[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81582-en":3,"doc-seo-81582-105":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},81582,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Solving Hamilton-Jacobi Equations by Residual Minimization of Monotone Finite-Difference Discretizations","A method solves inviscid and viscous Hamilton–Jacobi equations by minimizing squared residuals from monotone finite-difference discretizations on grids with varying resolution. The approach targets high-dimensional PDEs and is formulated as minimizing residual functionals via gradient-based optimization, enabling neural-network and GPU implementations. A wellposedness result states that any critical point of the finite-difference loss satisfies the monotone scheme with Dirichlet boundary data, and approximation error is controlled by the residual. Convergence rates for gradient flow and a multilevel training algorithm are provided, ensuring convergence to the unique viscosity solution as the grid refines, validated on eikonal, level-set, and Hamilton–Jacobi–Isaacs problems up to dimension eight.","arXiv :2601 .21764v3 [math .NA] 10 Jul 2026  \nSolving Hamilton-Jacobi equations by residual minimization of monotone finite-difference discretizations  \nO. Bokanowski∗1, C. Esteve-Yag¨ue†2, and R. Tsai‡3  \n1 Laboratoire Jacques-Louis Lions, Universit´e Paris Cit´e, France  \n2 Departamento de Matem´aticas, Universidad de Alicante, Spain  \n3 Department of Mathematics and Oden Institute, The University of Texas at Austin, USA  \nJuly 13, 2026  \nAbstract  \nWe introduce a method for solving Hamilton–Jacobi equations, both inviscid and viscous, by minimizing the squared residuals of monotone finite-difference discretizations on grids of varying resolution. The method is designed to leverage neural networks and modern GPUs to solve these equations in higher dimensions; consequently, the setting for our analysis is the minimization of the residual functionals via gradient-based optimization. We establish a wellposedness theory for this approach: any critical point of the finite-difference loss solves the monotone scheme together with the prescribed Dirichlet boundary conditions, and the error of an approximation is controlled by its residual. We then derive the rate of convergence of the gradient flow that minimizes the residual in several settings, noting that the rate may depend on the grid resolution and the domain dimension. Building on this foundation, we propose a multilevel training algorithm that exploits the faster convergence available on the coarser grids of the discretization. Combined with the convergence theorem of Barles and Souganidis for monotone and consistent schemes, our results guarantee convergence to the unique viscosity solution asthe grid is refined. We illustrate the approach on eikonal equations, level-set problems, and a Hamilton–Jacobi–Isaacs equation arising from a stochastic differential game, in dimensions up to eight.  \nKeywords: Hamilton-Jacobi equations, monotone finite-difference schemes, high dimensional PDEs, residual minimization, deep learning  \n∗ [olivier.bokanowski@u-paris.fr](olivier.bokanowski@u-paris.fr)[ ](olivier.bokanowski@u-paris.fr)† c .esteve@ua .es  \n‡[ytsai@math.utexas.edu](ytsai@math.utexas.edu)  \nContents  \n1 Introduction 2  \n2 Monotone schemes and their residual functionals 6  \n2.1 Hypotheses on the numerical Hamiltonian ........................ 6  \n2.2 Residual function and well–posedness ........................... 7  \n2.3 Alternative proof using a fixed point argument ..................... 13  \n3 Error estimates and Polyak–Lojasiewicz inequalities 14  \n3.1 Proper discretizations ................................... 14  \n3.2 Elliptic discretizations ................................... 17  \n4 Time dependent problems 23  \n4.1 Implicit discretizations ................................... 24  \n4.2 Explicit discretizations ................................... 28  \n5 Examples of monotone numerical Hamiltonians 30  \n6 Numerical algorithm 32  \n7 Numerical examples 35  \n7.1 Impact of discretization on training efficiency ...................... 36  \n7.1.1 Deterministic gradient descent over grid functions ............... 36  \n7.1.2 Implementation with neural networks and uniform sampling .......... 37  \n7.2 Level set method ...................................... 39  \n7.2.1 Front propagation ................................. 39  \n7.2.2 Non-linear advection with an obstacle ...................... 42  \n7.3 A viscous Hamilton–Jacobi–Isaacs equation ....................... 43  \n8 Conclusion 45  \n1 Introduction  \nHamilton–Jacobi (HJ) equations are ubiquitous in mathematical physics and control theory, governing systems ranging from level-set dynamics and front propagation to optimal control, differential games, and mathematical finance. We consider fully nonlinear equations of the general form  \nH (x, u, Du, D2 u) = 0, x ∈ Ω ⊂ Rd , (1)  \nwith boundary data u| Γ = g, and the time-dependent version  \n∂tu + H(x, u, Du, D2 u) = 0, x ∈ Ω ⊂ Rd, t > 0 , (2)  \nwith initial condition u (x, 0) = u0 (x) and ","cbCaifWYovsA5PwF","https://ap.wps.com/l/cbCaifWYovsA5PwF","pdf",1990742,3,1,48,"English","en",105,"# Introduction\n# Monotone schemes and their residual functionals\n## Hypotheses on the numerical Hamiltonian\n## Residual function and well–posedness\n## Alternative proof using a fixed point argument\n# Error estimates and Polyak–Lojasiewicz inequalities\n## Proper discretizations\n## Elliptic discretizations\n# Time dependent problems\n## Implicit discretizations\n## Explicit discretizations\n# Examples of monotone numerical Hamiltonians\n# Numerical algorithm\n# Numerical examples\n## Impact of discretization on training efficiency\n## Level set method\n## A viscous Hamilton–Jacobi–Isaacs equation\n# Conclusion","[{\"question\":\"How does the proposed method solve Hamilton–Jacobi equations?\",\"answer\":\"It minimizes the squared residuals of monotone finite-difference discretizations on grids, treating the process as gradient-based optimization, suitable for neural-network and GPU execution.\"},{\"question\":\"What does the wellposedness theory guarantee in this framework?\",\"answer\":\"Any critical point of the finite-difference loss satisfies the monotone scheme together with the prescribed Dirichlet boundary conditions, and the approximation error is bounded by the residual.\"},{\"question\":\"How is convergence to the viscosity solution ensured as the grid is refined?\",\"answer\":\"The results combine the paper’s convergence analysis with the Barles–Souganidis theorem for monotone and consistent schemes, guaranteeing convergence to the unique viscosity solution when the discretization is refined.\"}]",1784174501,121,{"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},"solving-hamilton-jacobi-equations-by-residual-minimization-of-monotone-finite-difference-discretizations","",{"@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/solving-hamilton-jacobi-equations-by-residual-minimization-of-monotone-finite-difference-discretizations/81582/",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","2026-07-25","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},"How does the proposed method solve Hamilton–Jacobi equations?","Question",{"text":75,"@type":76},"It minimizes the squared residuals of monotone finite-difference discretizations on grids, treating the process as gradient-based optimization, suitable for neural-network and GPU execution.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What does the wellposedness theory guarantee in this framework?",{"text":80,"@type":76},"Any critical point of the finite-difference loss satisfies the monotone scheme together with the prescribed Dirichlet boundary conditions, and the approximation error is bounded by the residual.",{"name":82,"@type":73,"acceptedAnswer":83},"How is convergence to the viscosity solution ensured as the grid is refined?",{"text":84,"@type":76},"The results combine the paper’s convergence analysis with the Barles–Souganidis theorem for monotone and consistent schemes, guaranteeing convergence to the unique viscosity solution when the discretization is 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