[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81775-en":3,"doc-seo-81775-105":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},81775,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","A Multi-Resolution Finite-Volume-inspired Deep Learning Framework for Spatiotemporal Dynamics Prediction","Predicting complex spatiotemporal dynamics in physical processes often requires costly numerical solvers or data-driven neural networks that incur high training expenses, accumulate errors, and generalize poorly to unseen parameters. The paper introduces MuRFiV, a Multi-Resolution Finite-Volume-inspired network that merges finite-volume conservative global behavior with deep learning local expressiveness. Across PDE-governed systems, MuRFiV delivers strong long-term accuracy and stability during extended autoregressive rollouts, outperforming data-driven baselines.","arXiv :2607 .00460v 1 [ cs .CE] 1 Jul 2026  \nA MULTI-RESOLUTION FINITE-VOLUME INSPIRED DEEP LEARNING FRAMEWORK FOR SPATIOTEMPORAL DYNAMICS  \nPREDICTION  \nA PREPRINT  \nXin-Yang Liu1 , Xiantao Fan1, 2 , and Jian-Xun Wang1,2,*  \n1Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, USA 2 Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY, USA  \n*Corresponding author: [jw2837@cornell.edu](jw2837@cornell.edu)  \nJuly 2, 2026  \nABSTRACT  \nPredicting complex spatiotemporal dynamics in physical processes often demands computationally expensive numerical methods or data-driven neural networks that suffer from high training costs, error accumulation, and limited generalizability to unseen parameters. An effective approach to address these challenges is leveraging physics priors in training neural networks, known as physics-informed deep learning (PiDL) . In this work, we introduce the Multi-Resolution Finite-Volume-inspired network, MuRFiV, designed to capitalize on the conservative property of finite volume on the global scale and the expressive power of deep learning on the local scale. We demonstrate the effectiveness of MuRFiV on several spatio-temporal systems governed by partial differential equations (PDEs), including Burgers’ equation, shallow water equations, and incompressible Navier-Stokes equations.  \nBy embedding PDE information into the deep learning architecture, MuRFiV achieves strong longterm prediction accuracy and remains stable over very long autoregressive rollouts, significantly outperforming data-driven neural network baselines. This result highlights the promise of combining multiresolution learning with finite-volume-inspired inductive bias for accurate and robust long-term prediction of complex dynamics.  \nKeywords Deep Learning · Scientific Machine Learning · Multi-Resolution · Computational Mechanics  \n1 Introduction  \nSpatiotemporal dynamics are central to many problems in science and engineering, including shock formation, transport and mixing, turbulence, and large-scale environmental prediction. Many of these systems are naturally posed as initial-boundary value problems for partial differential equations (PDEs) . Obtaining numerically reliable solutions at useful spatial and temporal resolutions, however, remains computationally demanding. Stability restrictions such as the Courant–Friedrichs–Lewy condition, the need to resolve disparate scales, and repeated simulations for design, control, or uncertainty quantification can together lead to large wall-clock costs even on modern hardware. Classical finite-difference, finite-volume, finite-element, and spectral methods offer well-established accuracy and stability properties, but they often require dense meshes and small time steps to control truncation error over long horizons. Deep-learning-based surrogate models seek to amortize this cost by replacing repeated numerical solves with a training stage followed by fast inference. Data-driven models have achieved strong predictive accuracy on canonical PDE benchmarks, including convolutional neural network (CNN) models for Cartesian grids [1], graph neural network (GNN) simulators for irregular meshes [2, 3], and neural operators for learning mappings between function spaces across discretizations [4, 5] . Despite these successes, purely data-driven surrogates are often black-box models that rely on large amounts of labeled trajectory data and can generalize poorly outside the training distribution. This limitation  \nis especially consequential for autoregressive rollout, where small one-step errors can compound into unstable or physically inconsistent long-term predictions.  \nPhysics-informed deep learning (PiDL) addresses these limitations by incorporating knowledge of governing laws into the learning problem. Physics-informed neural networks (PINNs) [6] and physics-constrained surrogate models [7] reduce the dependence on labeled d","cbCait3T9zXVUCxB","https://ap.wps.com/l/cbCait3T9zXVUCxB","pdf",7372827,3,1,19,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"What problem does MuRFiV address in spatiotemporal dynamics prediction?\",\"answer\":\"MuRFiV targets the high cost of traditional numerical methods and the limitations of purely data-driven models, including expensive training, error accumulation, and weak generalization to new parameters.\"},{\"question\":\"How does MuRFiV combine physical knowledge with deep learning?\",\"answer\":\"MuRFiV embeds partial differential equation information into the network design, using a finite-volume-inspired inductive bias to capture conservative behavior while leveraging deep learning for local representation.\"},{\"question\":\"Which types of PDE systems are evaluated in the study?\",\"answer\":\"The document states evaluations on several PDE-governed spatiotemporal systems, including Burgers’ equation, shallow water equations, and incompressible Navier–Stokes 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problem does MuRFiV address in spatiotemporal dynamics prediction?","Question",{"text":75,"@type":76},"MuRFiV targets the high cost of traditional numerical methods and the limitations of purely data-driven models, including expensive training, error accumulation, and weak generalization to new parameters.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does MuRFiV combine physical knowledge with deep learning?",{"text":80,"@type":76},"MuRFiV embeds partial differential equation information into the network design, using a finite-volume-inspired inductive bias to capture conservative behavior while leveraging deep learning for local representation.",{"name":82,"@type":73,"acceptedAnswer":83},"Which types of PDE systems are evaluated in the study?",{"text":84,"@type":76},"The document states evaluations on several PDE-governed spatiotemporal systems, including Burgers’ equation, shallow water equations, and incompressible Navier–Stokes 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