[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84473-en":3,"doc-seo-84473-105":29,"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":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},84473,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782698725881665579",8,"Research & Report","A Review of Shape-Morphing Solutions and Evolutional Neural Networks for Spatiotemporal Dynamics","Shape-morphing solutions (SMS) provide nonlinear-parameter approximate solutions to partial differential equations by allowing time-dependent basis or trial functions to evolve with the dynamics. This flexibility extends Galerkin truncations and supports reduced-order modeling and high-fidelity simulation of multiscale systems with localized time-varying features, including vortices, dispersive wave packets, and shocks. Mesh-free SMS scale to higher spatial dimensions and can be implemented as evolutional neural networks, whose weights depend on time. The SMS parameter evolution follows an SMS equation derived from the Dirac–Frenkel variational principle.","arXiv :2603 . 19526v2 [math .NA] 10 Jul 2026  \nA review of shape-morphing solutions and evolutional neural networks for spatiotemporal  \ndynamics  \nMohammad Farazmand∗  \nDepartment of Mathematics, North Carolina State University, 2311 Stinson Drive, Raleigh, NC 27695-8205, USA  \nAbstract  \nShape-morphing solutions (SMS) refer to a class of approximate solutions of partial differential equations (PDEs) with the distinguishing feature that they depend nonlinearly on a set of time-dependent parameters. They generalize Galerkin truncations by allowing the basis (or trial functions) to evolve in time in order to adapt to the solution of the PDE. As such, SMS are particularly suitable for reduced-order modeling as well as high fidelity simulation of multiscale systems which exhibit localized time-dependent features, such as vortices, dispersive wave packets, and shocks. Furthermore, being mesh-free, SMS is scalable for solving PDEs in higher spatial dimensions. As a special case, SMS allows the approximation of the PDE’s solution by a neural network whose weights and biases depend on time. Such neural networks are known as evolutional neural networks or neural Galerkin schemes. The evolution of SMS parameters is dictated by the SMS equation, a set of ordinary differential equations derived from the Dirac–Frenkel variational principle. Over the past five years, contributions to the theory and computation of SMS have been growing rapidly. Here, we survey these developments, showcase some applications of SMS, and highlight important open problems for future research. At the same time, this review is structured to serve as a tutorial for applied mathematicians, physicist, and engineers who wish to enter this field.  \nContents  \n1 Introduction 2  \n1.1 Historical background ........................ 4  \n1.2 Outline ................................ 6  \n∗ Corresponding author: [farazmand@ncsu.edu](farazmand@ncsu.edu)  \n2 Working example 6  \n3 Shape-morphing solutions 7  \n3.1 Examples of SMS ........................... 8  \n3.2 Evolution of SMS ........................... 9  \n3.3 Enforcing conserved quantities ................... 12  \n4 Geometric interpretation of SMS 14  \n5 Neural networks as SMS 16  \n6 Numerical examples 17  \n6.1 Nonlinear Schr¨odinger equation ................... 17  \n6.2 Navier–Stokes equation ....................... 19  \n6.3 Kuramoto–Sivashinsky equation ................... 20  \n6.4 Fokker–Planck equation ....................... 22  \n7 Computational aspects 24  \n7.1 Symbolic SMS ............................ 24  \n7.2 Collocation SMS ........................... 26  \n7.3 Regularized SMS ........................... 27  \n7.4 Boundary conditions ......................... 28  \n8 Conclusions and future directions 30  \n1 Introduction  \nConsider a general evolutionary partial differential equation (PDE),  \n∂tu = F (u), u (x, 0) = u0 (x), (1)  \nwhere F is a potentially nonlinear differential operator. We assume that this initial value problem is well-posed and has a unique solution (in an appropriate sense) for all times t ≥ 0. A conventional Galerkin-type approximation of the solution u(x, t) takes the form  \nr  \nu (x, t) ≃ X αi (t)ϕi (x), (2)  \ni=1  \nwhere αi (t) are the time-dependent parameters whose evolution needs to be determined. The choice of the basis (or trial) functions ϕi (x), which are independent of time, results in different numerical methods. For example, choosing Fourier modes as the basis leads to the Fourier (pseudo-) spectral method and piecewise linear (hat) functions lead to the finite element method.  \nHowever, the static nature of the basis ϕi may lead to computational inefficiencies when the solution comprises time-varying localized structures. Examples include vortices in fluid dynamics, shock waves, jet streams in the atmosphere, and dispersive wave packets. For such PDEs, an unnecessarily large  \n\n| \u003Cbr>(a) |  | (b)\u003Cbr> |\n| --- | --- | --- |\n| \u003Cbr>(c) |  |  |\n\nFigure 1: Three examples of shape-morphing s","cbCaie3sF8E8EIa0","https://ap.wps.com/l/cbCaie3sF8E8EIa0","pdf",4351493,1,41,"English","en",105,"# Introduction\n## Historical background\n## Outline\n# Working example\n# Shape-morphing solutions\n## Examples of SMS\n## Evolution of SMS\n## Enforcing conserved quantities\n# Geometric interpretation of SMS\n# Neural networks as SMS\n# Numerical examples\n## Nonlinear Schrödinger equation\n## Navier–Stokes equation\n## Kuramoto–Sivashinsky equation\n## Fokker–Planck equation\n# Computational aspects\n## Symbolic SMS\n## Collocation SMS\n## Regularized SMS\n## Boundary conditions\n# Conclusions and future directions","[{\"question\":\"What makes shape-morphing solutions different from standard Galerkin truncations?\",\"answer\":\"SMS depend nonlinearly on time-dependent parameters by letting the basis (trial functions) evolve in time, adapting to the PDE solution. Standard Galerkin bases are fixed in time.\"},{\"question\":\"How are evolutional neural networks related to shape-morphing solutions?\",\"answer\":\"An SMS can approximate a PDE solution using a neural network whose weights and biases depend on time. Such networks are called evolutional neural networks or neural Galerkin schemes.\"},{\"question\":\"How are the dynamics of SMS parameters determined?\",\"answer\":\"The evolution of SMS parameters is dictated by the SMS equation, a set of ordinary differential equations derived from the Dirac–Frenkel variational principle.\"}]",1784195871,103,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"a-review-of-shape-morphing-solutions-and-evolutional-neural-networks-for-spatiotemporal-dynamics","",{"@graph":35,"@context":85},[36,53,68],{"@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/a-review-of-shape-morphing-solutions-and-evolutional-neural-networks-for-spatiotemporal-dynamics/84473/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 makes shape-morphing solutions different from standard Galerkin truncations?","Question",{"text":75,"@type":76},"SMS depend nonlinearly on time-dependent parameters by letting the basis (trial functions) evolve in time, adapting to the PDE solution. Standard Galerkin bases are fixed in time.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are evolutional neural networks related to shape-morphing solutions?",{"text":80,"@type":76},"An SMS can approximate a PDE solution using a neural network whose weights and biases depend on time. Such networks are called evolutional neural networks or neural Galerkin schemes.",{"name":82,"@type":73,"acceptedAnswer":83},"How are the dynamics of SMS parameters determined?",{"text":84,"@type":76},"The evolution of SMS parameters is dictated by the SMS equation, a set of ordinary differential equations derived from the Dirac–Frenkel variational principle.","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":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]