{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T12:13:37Z","timestamp":1783944817931,"version":"3.55.0"},"reference-count":86,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["42002134"],"award-info":[{"award-number":["42002134"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2021T140735"],"award-info":[{"award-number":["2021T140735"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Neurocomputing"],"published-print":{"date-parts":[[2026,9]]},"DOI":"10.1016\/j.neucom.2026.133882","type":"journal-article","created":{"date-parts":[[2026,5,9]],"date-time":"2026-05-09T15:32:48Z","timestamp":1778340768000},"page":"133882","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["Physics-informed deep kernel method for partial differential equations"],"prefix":"10.1016","volume":"694","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8204-7336","authenticated-orcid":false,"given":"Shaoqun","family":"Dong","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leting","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jingru","family":"Hao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tao","family":"Xu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yao","family":"Cai","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yan","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yidi","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huangshuai","family":"Kong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xu","family":"Yang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.neucom.2026.133882_bib1","doi-asserted-by":"crossref","first-page":"218","DOI":"10.3390\/sym15010218","article-title":"Double formable integral transform for solving heat equations","volume":"15","author":"Saadeh","year":"2023","journal-title":"Symmetry-Basel"},{"key":"10.1016\/j.neucom.2026.133882_bib2","doi-asserted-by":"crossref","first-page":"2150","DOI":"10.1137\/20M1383550","article-title":"Convergence analysis of a fully discrete energy-stable numerical scheme for the Q-tensor flow of liquid crystals","volume":"60","author":"Gudibanda","year":"2022","journal-title":"SIAM J. Numer. Anal."},{"key":"10.1016\/j.neucom.2026.133882_bib3","doi-asserted-by":"crossref","first-page":"34001","DOI":"10.1088\/1741-4326\/ad1d77","article-title":"On multiple solutions of the Grad-Shafranov equation","volume":"64","author":"Ham","year":"2024","journal-title":"Nucl. Fusion"},{"key":"10.1016\/j.neucom.2026.133882_bib4","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1007\/s10092-023-00562-0","article-title":"Analytic regularity and solution approximation for a semilinear elliptic partial differential equation in a polygon","volume":"61","author":"He","year":"2024","journal-title":"Calcolo"},{"key":"10.1016\/j.neucom.2026.133882_bib5","doi-asserted-by":"crossref","first-page":"4458","DOI":"10.1002\/mp.17005","article-title":"A semi-analytical procedure to determine the ion recombination correction factor in high dose-per-pulse beams","volume":"51","author":"Bancheri","year":"2024","journal-title":"Med. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib6","first-page":"1","article-title":"Improved analytic energy operator and novel three-spectral line interpolation DFT method for parameter estimation of voltage flicker","volume":"20","author":"Li","year":"2024","journal-title":"IEEE Trans. Ind. Inf."},{"key":"10.1016\/j.neucom.2026.133882_bib7","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.matcom.2023.04.011","article-title":"Error analysis of a finite difference method for the distributed order sub-diffusion equation using discrete comparison principle","volume":"211","author":"Cao","year":"2023","journal-title":"Math. Comput. Simul."},{"key":"10.1016\/j.neucom.2026.133882_bib8","first-page":"31006","article-title":"Solutions of time-space fractional partial differential equations using Picard's iterative method","volume":"19","author":"Kumar","year":"2024","journal-title":"Am. Soc. Mech. Eng."},{"key":"10.1016\/j.neucom.2026.133882_bib9","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1007\/s12190-023-01977-2","article-title":"Uniform convergence analysis of a new adaptive upwind finite difference method for singularly perturbed convection-reaction-diffusion boundary value problems","volume":"70","author":"Zheng","year":"2024","journal-title":"J. Appl. Math. Comput."},{"key":"10.1016\/j.neucom.2026.133882_bib10","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","article-title":"Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations","volume":"378","author":"Raissi","year":"2019","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib11","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2019.108925","article-title":"PDE-Net 2.0: learning PDEs from data with a numeric-symbolic hybrid deep network","volume":"399","author":"Long","year":"2019","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib12","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2019.108968","article-title":"Efficient deep learning techniques for multiphase flow simulation in heterogeneous porousc media","volume":"401","author":"Wang","year":"2020","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib13","doi-asserted-by":"crossref","DOI":"10.1126\/sciadv.1602614","article-title":"Data-driven discovery of partial differential equations","volume":"3","author":"Rudy","year":"2017","journal-title":"Sci. Adv."},{"key":"10.1016\/j.neucom.2026.133882_bib14","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2021.113722","article-title":"DiscretizationNet: a machine-learning based solver for Navier\u2013Stokes equations using finite volume discretization","volume":"378","author":"Ranade","year":"2021","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"10.1016\/j.neucom.2026.133882_bib15","doi-asserted-by":"crossref","first-page":"1026","DOI":"10.1126\/science.aaw4741","article-title":"Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations","volume":"367","author":"Raissi","year":"2020","journal-title":"Science"},{"key":"10.1016\/j.neucom.2026.133882_bib16","doi-asserted-by":"crossref","first-page":"73607","DOI":"10.1063\/5.0159224","article-title":"Radial basis function-differential quadrature-based physics-informed neural network for steady incompressible flows","volume":"35","author":"Xiao","year":"2023","journal-title":"Phys. Fluids"},{"key":"10.1016\/j.neucom.2026.133882_bib17","doi-asserted-by":"crossref","DOI":"10.1016\/j.tws.2024.112495","article-title":"Physics-informed neural networks (PINN) for computational solid mechanics: Numerical frameworks and applications","volume":"205","author":"Hu","year":"2024","journal-title":"Thin-Walled Struct."},{"key":"10.1016\/j.neucom.2026.133882_bib18","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1016\/j.neunet.2023.03.014","article-title":"Deep learning-accelerated computational framework based on physics informed neural network for the solution of linear elasticity","volume":"162","author":"Roy","year":"2023","journal-title":"Neural Netw."},{"key":"10.1016\/j.neucom.2026.133882_bib19","doi-asserted-by":"crossref","first-page":"1550","DOI":"10.1111\/sapm.12640","article-title":"Some exact and approximate solutions to a generalized Maxwell\u2013Cattaneo equation","volume":"151","author":"Herron","year":"2023","journal-title":"Stud. Appl. Math."},{"key":"10.1016\/j.neucom.2026.133882_bib20","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.camwa.2022.12.008","article-title":"Physics-informed neural networks combined with polynomial interpolation to solve nonlinear partial differential equations","volume":"132","author":"Tang","year":"2023","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.neucom.2026.133882_bib21","first-page":"1","article-title":"Automatic differentiation in machine learning: a survey","volume":"18","author":"Baydin","year":"2018","journal-title":"J. Mach. Learn. Res."},{"key":"10.1016\/j.neucom.2026.133882_bib22","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/s10483-019-2429-8","article-title":"Neural network as a function approximator and its application in solving differential equations","volume":"40","author":"Liu","year":"2019","journal-title":"Appl. Math. Mech. -Engl. Ed."},{"key":"10.1016\/j.neucom.2026.133882_bib23","author":"Kingma","year":"2014","journal-title":"Adam A Method Stoch. Optim. 3rd Int. Conf. Learn. Represent."},{"key":"10.1016\/j.neucom.2026.133882_bib24","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2020.109913","article-title":"B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data","volume":"425","author":"Yang","year":"2021","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib25","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2024.117075","article-title":"Data-driven physics-informed neural networks: a digital twin perspective","volume":"428","author":"Yang","year":"2024","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"10.1016\/j.neucom.2026.133882_bib26","article-title":"Adaptive trajectories sampling for solving PDEs with deep learning methods","volume":"481","author":"Chen","year":"2024","journal-title":"Appl. Math. Comput."},{"key":"10.1016\/j.neucom.2026.133882_bib27","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2024.113561","article-title":"Annealed adaptive importance sampling method in PINNs for solving high dimensional partial differential equations","volume":"521","author":"Zhang","year":"2025","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib28","doi-asserted-by":"crossref","first-page":"962","DOI":"10.1111\/mice.12685","article-title":"Efficient training of physics-informed neural networks via importance sampling","volume":"36","author":"Nabian","year":"2021","journal-title":"Comput. -Aided Civ. Infrastruct. Eng."},{"key":"10.1016\/j.neucom.2026.133882_bib29","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1137\/19M1274067","article-title":"DeepXDE: a deep learning library for solving differential equations","volume":"63","author":"Lu","year":"2021","journal-title":"SIAM Rev."},{"key":"10.1016\/j.neucom.2026.133882_bib30","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2022.115671","article-title":"A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks","volume":"403","author":"Wu","year":"2023","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"10.1016\/j.neucom.2026.133882_bib31","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2022.111868","article-title":"DAS-PINNs: a deep adaptive sampling method for solving high-dimensional partial differential equations","volume":"476","author":"Tang","year":"2023","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib32","doi-asserted-by":"crossref","DOI":"10.1016\/j.neunet.2024.106706","article-title":"Moving sampling physics-informed neural networks induced by moving mesh PDE","volume":"180","author":"Yang","year":"2024","journal-title":"Neural Netw."},{"key":"10.1016\/j.neucom.2026.133882_bib33","doi-asserted-by":"crossref","first-page":"21914","DOI":"10.1063\/5.0188830","article-title":"Physics-informed neural networks with domain decomposition for the incompressible Navier\u2013Stokes equations","volume":"36","author":"Gu","year":"2024","journal-title":"Phys. Fluids"},{"key":"10.1016\/j.neucom.2026.133882_bib34","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1007\/s44207-024-00003-y","article-title":"Machine learning and domain decomposition methods - a survey","volume":"1","author":"Klawonn","year":"2024","journal-title":"Comput. Sci. Eng."},{"key":"10.1016\/j.neucom.2026.133882_bib35","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2020.113547","article-title":"hp-VPINNs: Variational physics-informed neural networks with domain decomposition","volume":"374","author":"Kharazmi","year":"2021","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"10.1016\/j.neucom.2026.133882_bib36","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2020.113028","article-title":"Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems","volume":"365","author":"Jagtap","year":"2020","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"10.1016\/j.neucom.2026.133882_bib37","doi-asserted-by":"crossref","first-page":"2002","DOI":"10.4208\/cicp.OA-2020-0164","article-title":"Extended physics-informed neural networks (XPINNs): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations","volume":"28","author":"D. Jagtap","year":"2020","journal-title":"Commun. Comput. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib38","doi-asserted-by":"crossref","DOI":"10.1016\/j.engappai.2023.107183","article-title":"Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology","volume":"126","author":"Hu","year":"2023","journal-title":"Eng. Appl. Artif. Intell."},{"key":"10.1016\/j.neucom.2026.133882_bib39","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2023.112464","article-title":"A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions","volume":"493","author":"Penwarden","year":"2023","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib40","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2024.117036","article-title":"CEENs: Causality-enforced evolutional networks for solving time-dependent partial differential equations","volume":"427","author":"Jung","year":"2024","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"10.1016\/j.neucom.2026.133882_bib41","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2024.117222","article-title":"Multistep asymptotic pre-training strategy based on PINNs for solving steep boundary singular perturbation problems","volume":"431","author":"Cao","year":"2024","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"10.1016\/j.neucom.2026.133882_bib42","doi-asserted-by":"crossref","first-page":"930","DOI":"10.4208\/cicp.OA-2020-0086","article-title":"Solving Allen-Cahn and Cahn-Hilliard Equations Using the Adaptive Physics Informed Neural Networks","volume":"29","author":"L. Wight","year":"2021","journal-title":"Commun. Comput. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib43","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2023.112258","article-title":"Pre-training strategy for solving evolution equations based on physics-informed neural networks","volume":"489","author":"Guo","year":"2023","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib44","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2021.114474","article-title":"A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations","volume":"390","author":"Mattey","year":"2022","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"10.1016\/j.neucom.2026.133882_bib45","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/j.camwa.2023.09.030","article-title":"Physics-informed neural networks with parameter asymptotic strategy for learning singularly perturbed convection-dominated problem","volume":"150","author":"Cao","year":"2023","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.neucom.2026.133882_bib46","doi-asserted-by":"crossref","first-page":"2453","DOI":"10.1103\/PhysRevE.104.045303","article-title":"Evolutional Deep Neural Network","volume":"104","author":"Du","year":"2021","journal-title":"Phys. Rev. E."},{"key":"10.1016\/j.neucom.2026.133882_bib47","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2022.111024","article-title":"Physics-informed neural networks for the shallow-water equations on the sphere","volume":"456","author":"Bihlo","year":"2022","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib48","article-title":"Characterizing possible failure modes in physics-informed neural networks","author":"Krishnapriyan","year":"2021","journal-title":"Neural Inf. Process. Syst."},{"key":"10.1016\/j.neucom.2026.133882_bib49","doi-asserted-by":"crossref","first-page":"A3055","DOI":"10.1137\/20M1318043","article-title":"P.P.U.S. Univ. Of Pennsylvania, Understanding and mitigating gradient flow pathologies in physics-informed neural networks","volume":"43","author":"Wang","year":"2021","journal-title":"SIAM J. Sci. Comput."},{"key":"10.1016\/j.neucom.2026.133882_bib50","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2021.110768","article-title":"When and why PINNs fail to train: A neural tangent kernel perspective","volume":"449","author":"Wang","year":"2022","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib51","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2022.111722","article-title":"Self-adaptive physics-informed neural networks","volume":"474","author":"McClenny","year":"2023","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib52","doi-asserted-by":"crossref","DOI":"10.1016\/j.asoc.2024.112632","article-title":"wbPINN: Weight balanced physics-informed neural networks for multi-objective learning","volume":"170","author":"Cao","year":"2025","journal-title":"Appl. Soft Comput."},{"key":"10.1016\/j.neucom.2026.133882_bib53","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2025.114297","article-title":"Adaptive residual splitting in PINNs for solving complex PDEs","volume":"540","author":"Cao","year":"2025","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib54","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2022.114823","article-title":"Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems","volume":"393","author":"Yu","year":"2022","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"10.1016\/j.neucom.2026.133882_bib55","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2023.113169","article-title":"Generalized conditional symmetry enhanced physics-informed neural network and application to the forward and inverse problems of nonlinear diffusion equations","volume":"168","author":"Zhang","year":"2023","journal-title":"Chaos Solitons & Fractals"},{"key":"10.1016\/j.neucom.2026.133882_bib56","doi-asserted-by":"crossref","first-page":"1137","DOI":"10.1007\/s00466-024-02554-5","article-title":"MRF-PINN: a multi-receptive-field convolutional physics-informed neural network for solving partial differential equations","volume":"75","author":"Zhang","year":"2025","journal-title":"Comput. Mech."},{"key":"10.1016\/j.neucom.2026.133882_bib57","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1016\/j.neunet.2021.11.022","article-title":"Transformers for modeling physical systems","volume":"146","author":"Geneva","year":"2022","journal-title":"Neural Netw."},{"key":"10.1016\/j.neucom.2026.133882_bib58","doi-asserted-by":"crossref","DOI":"10.1016\/j.rser.2024.114898","article-title":"Hydrogen jet and diffusion modeling by physics-informed graph neural network","volume":"207","author":"Zhang","year":"2025","journal-title":"Renew. Sust. Energ. Rev."},{"key":"10.1016\/j.neucom.2026.133882_bib59","doi-asserted-by":"crossref","DOI":"10.1016\/j.cnsns.2024.107911","article-title":"Physics-informed ConvNet: Learning physical field from a shallow neural network","volume":"132","author":"Shi","year":"2024","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"10.1016\/j.neucom.2026.133882_bib60","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1038\/s42256-023-00685-7","article-title":"Encoding physics to learn reaction\u2013diffusion processes","volume":"5","author":"Rao","year":"2023","journal-title":"Nat. Mach. Intell."},{"key":"10.1016\/j.neucom.2026.133882_bib61","doi-asserted-by":"crossref","first-page":"5514","DOI":"10.1109\/TNNLS.2021.3070878","article-title":"A High-Efficient Hybrid Physics-Informed Neural Networks Based on Convolutional Neural Network","volume":"33","author":"Fang","year":"2022","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"10.1016\/j.neucom.2026.133882_bib62","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.jcp.2020.110079","article-title":"PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain","volume":"428","author":"Gao","year":"2021","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib63","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2023.115944","article-title":"An unsupervised latent\/output physics-informed convolutional-LSTM network for solving partial differential equations using peridynamic differential operator","volume":"407","author":"Mavi","year":"2023","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"10.1016\/j.neucom.2026.133882_bib64","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.jcp.2019.109056","article-title":"Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks","volume":"403","author":"Geneva","year":"2020","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib65","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/j.cpc.2024.109462","article-title":"Combining physics-informed graph neural network and finite difference for solving forward and inverse spatiotemporal PDEs","volume":"308","author":"Zhang","year":"2025","journal-title":"Comput. Phys. Commun."},{"key":"10.1016\/j.neucom.2026.133882_bib66","article-title":"Physics\u2011informed attention\u2011based neural network for hyperbolic partial diferential equations: application to the Buckley\u2013Leverett problem","author":"Torrado","year":"2022","journal-title":"Sci. Rep."},{"key":"10.1016\/j.neucom.2026.133882_bib67","doi-asserted-by":"crossref","first-page":"1411","DOI":"10.1016\/j.petsci.2022.11.027","article-title":"A deep kernel method for lithofacies identification using conventional well logs","volume":"20","author":"Dong","year":"2023","journal-title":"Pet. Sci."},{"key":"10.1016\/j.neucom.2026.133882_bib68","doi-asserted-by":"crossref","DOI":"10.1016\/j.engeos.2024.100300","article-title":"Fracture identification of carbonate reservoirs by deep forest model: An example from the D oilfield in Zagros Basin","volume":"5","author":"Ji","year":"2024","journal-title":"Energy Geosci."},{"key":"10.1016\/j.neucom.2026.133882_bib69","doi-asserted-by":"crossref","DOI":"10.1016\/j.asoc.2025.114048","article-title":"Deeper-PINNs: Unlocking the power of deep physics-informed neural networks","volume":"185","author":"Jiang","year":"2025","journal-title":"Appl. Soft Comput."},{"key":"10.1016\/j.neucom.2026.133882_bib70","article-title":"Derivative-free optimization via classification","author":"Yu","year":"2016","journal-title":"Thirtieth AAAI Conf. Artif. Intell."},{"key":"10.1016\/j.neucom.2026.133882_bib71","doi-asserted-by":"crossref","first-page":"1809","DOI":"10.1007\/s11081-022-09753-0","article-title":"Deep Gaussian process for multi-objective bayesian optimization","volume":"24","author":"Hebbal","year":"2023","journal-title":"Optim. Eng."},{"key":"10.1016\/j.neucom.2026.133882_bib72","doi-asserted-by":"crossref","DOI":"10.1016\/j.compchemeng.2023.108194","article-title":"Combining multi-fidelity modelling and asynchronous batch Bayesian Optimization","volume":"172","author":"Folch","year":"2023","journal-title":"Comput. Chem. Eng."},{"key":"10.1016\/j.neucom.2026.133882_bib73","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2023.115501","article-title":"Corrector estimates and numerical simulations of a system of diffusion\u2013reaction\u2013dissolution\u2013precipitation model in a porous medium","volume":"440","author":"Ghosh","year":"2024","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.neucom.2026.133882_bib74","doi-asserted-by":"crossref","DOI":"10.1016\/j.cnsns.2025.108973","article-title":"Non-Lie non-classical symmetry solutions of a class of nonlinear reaction\u2013diffusion equations","author":"Plenty","year":"2025","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"10.1016\/j.neucom.2026.133882_bib75","doi-asserted-by":"crossref","DOI":"10.1016\/j.aim.2024.109948","article-title":"Virtual linearity for KPP reaction-diffusion equations","volume":"458","author":"Zlato\u0161","year":"2024","journal-title":"Adv. Math."},{"key":"10.1016\/j.neucom.2026.133882_bib76","doi-asserted-by":"crossref","DOI":"10.1016\/j.cpc.2025.109599","article-title":"Discovery and inversion of the viscoelastic wave equation in inhomogeneous media","volume":"312","author":"Chen","year":"2025","journal-title":"Comput. Phys. Commun."},{"key":"10.1016\/j.neucom.2026.133882_bib77","article-title":"Boundary output feedback stabilization for a cascaded wave PDE-ODE system with velocity recirculation and matched disturbance","volume":"444","author":"Li","year":"2023","journal-title":"Appl. Math. Comput."},{"key":"10.1016\/j.neucom.2026.133882_bib78","article-title":"Study of nonlinear wave equation of optical field for solotonic type results","volume":"13","author":"Ullah","year":"2025","journal-title":"Partial Differ. Equ. Appl. Math."},{"key":"10.1016\/j.neucom.2026.133882_bib79","doi-asserted-by":"crossref","first-page":"71103","DOI":"10.1063\/5.0153756","article-title":"Polarization consistent dielectric screening in polarizable continuum model calculations of solvation energies","volume":"159","author":"Khatri","year":"2023","journal-title":"J. Chem. Phys."},{"key":"10.1016\/j.neucom.2026.133882_bib80","doi-asserted-by":"crossref","first-page":"16581","DOI":"10.1007\/s00521-024-09935-0","article-title":"Application of machine learning to model the pressure poisson equation for fluid flow on generic geometries","volume":"36","author":"Sousa","year":"2024","journal-title":"Neural Comput. Appl."},{"key":"10.1016\/j.neucom.2026.133882_bib81","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2025.117956","article-title":"SK-PINN: Accelerated physics-informed deep learning by smoothing kernel gradients","volume":"440","author":"Pan","year":"2025","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"10.1016\/j.neucom.2026.133882_bib82","series-title":"Proceedings of the 27th International Conference on Machine Learning","article-title":"Gaussian Process Optimization in the Bandit Setting: No Regret and Experimental Design","author":"Srinivas","year":"2010"},{"key":"10.1016\/j.neucom.2026.133882_bib83","first-page":"2879","article-title":"Convergence rates of efficient global optimization algorithms","volume":"12","author":"Bull","year":"2011","journal-title":"J. Mach. Learn. Res."},{"key":"10.1016\/j.neucom.2026.133882_bib84","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/j.neucom.2020.07.061","article-title":"On hyperparameter optimization of machine learning algorithms: Theory and practice","volume":"415","author":"Yang","year":"2020","journal-title":"Neurocomputing"},{"key":"10.1016\/j.neucom.2026.133882_bib85","doi-asserted-by":"crossref","first-page":"2148","DOI":"10.1109\/TAI.2025.3540799","article-title":"High-Dimensional Hyperparameter Optimization via Adjoint Differentiation","volume":"6","author":"Dou","year":"2025","journal-title":"IEEE Trans. Artif. Intell."},{"key":"10.1016\/j.neucom.2026.133882_bib86","doi-asserted-by":"crossref","first-page":"8771","DOI":"10.1007\/s00603-023-03483-0","article-title":"Microseismic Location in Hardrock Metal Mines by Machine Learning Models Based on Hyperparameter Optimization Using Bayesian Optimizer","volume":"56","author":"Zhou","year":"2023","journal-title":"Rock. Mech. Rock. Eng."}],"container-title":["Neurocomputing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0925231226012798?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0925231226012798?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T11:49:46Z","timestamp":1783943386000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0925231226012798"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9]]},"references-count":86,"alternative-id":["S0925231226012798"],"URL":"https:\/\/doi.org\/10.1016\/j.neucom.2026.133882","relation":{},"ISSN":["0925-2312"],"issn-type":[{"value":"0925-2312","type":"print"}],"subject":[],"published":{"date-parts":[[2026,9]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Physics-informed deep kernel method for partial differential equations","name":"articletitle","label":"Article Title"},{"value":"Neurocomputing","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.neucom.2026.133882","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"133882"}}