{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T18:06:32Z","timestamp":1781114792829,"version":"3.54.1"},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T00:00:00Z","timestamp":1769904000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T00:00:00Z","timestamp":1769904000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2026,2]]},"DOI":"10.1007\/s00366-025-02258-1","type":"journal-article","created":{"date-parts":[[2026,2,8]],"date-time":"2026-02-08T09:00:56Z","timestamp":1770541256000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["RBF-fPINNs: radial basis function-enhanced fractional physics-informed neural networks"],"prefix":"10.1007","volume":"42","author":[{"given":"Maryam","family":"Mohammadi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Reza","family":"Mokhtari","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mohadese","family":"Ramezani","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,2,8]]},"reference":[{"issue":"3","key":"2258_CR1","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1007\/s10915-022-01939-z","volume":"92","author":"S Cuomo","year":"2022","unstructured":"Cuomo S, Di Cola VS, Giampaolo F, Rozza G, Raissi M, Piccialli F (2022) Scientific machine learning through physics\u2013informed neural networks: Where we are and what\u2019s next. J Sci Comput 92(3):88","journal-title":"J Sci Comput"},{"issue":"3731","key":"2258_CR2","first-page":"34","volume":"153","author":"R Bellman","year":"1966","unstructured":"Bellman R (1966) Dynamic Program Sci 153(3731):34\u201337","journal-title":"Dynamic Program Sci"},{"issue":"3","key":"2258_CR3","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1038\/s42256-021-00302-5","volume":"3","author":"L Lu","year":"2021","unstructured":"Lu L, Jin P, Pang G, Zhang Z, Karniadakis GE (2021) Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature Mach Intell 3(3):218\u2013229","journal-title":"Nature Mach Intell"},{"key":"2258_CR4","unstructured":"Li Z, Kovachki N, Azizzadenesheli K, Liu B, Bhattacharya K, Stuart A, Anandkumar A (2020) Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895"},{"key":"2258_CR5","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","volume":"378","author":"M Raissi","year":"2019","unstructured":"Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys 378:686\u2013707","journal-title":"J Comput Phys"},{"issue":"153","key":"2258_CR6","first-page":"1","volume":"18","author":"AG Baydin","year":"2018","unstructured":"Baydin AG, Pearlmutter BA, Radul AA, Siskind JM (2018) Automatic differentiation in machine learning: a survey. J Mach Learn Res 18(153):1\u201343","journal-title":"J Mach Learn Res"},{"key":"2258_CR7","doi-asserted-by":"crossref","first-page":"112258","DOI":"10.1016\/j.jcp.2023.112258","volume":"489","author":"J Guo","year":"2023","unstructured":"Guo J, Yao Y, Wang H, Gu T (2023) Pre-training strategy for solving evolution equations based on physics-informed neural networks. J Comput Phys 489:112258","journal-title":"J Comput Phys"},{"key":"2258_CR8","doi-asserted-by":"crossref","first-page":"133945","DOI":"10.1016\/j.physd.2023.133945","volume":"456","author":"Z Miao","year":"2023","unstructured":"Miao Z, Chen Y (2023) VC-PINN: Variable coefficient physics-informed neural network for forward and inverse problems of PDEs with variable coefficient. Physica D 456:133945","journal-title":"Physica D"},{"key":"2258_CR9","doi-asserted-by":"crossref","first-page":"109428","DOI":"10.1016\/j.cpc.2024.109428","volume":"307","author":"J Zhang","year":"2024","unstructured":"Zhang J, Ding C (2024) Simple yet effective adaptive activation functions for physics-informed neural networks. Comput Phys Commun 307:109428","journal-title":"Comput Phys Commun"},{"key":"2258_CR10","first-page":"1","volume":"41","author":"A Dekhovich","year":"2024","unstructured":"Dekhovich A, Sluiter MH, Tax DM, Bessa MA (2024) iPINNs: Incremental learning for physics-informed neural networks. Eng Comput 41:1\u201314","journal-title":"Eng Comput"},{"issue":"1","key":"2258_CR11","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s00366-024-01985-1","volume":"41","author":"O Kianian","year":"2025","unstructured":"Kianian O, Sarrami S, Movahedian B, Azhari M (2025) PINN-based forward and inverse bending analysis of nanobeams on a three-parameter nonlinear elastic foundation including hardening and softening effect using nonlocal elasticity theory. Eng Comput 41(1):71\u201397","journal-title":"Eng Comput"},{"key":"2258_CR12","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1016\/j.camwa.2024.01.021","volume":"159","author":"J Deng","year":"2024","unstructured":"Deng J, Wu J, Zhang S, Li W, Wang YG (2024) Physical informed neural networks with soft and hard boundary constraints for solving advection-diffusion equations using Fourier expansions. Comput Math Appl 159:60\u201375","journal-title":"Comput Math Appl"},{"key":"2258_CR13","doi-asserted-by":"crossref","first-page":"109422","DOI":"10.1016\/j.cpc.2024.109422","volume":"307","author":"FM Rohrhofer","year":"2025","unstructured":"Rohrhofer FM, Posch S, G\u00f6\u00dfnitzer C, Geiger BC (2025) Approximating families of sharp solutions to Fisher\u2019s equation with physics-informed neural networks. Comput Phys Commun 307:109422","journal-title":"Comput Phys Commun"},{"issue":"3","key":"2258_CR14","doi-asserted-by":"crossref","first-page":"1603","DOI":"10.1007\/s00366-023-01883-y","volume":"40","author":"H Tarbiyati","year":"2024","unstructured":"Tarbiyati H, Nemati Saray B (2024) Weight initialization algorithm for physics-informed neural networks using finite differences. Eng Comput 40(3):1603\u20131619","journal-title":"Eng Comput"},{"key":"2258_CR15","doi-asserted-by":"crossref","first-page":"104833","DOI":"10.1016\/j.cageo.2021.104833","volume":"155","author":"UB Waheed","year":"2021","unstructured":"Waheed UB, Haghighat E, Alkhalifah T, Song C, Hao Q (2021) PINNeik: Eikonal solution using physics-informed neural networks. Comput Geosci 155:104833","journal-title":"Comput Geosci"},{"key":"2258_CR16","doi-asserted-by":"crossref","first-page":"116290","DOI":"10.1016\/j.cma.2023.116290","volume":"415","author":"J Bai","year":"2023","unstructured":"Bai J, Liu GR, Gupta A, Alzubaidi L, Feng XQ, Gu Y (2023) Physics-informed radial basis network (PIRBN): A local approximating neural network for solving nonlinear partial differential equations. Comput Methods Appl Mech Eng 415:116290","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"2","key":"2258_CR17","doi-asserted-by":"crossref","first-page":"241","DOI":"10.3390\/math12020241","volume":"12","author":"D Stenkin","year":"2024","unstructured":"Stenkin D, Gorbachenko V (2024) Mathematical modeling on a physics-informed radial basis function network. Mathematics 12(2):241","journal-title":"Mathematics"},{"issue":"6","key":"2258_CR18","doi-asserted-by":"crossref","first-page":"4003","DOI":"10.1007\/s00366-024-01957-5","volume":"41","author":"FV Difonzo","year":"2025","unstructured":"Difonzo FV, Lopez L, Pellegrino SF (2025) Physics-informed neural networks for an inverse problem in peridynamic models. Eng Comput 41(6):4003\u20134012","journal-title":"Eng Comput"},{"issue":"6","key":"2258_CR19","doi-asserted-by":"crossref","first-page":"1403","DOI":"10.1029\/2000WR900031","volume":"36","author":"DA Benson","year":"2000","unstructured":"Benson DA, Wheatcraft SW, Meerschaert MM (2000) Application of a fractional advection-dispersion equation. Water Resour Res 36(6):1403\u20131412","journal-title":"Water Resour Res"},{"issue":"4","key":"2258_CR20","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1016\/j.ultrasmedbio.2013.09.033","volume":"40","author":"S Holm","year":"2014","unstructured":"Holm S, N\u00e4sholm SP (2014) Comparison of fractional wave equations for power law attenuation in ultrasound and elastography. Ultrasound Med Bio 40(4):695\u2013703","journal-title":"Ultrasound Med Bio"},{"key":"2258_CR21","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1016\/j.cma.2016.03.018","volume":"305","author":"F Song","year":"2016","unstructured":"Song F, Xu C, Karniadakis GE (2016) A fractional phase-field model for two-phase flows with tunable sharpness: algorithms and simulations. Comput Methods Appl Mech Eng 305:376\u2013404","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2258_CR22","doi-asserted-by":"crossref","DOI":"10.1142\/p926","volume-title":"Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models","author":"F Mainardi","year":"2022","unstructured":"Mainardi F (2022) Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. World Scientific, Singapore"},{"key":"2258_CR23","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.cpc.2018.11.010","volume":"237","author":"S Guo","year":"2019","unstructured":"Guo S, Mei L, Hou Y, Zhang Z (2019) An efficient finite difference\/Hermite\u2013Galerkin spectral method for time-fractional coupled sine\u2013Gordon equations on multidimensional unbounded domains and its application in numerical simulations of vector solitons. Comput Phys Commun 237:110\u2013128","journal-title":"Comput Phys Commun"},{"issue":"1","key":"2258_CR24","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1137\/19M1301230","volume":"60","author":"RL Du","year":"2022","unstructured":"Du RL, Sun ZZ, Wang H (2022) Temporal second-order finite difference schemes for variable-order time-fractional wave equations. SIAM J Numer Anal 60(1):104\u2013132","journal-title":"SIAM J Numer Anal"},{"issue":"4","key":"2258_CR25","doi-asserted-by":"crossref","first-page":"2352","DOI":"10.1093\/imanum\/drac045","volume":"43","author":"J Markus Melenk","year":"2023","unstructured":"Markus Melenk J, Rieder A (2023) An exponentially convergent discretization for space\u2013time fractional parabolic equations using hp-FEM. IMA J Numer Anal 43(4):2352\u20132376","journal-title":"IMA J Numer Anal"},{"key":"2258_CR26","first-page":"1","volume":"27","author":"X Qi","year":"2024","unstructured":"Qi X, Xu C (2024) An efficient numerical method to the stochastic fractional heat equation with random coefficients and fractionally integrated multiplicative noise. Fract Calc Appl Anal 27:1\u201327","journal-title":"Fract Calc Appl Anal"},{"issue":"3","key":"2258_CR27","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s10915-024-02625-y","volume":"100","author":"M Ramezani","year":"2024","unstructured":"Ramezani M, Mokhtari R, Yan Y (2024) Correction of a high-order numerical method for approximating time-fractional wave equation. J Sci Comput 100(3):71","journal-title":"J Sci Comput"},{"issue":"4","key":"2258_CR28","doi-asserted-by":"crossref","first-page":"A2603","DOI":"10.1137\/18M1229845","volume":"41","author":"G Pang","year":"2019","unstructured":"Pang G, Lu L, Karniadakis GE (2019) fPINNs: Fractional physics-informed neural networks. SIAM J Sci Comput 41(4):A2603\u2013A2626","journal-title":"SIAM J Sci Comput"},{"key":"2258_CR29","doi-asserted-by":"crossref","first-page":"115523","DOI":"10.1016\/j.cma.2022.115523","volume":"400","author":"L Guo","year":"2022","unstructured":"Guo L, Wu H, Yu X, Zhou T (2022) Monte Carlo fPINNs: Deep learning method for forward and inverse problems involving high dimensional fractional partial differential equations. Comput Methods Appl Mech Eng 400:115523","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2258_CR30","doi-asserted-by":"crossref","first-page":"114238","DOI":"10.1016\/j.chaos.2023.114238","volume":"177","author":"Z Ma","year":"2023","unstructured":"Ma Z, Hou J, Zhu W, Peng Y, Li Y (2023) PMNN: Physical model-driven neural network for solving time-fractional differential equations. Chaos Solitons Fractal 177:114238","journal-title":"Chaos Solitons Fractal"},{"issue":"30","key":"2258_CR31","first-page":"19097","volume":"36","author":"J Shi","year":"2024","unstructured":"Shi J, Yang X, Liu X (2024) A novel fractional physics-informed neural networks method for solving the time-fractional Huxley equation. Neural Comput Appl 36(30):19097\u201319119","journal-title":"Neural Comput Appl"},{"key":"2258_CR32","doi-asserted-by":"crossref","first-page":"106418","DOI":"10.1016\/j.enganabound.2025.106418","volume":"179","author":"M Ramezani","year":"2025","unstructured":"Ramezani M, Mohammadi M, Mokhtari R (2025) dPINNs: A physics-informed framework for forward and inverse problems governed by distributed-order derivatives. Eng Anal Boundary Elem 179:106418","journal-title":"Eng Anal Boundary Elem"},{"key":"2258_CR33","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/j.camwa.2025.02.012","volume":"183","author":"L Qiu","year":"2025","unstructured":"Qiu L, Wang F, Liang Y, Qin QH (2025) Physics-informed radial basis function network based on Hausdorff fractal distance for solving Hausdorff derivative elliptic problems. Comput Math Appl 183:271\u2013286","journal-title":"Comput Math Appl"},{"key":"2258_CR34","doi-asserted-by":"crossref","first-page":"013605","DOI":"10.1063\/5.0249560","volume":"37","author":"S Lv","year":"2025","unstructured":"Lv S, Li D, Zha W, Xing Y (2025) Physics-informed radial basis function neural network for efficiently modeling oil\u2013water two-phase Darcy flow. Phys Fluid 37:013605","journal-title":"Phys Fluid"},{"key":"2258_CR35","first-page":"129043","volume":"486","author":"Z Tan","year":"2025","unstructured":"Tan Z (2025) Second-order non-uniform and fast two-grid finite element methods for non-linear time-fractional mobile\/immobile equations with weak regularity. Appl Math Comput 486:129043","journal-title":"Appl Math Comput"},{"issue":"1","key":"2258_CR36","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1007\/s12190-023-01973-6","volume":"70","author":"H Qiao","year":"2024","unstructured":"Qiao H, Cheng A (2024) A fast finite difference method for 2D time variable fractional mobile\/immobile equation. J Appl Math Comput 70(1):551\u2013577","journal-title":"J Appl Math Comput"},{"issue":"2","key":"2258_CR37","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/s10915-024-02505-5","volume":"99","author":"ZY Zheng","year":"2024","unstructured":"Zheng ZY, Wang YM (2024) Fast high-order compact finite difference methods based on the averaged L1 formula for a time-fractional mobile-immobile diffusion problem. J Sci Comput 99(2):43","journal-title":"J Sci Comput"},{"key":"2258_CR38","first-page":"1","volume":"34","author":"Q Jiang","year":"2022","unstructured":"Jiang Q, Zhu L, Shu C, Sekar V (2022) An efficient multilayer RBF neural network and its application to regression problems. Neural Comput Appl 34:1\u201318","journal-title":"Neural Comput Appl"},{"key":"2258_CR39","volume-title":"Radial basis functions, multi-variable functional interpolation, and adaptive networks","author":"DS Boomhead","year":"1988","unstructured":"Boomhead DS, Lowe D (1988) Radial basis functions, multi-variable functional interpolation, and adaptive networks. No. RSREMEMO4148"},{"issue":"1","key":"2258_CR40","first-page":"75","volume":"95","author":"X Li","year":"1998","unstructured":"Li X (1998) On simultaneous approximations by radial basis function neural networks. Appl Math Comput 95(1):75\u201389","journal-title":"Appl Math Comput"},{"issue":"6","key":"2258_CR41","doi-asserted-by":"crossref","first-page":"3987","DOI":"10.1007\/s41980-022-00729-5","volume":"48","author":"M Ramezani","year":"2022","unstructured":"Ramezani M, Mokhtari R (2022) A novel high-order finite-difference method for the time-fractional diffusion equation with smooth\/nonsmooth solutions. Bull Iran Math Soc 48(6):3987\u20134013","journal-title":"Bull Iran Math Soc"},{"key":"2258_CR42","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1016\/j.apnum.2020.02.015","volume":"153","author":"M Ramezani","year":"2020","unstructured":"Ramezani M, Mokhtari R, Haase G (2020) Some high order formulae for approximating Caputo fractional derivatives. Appl Numer Math 153:300\u2013318","journal-title":"Appl Numer Math"},{"key":"2258_CR43","volume-title":"The fractional calculus","author":"KB Oldham","year":"1974","unstructured":"Oldham KB, Spanier J (1974) The fractional calculus. Academic Press, New York, London"},{"issue":"1","key":"2258_CR44","first-page":"503","volume":"45","author":"DC Liu","year":"1989","unstructured":"Liu DC, Nocedal J (1989) On the limited memory BFGS method for large scale optimization. Math Program 45(1):503\u2013528","journal-title":"Math Program"},{"issue":"4","key":"2258_CR45","doi-asserted-by":"crossref","first-page":"945","DOI":"10.4208\/nmtma.OA-2021-0020","volume":"14","author":"M Ramezani","year":"2021","unstructured":"Ramezani M, Mokhtari R, Haase G (2021) Stability and convergence analyses of the FDM based on some L-type formulae for solving the subdiffusion equation. Num Math Theory Method Appl 14(4):945\u2013971","journal-title":"Num Math Theory Method Appl"},{"key":"2258_CR46","doi-asserted-by":"publisher","DOI":"10.1007\/s42967-025-00501-6","author":"M Ramezani","year":"2025","unstructured":"Ramezani M, Mokhtari R (2025) Numerical solution of distributed-order fractional diffusion equations using a high-order temporal scheme. Commun Appl Math Comput. https:\/\/doi.org\/10.1007\/s42967-025-00501-6","journal-title":"Commun Appl Math Comput"},{"key":"2258_CR47","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1553\/etna_vol55s568","volume":"55","author":"M Ramezani","year":"2022","unstructured":"Ramezani M, Mokhtari R, Haase G (2022) Analysis of stability and convergence for L-type formulas combined with a spatial finite element method for solving subdiffusion problems. Electron Trans Numer Anal 55:568\u2013584","journal-title":"Electron Trans Numer Anal"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-025-02258-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-025-02258-1","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-025-02258-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T09:01:36Z","timestamp":1777021296000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-025-02258-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2]]},"references-count":47,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,2]]}},"alternative-id":["2258"],"URL":"https:\/\/doi.org\/10.1007\/s00366-025-02258-1","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2]]},"assertion":[{"value":"4 April 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 October 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 February 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"The authors declare no Conflict of interest.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"42"}}