameterized diagonal scaling layers, without penalty terms or projections.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,109,114,119,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":29,"slug":108},"Comic","comic",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":20,"slug":137},19,"General","general"]