[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84987-en":3,"doc-seo-84987-105":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},84987,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","JAX-FVM: A Differentiable, Entropy-Stable Finite Volume Solver on Unstructured Meshes for Compressible Flows","JAX-FVM introduces an open-source, fully differentiable finite volume method for the two-dimensional compressible Euler and Navier-Stokes equations on unstructured triangular meshes. Implemented entirely in JAX, the solver compiles mesh connectivity, flux evaluation, slope limiting, and time integration with end-to-end automatic differentiation, executing transparently on CPU or GPU. It combines entropy-conservative two-point flux with entropy-variable Rusanov/Roe dissipation, MUSCL reconstruction with least-squares gradients and Venkatakrishnan limiting, and explicit/implicit integrators using AD-derived Jacobian actions, alongside standard verification cases.","arXiv :2607 .07385v1 [math .NA] 8 Jul 2026  \nJAX-FVM: A differentiable, entropy-stable finite volume solver on unstructured meshes for compressible flows  \nGuillaume de Romémonta,∗  \naINRIA, [200 Av. de](200 Av. de) la Vieille Tour, Talence, 33400 , France  \nAbstract  \nWe present JAX-FVM, an open-source, fully differentiable finite volume method (FVM) for the twodimensional compressible Euler and Navier-Stokes equations on unstructured triangular meshes. The solveris written entirely in JAX, so that every operation : mesh connectivity, flux evaluation, slope limiting, and time integration is just-in-time compiled, vectorised, and end-to-end differentiable through automatic differentiation (AD), and runs transparently on CPU or GPU. On the numerical side, JAX-FVM is built around an entropy-conservative Tadmor/Ismail-Roe two-point flux supplemented with entropy-variable Rusanov or Roe dissipation, second-order MUSCL reconstruction of primitive variables with least-squares gradientsand Venkatakrishnan limiting, and a family of explicit (RK2-4) and matrix-free implicit (Newton, SDIRK2) time integrators whose Jacobian actions are obtained by AD. The combination of an unstructured-mesh compressible FVM with end-to-end differentiability fills a gap left by existing differentiable CFD frameworks, which are almost exclusively restricted to structured grids or spectral discretisations. We describe the governing equations, the discretisation, the software architecture, and a set of standard verification cases. The code is openly available at [https://github.com/guigzair/jax_fvm](https://github.com/guigzair/jax_fvm).  \nKeywords: differentiable programming, JAX, finite volume method, unstructured meshes, entropy stability, compressible Euler and Navier-Stokes equations  \n1. Introduction  \nComputational fluid dynamics (CFD) has become an indispensable tool across science and engineering, from aerodynamics and combustion to biomedical flows (Brunton et al., 2020) . The canonical use of a CFD solver is a forward one: given a geometry, boundary conditions, and physical parameters, one integrates the governing equations to obtain the flow field. Yet many of the most consequential problems are inverse in nature, since boundary conditions are only partially observed, constitutive or model parameters are poorly characterised, and turbulence closures carry model-form uncertainty. Such problems, for instance recovering vascular resistance and compliance from sparse clinical measurements (Pant et al., 2017), inferring geometry from surface pressure in shape optimisation, or identifying turbulence-closure corrections from reference data (Duraisamy et al., 2019), are naturally cast as PDE-constrained optimisation, and solving them at scale requires the gradient of the solver output with respect to its inputs. Finite-difference or derivative-free strategies become prohibitive in high dimension, so scalable sensitivity computation, historically via adjoint methods, is essential.  \nTwo developments have reshaped how these gradients are obtained. First, differentiable programming and automatic differentiation (AD) frameworks such as JAX (Bradbury et al., 2018), PyTorch (Paszke et al., 2019), TensorFlow (Abadi et al., 2016), and Julia (Bezanson et al., 2017) allow the derivative of an  \n⋆  \n∗ Corresponding author.  \nEmail address: guillaume dot romemont at protonmail dot com (Guillaume de Romémont)  \nentire program to be evaluated to machine precision without hand-derived adjoints (Sapienza et al., 2024) . Traditional adjoint CFD implementations, e.g. SU2 (Economon et al., 2016) or dolfin-adjoint (Mitusch et al. , 2019), remain code-intrusive and are architecturally decoupled from the AD ecosystems in which modern machine-learning components are built. Second, the hybridisation of machine learning (ML) with CFD has grown into a dominant paradigm (Brunton et al., 2020; Karniadakis et al., 2021) . Early hybrid models embedded offline-trained closures into con","cbCaivqfOzBmRZY1","https://ap.wps.com/l/cbCaivqfOzBmRZY1","pdf",1065246,1,11,"English","en",105,"# Introduction\n## Differentiable programming and AD for CFD gradients\n## Fully coupled differentiable CFD and ML integration\n## GPU-native differentiable CFD platforms and their grid limitations\n## Unstructured meshes as a missing piece","[{\"question\":\"What equations does JAX-FVM solve and on what mesh type?\",\"answer\":\"JAX-FVM solves the two-dimensional compressible Euler and Navier-Stokes equations on unstructured triangular meshes.\"},{\"question\":\"How does JAX-FVM achieve end-to-end differentiability?\",\"answer\":\"Every solver operation—mesh connectivity, flux evaluation, slope limiting, and time integration—is written in JAX and compiled with just-in-time vectorization, then differentiated through automatic differentiation.\"},{\"question\":\"What numerical ingredients ensure entropy stability?\",\"answer\":\"The method builds on an entropy-conservative Tadmor/Ismail-Roe two-point flux, augmented with entropy-variable Rusanov or Roe dissipation, and uses MUSCL reconstruction with least-squares gradients plus Venkatakrishnan limiting.\"}]",1784200059,28,{"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},"jax-fvm-a-differentiable-entropy-stable-finite-volume-solver-on-unstructured-meshes-for-compressible-flows","",{"@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/jax-fvm-a-differentiable-entropy-stable-finite-volume-solver-on-unstructured-meshes-for-compressible-flows/84987/",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 equations does JAX-FVM solve and on what mesh type?","Question",{"text":75,"@type":76},"JAX-FVM solves the two-dimensional compressible Euler and Navier-Stokes equations on unstructured triangular meshes.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does JAX-FVM achieve end-to-end differentiability?",{"text":80,"@type":76},"Every solver operation—mesh connectivity, flux evaluation, slope limiting, and time integration—is written in JAX and compiled with just-in-time vectorization, then differentiated through automatic differentiation.",{"name":82,"@type":73,"acceptedAnswer":83},"What numerical ingredients ensure entropy stability?",{"text":84,"@type":76},"The method builds on an entropy-conservative Tadmor/Ismail-Roe two-point flux, augmented with entropy-variable Rusanov or Roe dissipation, and uses MUSCL reconstruction with least-squares gradients plus Venkatakrishnan 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