[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83908-en":3,"doc-seo-83908-105":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},83908,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","PDEFlow: Autonomous Agentic PDE Pipelines for Neural Operator Learning and Solver-Free Inference","PDEFlow proposes an autonomous agentic framework that converts user-level ODE/PDE descriptions into solver-backed neural-operator pipelines. The workflow links problem specification, parameter sampling, FEniCSx-based finite-element solution generation, operator training, and checkpoint-based solver-free inference. A stateful input graph turns multi-turn natural-language requests and user edits into validated JSON specifications via JSON patches with drift prevention. The current implementation uses a multi-branch Bayesian DeepONet. Benchmarks on ODE and PDE tasks demonstrate valid specification construction, solver-backed dataset generation, and fast predictions from saved checkpoints.","arXiv :2607 .05 134v 1 [ cs .LG] 6 Jul 2026  \n\n| PDEFlow: Autonomous Agentic PDE Pipelines for Neural Operator Learning and Solver-Free Inference |\n| --- |\n| Akshat Jani* Prathamesh Gadekar* Sakhinana Sagar Srinivas Venkataramana Runkana\u003Cbr>Tata Research Development and Design Center, Pune, India 411057\u003Cbr>Abstract\u003Cbr>We present PDEFlow, an autonomous agentic framework that turns user-level ODE and PDE descriptions into solver-backed neural-operator pipelines. The workflow links problem specification, data generation, operator training, and checkpoint-based inference. A stateful input graph converts multi-turn natural-language input and user edits into validated problem specifications. The data-generation module then samples parameters, solves the configured governing-equation with FEniCSx finite-element backend, and stores the solutions as operator-ready tensors. The training and inference stages use a registry-based interface, allowing different neural operators to be trained and deployed without changing the surrounding pipeline. In the current implementation, we instantiate this interface with a multi-branch Bayesian DeepONet. Experiments on benchmark ODE and PDE tasks show that PDEFlow can construct valid specifications, generate solver-backed datasets, train neural operators across steady and transient problem classes, and provide solver-free predictions from saved checkpoints. The framework is designed for repeatable scientific and engineering workflows where many related physics configurations must be specified, simulated, learned, and queried with minimal manual intervention.\u003Cbr>1 Introduction\u003Cbr>Many scientific design tasks involve repeated evaluation of the same governing physics under changing conditions. An airfoil is tested across angles of attack and geometric variants; a heat exchanger is studied under different material parameters; a reactor model is run across sampled coefficients for uncertainty quantification. The pattern is simple. The same equation is solved many times.\u003Cbr>This repetition defines downstream workflows such as design optimisation, uncertainty quantification and design-space exploration. Each query requires a valid specification, a solver run, and postprocessing. The cost is high. Not only in computation, but also in the process of turning a researcher’s description into an executable form. That step is often manual. It is also fragile.\u003Cbr>A user rarely writes a complete specification in one attempt. The equation may come first. Boundary conditions are corrected later. Coefficient distributions are added after initial runs. Small edits accumulate. A wrong boundary edge or an outdated parameter can silently affect all subsequent results. These inconsistencies are easy to miss because most workflows treat every update as a fresh configuration rather than a controlled modification of state.\u003Cbr>Neural operators address the cost of repeated solves. Once trained, a model can approximate solutions over a range of inputs without calling the numerical solver. This is useful when exploring many configurations. But operator learning usually begins after two difficult steps have already been completed: the problem specification has been made valid, and the solver-backed dataset has been generated. The earlier workflow remains unresolved. Can an agentic system reliably construct, revise, and audit differential-equation specifications across multiple user interactions? Can the same workflow automatically generate solver-backed training data, train a neural operator, and reuse the |\n\n* Equal Contribution. Correspondence to: Sakhinana Sagar Srinivas \u003C[sagar.sakhinana@tcs.com](sagar.sakhinana@tcs.com)> .  \nNeurIPS 2026 Workshop  \nsaved checkpoint for solver-free inference? More broadly, can natural-language problem entry be connected to an end-to-end pipeline for specification, simulation, learning, and fast prediction?  \nWe introduce a system that connects specification, data generation, model training and ","cbCaiu2lwdZCyIYA","https://ap.wps.com/l/cbCaiu2lwdZCyIYA","pdf",13884587,4,1,39,"English","en",105,"# Introduction\n# Related Works","[{\"question\":\"What problem does PDEFlow address in neural operator workflows?\",\"answer\":\"It targets the earlier, manual and fragile step of constructing and revising differential-equation specifications before generating solver-backed datasets. PDEFlow automates specification, simulation data generation, training, and inference in one workflow.\"},{\"question\":\"How does PDEFlow handle multi-turn user input and edits?\",\"answer\":\"It maintains a stateful input graph that converts multi-turn natural-language input into validated JSON specifications. User edits are represented as JSON patches that are checked and corrected as needed before being applied.\"},{\"question\":\"How does PDEFlow enable solver-free inference after training?\",\"answer\":\"After generating solver-backed datasets and training a neural operator, inference is performed from saved checkpoints without re-solving the governing equations.\"}]",1784191376,98,{"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},"pdeflow-autonomous-agentic-pde-pipelines-for-neural-operator-learning-and-solver-free-inference","",{"@graph":36,"@context":85},[37,53,68],{"@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":20},"https://docshare.wps.com/document/pdeflow-autonomous-agentic-pde-pipelines-for-neural-operator-learning-and-solver-free-inference/83908/",{"url":52,"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","2026-07-27","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 problem does PDEFlow address in neural operator workflows?","Question",{"text":75,"@type":76},"It targets the earlier, manual and fragile step of constructing and revising differential-equation specifications before generating solver-backed datasets. PDEFlow automates specification, simulation data generation, training, and inference in one workflow.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does PDEFlow handle multi-turn user input and edits?",{"text":80,"@type":76},"It maintains a stateful input graph that converts multi-turn natural-language input into validated JSON specifications. User edits are represented as JSON patches that are checked and corrected as needed before being applied.",{"name":82,"@type":73,"acceptedAnswer":83},"How does PDEFlow enable solver-free inference after training?",{"text":84,"@type":76},"After generating solver-backed datasets and training a neural operator, inference is performed from saved checkpoints without re-solving the governing equations.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]