[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-123617-en":3,"doc-seo-123617-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":11,"language":21,"language_code":22,"site_id":23,"html_lang":22,"table_of_contents":24,"faqs":25,"seo_title":26,"seo_description":14,"update_tm":27,"read_time":28},123617,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1785132997149421697",8,"Research & Report","Physics-informed machine learning for solving partial differential equations in porous media","Physical phenomena in nature are generally modeled by partial differential equations, yet solving two-phase flow PDEs in porous media is difficult due to infinite-dimensional complexity and high computational cost in traditional numerical simulation. Physics-informed neural networks offer a promising alternative, but existing PINN methods often depend on artificial diffusion, prior knowledge, dense sampling near shocks, or special activation designs. This study develops an LSTM-attention PINN for Buckley–Leverett equations, improving shallow feature learning without artificial diffusion and matching analytical solutions, including accurate shock-point approximations, outperforming traditional physics-informed machine learning.","Advances in  \nOrigiGeo-Enal articlenergy Research Vol. 8, No. 1, p. 37-44, 2023 Physics-informed machine learning for solving partial differential equations in porous media  \nLiqun Shan 1 ;2 , Chengqian Liu2 , Yanchang Liu2*, Yazhou Tu 1 , Linyu Dong3 , Xiali Hei 1  \n1 School of Computing and Informatics, University of Louisiana at Lafayette, Lafayette, LA 70503, USA  \n2 School of Physical and Electrical Engineering, Northeast Petroleum University, Daqing 163318, P. R. China  \n3 Retirement Management Center, Dagang Oilfield, Tianjin 300456, P. R. China  \n\n| Keywords:\u003Cbr>Porous media two-phase flow\u003Cbr>Buckley-Leverett equation physics-informed neural networks recurrent neural network attention mechanism\u003Cbr>Cited as:\u003Cbr>Shan, L., Liu, C., Liu, Y., Tu, Y., Dong, L., Hei, X. Physics-informed machine learning for solving partial differential equations in porous media. Advances in Geo-Energy Research, 2023, 8(1): 37-44 .\u003Cbr>[https://doi.org/10.46690/ager.2023.04.04](https://doi.org/10.46690/ager.2023.04.04) | Abstract:\u003Cbr>Physical phenomenon in nature is generally simulated by partial differential equations. Among different sorts of partial differential equations, the problem of two-phase flow in porous media has been paid intense attention. As a promising direction, physics-informed neural networks shed new light on the solution of partial differential equations. However, current physics-informed neural networks' ability to learn partial differential equations relies on adding artificial diffusion or using prior knowledge to increase the number of training points along the shock trajectory, or adaptive activation functions. To address these issues, this study proposes a physics-informed neural network with long short-term memory and attention mechanism, an ingenious method to solve the Buckley-Leverett partial differential equations representing two-phase flow in porous media. The designed network structure overcomes the dependency on artificial diffusion terms and enhances the importance of shallow features. The experimental results show that the proposed method isin good agreement with analytical solutions. Accurate approximations are shown even when encountering shock points in saturated fields of porous media. Furthermore, experiments show our innovative method outperforms existing traditional physics-informed machine learning approaches. |\n| --- | --- |\n\n1. Introduction  \nPhysical phenomena in nature are usually modeled by partial differential equations (PDEs) . Because the fields of PDEs are infinite dimensional spaces, they are quite difficult to solve those PDEs. In the past several decades, to accurately structure physical processes, predict physical phenomena, and push breakthroughs in geotechnical engineering discoveries, numerous researchers employed numerical methods to simulate physical systems, that is to say, numerical simulations. Finite-dimensional approximations have been developed, such as the finite element method, finite volume method, finite difference method, etc. Solutions to high-dimensional PDEs usually imply huge matrices, resulting in a large computational cost. In this case, the numerical schemes require dividing the space into multiple small grid blocks. Therefore, numerical  \nsimulations are not feasible for real-time and many query scenarios with high computational requirements.  \nWith the popularity of machine learning, it has been oneof the ubiquitous methods for solving physical engineering problems. At present, these techniques have been demonstrated to be a promising approach in solving the aforementioned issues (Cai et al., 2021; Jin et al., 2021; Almajid and AbuAl-Saud, 2022; Kemeth et al., 2022; Vinuesa and Brunton, 2022) . Machine learning approaches can be classified into two methodologies: data-driven methods and physics-informed neural networks (PINN) methods. Data-driven approaches cannot achieve good performance without large amounts of data. As a matter of fact, reservoir engineering data are class","cbCait5J3v5pBqw5","https://ap.wps.com/l/cbCait5J3v5pBqw5","pdf",1118070,1,"English","en",105,"# Introduction\n## Physics modeling with PDEs\n## Numerical simulation limitations\n## Machine learning for physical engineering\n## Data-driven methods vs. PINNs\n## Physics-informed neural networks and PINN formulation","[{\"question\":\"What problem does the study address in porous media modeling?\",\"answer\":\"It targets solving partial differential equations governing two-phase flow in porous media, represented by the Buckley–Leverett equation, where traditional numerical approaches can be computationally expensive.\"},{\"question\":\"How does the proposed model differ from conventional physics-informed neural networks?\",\"answer\":\"It uses a physics-informed neural network with an LSTM and an attention mechanism, reducing reliance on artificial diffusion terms and improving the learning of shallow features.\"},{\"question\":\"How do the experimental results validate the proposed approach?\",\"answer\":\"The method agrees with analytical solutions and maintains accurate approximations even at shock points in saturated porous-media fields, and it outperforms existing traditional physics-informed machine learning approaches.\"}]","Physics-informed machine learning for solving partial differential equations in porous media | 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problem does the study address in porous media modeling?","Question",{"text":74,"@type":75},"It targets solving partial differential equations governing two-phase flow in porous media, represented by the Buckley–Leverett equation, where traditional numerical approaches can be computationally expensive.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the proposed model differ from conventional physics-informed neural networks?",{"text":79,"@type":75},"It uses a physics-informed neural network with an LSTM and an attention mechanism, reducing reliance on artificial diffusion terms and improving the learning of shallow features.",{"name":81,"@type":72,"acceptedAnswer":82},"How do the experimental results validate the proposed approach?",{"text":83,"@type":75},"The method agrees with analytical solutions and maintains accurate approximations even at shock points in saturated porous-media fields, and it outperforms existing traditional physics-informed machine learning 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