{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,17]],"date-time":"2026-02-17T19:37:25Z","timestamp":1771357045324,"version":"3.50.1"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T00:00:00Z","timestamp":1751587200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T00:00:00Z","timestamp":1751587200000},"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":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2025,10]]},"DOI":"10.1007\/s12190-025-02588-9","type":"journal-article","created":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T10:44:25Z","timestamp":1751625865000},"page":"7407-7436","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Reproducing kernel neural networks for nonlinear integro-differential equations"],"prefix":"10.1007","volume":"71","author":[{"given":"Xirui","family":"Fu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiabao","family":"Yang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Boying","family":"Wu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yingqi","family":"Gao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0239-7382","authenticated-orcid":false,"given":"Huanmin","family":"Yao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,7,4]]},"reference":[{"key":"2588_CR1","doi-asserted-by":"publisher","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","volume":"378","author":"M. Raissi","year":"2019","unstructured":"Raissi, M., Perdikaris, P., Karniadakis, G.E.: 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 (2019). https:\/\/doi.org\/10.1016\/j.jcp.2018.10.045","journal-title":"J. Comput. Phys."},{"issue":"153","key":"2588_CR2","first-page":"1","volume":"18","author":"A.G. Baydin","year":"2018","unstructured":"Baydin, A.G., Pearlmutter, B.A., Radul, A.A., Siskind, J.M.: Automatic differentiation in machine learning: a survey. J. Mach. Learn. Res. 18(153), 1\u201343 (2018)","journal-title":"J. Mach. Learn. Res."},{"issue":"7","key":"2588_CR3","doi-asserted-by":"publisher","first-page":"1069","DOI":"10.1007\/s10483-023-2994-7","volume":"44","author":"Z.P. Mao","year":"2023","unstructured":"Mao, Z.P., Meng, X.H.: Physics-informed neural networks with residual\/gradient-based adaptive sampling methods for solving partial differential equations with sharp solutions. Appl. Math. Mech. 44(7), 1069\u20131084 (2023). https:\/\/doi.org\/10.1007\/s10483-023-2994-7","journal-title":"Appl. Math. Mech."},{"issue":"5","key":"2588_CR4","doi-asserted-by":"publisher","first-page":"3055","DOI":"10.1137\/20M1318043","volume":"43","author":"S. Wang","year":"2021","unstructured":"Wang, S., Teng, Y., Perdikaris, P.: Understanding and mitigating gradient flow pathologies in physics-informed neural networks. SIAM J. Sci. Comput. 43(5), 3055\u20133081 (2021). https:\/\/doi.org\/10.1137\/20M1318043","journal-title":"SIAM J. Sci. Comput."},{"key":"2588_CR5","doi-asserted-by":"publisher","first-page":"109136","DOI":"10.1016\/j.jcp.2019.109136","volume":"404","author":"A.D. Jagtap","year":"2020","unstructured":"Jagtap, A.D., Kawaguchi, K., Karniadakis, G.E.: Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. J. Comput. Phys. 404, 109136 (2020). https:\/\/doi.org\/10.1016\/j.jcp.2019.109136","journal-title":"J. Comput. Phys."},{"issue":"1","key":"2588_CR6","doi-asserted-by":"publisher","first-page":"64","DOI":"10.3390\/sym13010064","volume":"13","author":"J. Zhu","year":"2021","unstructured":"Zhu, J., Liu, Y., Cao, J.H.: Effects of second-order velocity slip and the different spherical nanoparticles on nanofluid flow. MDPI 13(1), 64 (2021). https:\/\/doi.org\/10.3390\/sym13010064","journal-title":"MDPI"},{"issue":"2","key":"2588_CR7","doi-asserted-by":"publisher","first-page":"256","DOI":"10.3390\/sym13020256","volume":"13","author":"A.A. Minakov","year":"2021","unstructured":"Minakov, A.A., Schick, C.: Integro-differential equation for the non-equilibrium thermal response of glass-forming materials: analytical solutions. Symmetry 13(2), 256 (2021). https:\/\/doi.org\/10.3390\/sym13020256","journal-title":"Symmetry"},{"key":"2588_CR8","doi-asserted-by":"publisher","first-page":"314","DOI":"10.1016\/j.cam.2016.07.027","volume":"311","author":"B.S.H. Kashkaria","year":"2017","unstructured":"Kashkaria, B.S.H., Syam, M.I.: Evolutionary computational intelligence in solving a class of nonlinear volterra-fredholm integro-differential equations. J. Comput. Appl. Math. 311, 314\u2013323 (2017). https:\/\/doi.org\/10.1016\/j.cam.2016.07.027","journal-title":"J. Comput. Appl. Math."},{"key":"2588_CR9","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1016\/j.asoc.2017.11.002","volume":"62","author":"M.A.Z. Raja","year":"2018","unstructured":"Raja, M.A.Z., Abbas, S., Syam, M.I., Wazwaz, A.M.: Design of neuro-evolutionary model for solving nonlinear singularly perturbed boundary value problems. Appl. Soft Comput. 62, 373\u2013394 (2018). https:\/\/doi.org\/10.1016\/j.asoc.2017.11.002","journal-title":"Appl. Soft Comput."},{"key":"2588_CR10","doi-asserted-by":"publisher","first-page":"110977","DOI":"10.1016\/j.chaos.2021.110977","volume":"147","author":"M.I. Syam","year":"2021","unstructured":"Syam, M.I., Sharadga, M., Hashim, I.: A numerical method for solving fractional delay differential equations based on the operational matrix method. Chaos, Solitons Fractals 147, 110977 (2021). https:\/\/doi.org\/10.1016\/j.chaos.2021.110977","journal-title":"Chaos, Solitons Fractals"},{"key":"2588_CR11","doi-asserted-by":"publisher","first-page":"103265","DOI":"10.1016\/j.rinp.2020.103265","volume":"18","author":"A.K. Alomari","year":"2020","unstructured":"Alomari, A.K., Syam, M.I., Anakira, N.R., Jameel, A.F.: Homotopy sumudu transform method for solving applications in physics. Results Phys. 18, 103265 (2020). https:\/\/doi.org\/10.1016\/j.rinp.2020.103265","journal-title":"Results Phys."},{"issue":"1","key":"2588_CR12","doi-asserted-by":"publisher","first-page":"2277738","DOI":"10.1080\/27690911.2023.2277738","volume":"31","author":"S.M. Syam","year":"2023","unstructured":"Syam, S.M., Siri, Z., Altoum, S.H., Aigo, M.A., And, R.M.K.: A novel study for solving systems of nonlinear fractional integral equations. Appl. Math. Sci. Eng. 31(1), 2277738 (2023). https:\/\/doi.org\/10.1080\/27690911.2023.2277738","journal-title":"Appl. Math. Sci. Eng."},{"issue":"6","key":"2588_CR13","doi-asserted-by":"publisher","first-page":"1263","DOI":"10.3390\/sym15061263","volume":"15","author":"S.M. Syam","year":"2023","unstructured":"Syam, S.M., Siri, Z., Altoum, S.H., Md. Kasmani, R.: Analytical and numerical methods for solving second-order two-dimensional symmetric sequential fractional integro-differential equations. Symmetry 15(6), 1263 (2023). https:\/\/doi.org\/10.3390\/sym15061263","journal-title":"Symmetry"},{"key":"2588_CR14","doi-asserted-by":"publisher","first-page":"100824","DOI":"10.1016\/j.padiff.2024.100824","volume":"11","author":"L. Abdelhaq","year":"2024","unstructured":"Abdelhaq, L., Syam, S.M., Syam, M.I.: An efficient numerical method for two-dimensional fractional integro-differential equations with modified atangana\u2013baleanu fractional derivative using operational matrix approach. Partial Differ. Equations In Appl. Math. 11, 100824 (2024). https:\/\/doi.org\/10.1016\/j.padiff.2024.100824","journal-title":"Partial Differ. Equations In Appl. Math."},{"key":"2588_CR15","doi-asserted-by":"publisher","first-page":"116160","DOI":"10.1016\/j.cam.2024.116160","volume":"453","author":"I. Amirali","year":"2025","unstructured":"Amirali, I., Fedakar, B., Amiraliyev, G.M.: Second-order numerical method for a neutral Volterra integro-differential equation. J. Comput. Appl. Math. 453, 116160 (2025). https:\/\/doi.org\/10.1016\/j.cam.2024.116160","journal-title":"J. Comput. Appl. Math."},{"key":"2588_CR16","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/s40314-022-01933-z","volume":"41","author":"M. Cakir","year":"2022","unstructured":"Cakir, M., Ekinci, Y., Cimen, E.: A numerical approach for solving nonlinear Fredholm integro-differential equation with boundary layer. Comput. Appl. Math. 41, 259 (2022). https:\/\/doi.org\/10.1007\/s40314-022-01933-z","journal-title":"Comput. Appl. Math."},{"key":"2588_CR17","doi-asserted-by":"publisher","first-page":"1800","DOI":"10.1134\/S0965542523100020","volume":"63","author":"M. Cakir","year":"2023","unstructured":"Cakir, M., Cimen, E.: A novel uniform numerical approach to solve a singularly perturbed Volterra integro-differential equation. Comput. Math. And Math. Phys. 63, 1800\u20131816 (2023). https:\/\/doi.org\/10.1134\/S0965542523100020","journal-title":"Comput. Math. And Math. Phys."},{"issue":"1","key":"2588_CR18","doi-asserted-by":"publisher","first-page":"208","DOI":"10.1137\/19M1274067","volume":"63","author":"L. Lu","year":"2021","unstructured":"Lu, L., Meng, X., Mao, Z., Karniadakis, G.E.: Deepxde: a deep learning library for solving differential equations. Soc. For Ind. And Appl. Math. 63(1), 208\u2013228 (2021). https:\/\/doi.org\/10.1137\/19M1274067","journal-title":"Soc. For Ind. And Appl. Math."},{"key":"2588_CR19","doi-asserted-by":"publisher","first-page":"111260","DOI":"10.1016\/j.jcp.2022.111260","volume":"462","author":"L. Yuan","year":"2022","unstructured":"Yuan, L., Ni, Y.Q., Deng, X.Y., Hao, S.: A-pinn: auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations. J. Comput. Phys. 462, 111260 (2022). https:\/\/doi.org\/10.1016\/j.jcp.2022.111260","journal-title":"J. Comput. Phys."},{"issue":"4","key":"2588_CR20","doi-asserted-by":"publisher","first-page":"2603","DOI":"10.1137\/18M1229845","volume":"41","author":"G.F. Pang","year":"2019","unstructured":"Pang, G.F., Lu, L., Karniadakis, G.E.: Fpinns: fractional physics-informed neural networks. Soc. For Ind. And Appl. Math. 41(4), 2603\u20132626 (2019). https:\/\/doi.org\/10.1137\/18M1229845","journal-title":"Soc. For Ind. And Appl. Math."},{"key":"2588_CR21","unstructured":"Burton, T.A.: Volterra Integral and Differential Equations, 2nd edn. Elsevier, Amsterdam (2005)"},{"key":"2588_CR22","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970852","volume-title":"Analytical and Numerical Methods for Volterra Equations","author":"P. Linz","year":"1985","unstructured":"Linz, P.: Analytical and Numerical Methods for Volterra Equations. Society for Industrial and Applied Mathematics, Philadelphia (1985)"},{"key":"2588_CR23","doi-asserted-by":"publisher","first-page":"54","DOI":"10.54671\/BMAA-2025-1-5","volume":"17","author":"A.S. Hassan","year":"2025","unstructured":"Hassan, A.S.: Existence and uniqueness results for nonlinear integral operators in a reproducing kernel Hilbert space. Bull. Of Math. Anal. And Appl. 17, 54\u201364 (2025). https:\/\/doi.org\/10.54671\/BMAA-2025-1-5","journal-title":"Bull. Of Math. Anal. And Appl."},{"issue":"2","key":"2588_CR24","doi-asserted-by":"publisher","first-page":"379","DOI":"10.4208\/cicp.OA-2019-0168","volume":"27","author":"B. Li","year":"2020","unstructured":"Li, B., Tang, S.S., Yu, H.J.: Better approximations of high dimensional smooth functions by deep neural networks with rectified power units. Commun Comput Phys 27(2), 379\u2013411 (2020). https:\/\/doi.org\/10.4208\/cicp.OA-2019-0168","journal-title":"Commun Comput Phys"},{"issue":"6","key":"2588_CR25","doi-asserted-by":"publisher","first-page":"1105","DOI":"10.1137\/21M1397908","volume":"43","author":"L. Lu","year":"2021","unstructured":"Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G.: Physics-informed neural networks with hard constraints for inverse design. Soc. For Ind. And Appl. Math. 43(6), 1105\u20131132 (2021). https:\/\/doi.org\/10.1137\/21M1397908","journal-title":"Soc. For Ind. And Appl. Math."},{"key":"2588_CR26","unstructured":"Arfken, G.B., Weber, H.J., Harris, F.E.: Mathematical Methods for Physicists (Seventh Edition), 7th edn. Academic, Boston (2013)"},{"issue":"3","key":"2588_CR27","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1080\/00207728308926462","volume":"14","author":"M.L. Wang","year":"1983","unstructured":"Wang, M.L., Chang, R.Y.: Solution of linear dynamic system with initial or boundary value conditions by shifted Legendre approximations. Int J Syst Sci 14(3), 343\u2013353 (1983). https:\/\/doi.org\/10.1080\/00207728308926462","journal-title":"Int J Syst Sci"},{"key":"2588_CR28","doi-asserted-by":"publisher","first-page":"108416","DOI":"10.1016\/j.aml.2022.108416","volume":"135","author":"X.T. Su","year":"2023","unstructured":"Su, X.T., Yang, J.B., Yao, H.M.: Shifted Legendre reproducing kernel Galerkin method for the quasilinear degenerate ppgarabolic problem. Appl. Math. Lett. 135, 108416 (2023). https:\/\/doi.org\/10.1016\/j.aml.2022.108416","journal-title":"Appl. Math. Lett."},{"key":"2588_CR29","volume-title":"Nonlinear Numerical Analysis in Reproducing Kernel Space","author":"M. Cui","year":"2009","unstructured":"Cui, M., Lin, Y.: Nonlinear Numerical Analysis in Reproducing Kernel Space. Nova Science Publishers, Inc., USA (2009)"},{"issue":"4","key":"2588_CR30","doi-asserted-by":"publisher","first-page":"1971","DOI":"10.1137\/22M152776","volume":"45","author":"Z. Gao","year":"2023","unstructured":"Gao, Z., Yan, L., Zhou, T.: Failure-informed adaptive sampling for pinns. Methods And Algoritms For Sci. Comput. 45(4), 1971\u20131994 (2023). https:\/\/doi.org\/10.1137\/22M152776","journal-title":"Methods And Algoritms For Sci. Comput."},{"key":"2588_CR31","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1016\/bs.hna.2016.07.002","volume-title":"Handbook of Numerical Methods for Hyperbolic Problems","author":"P. Bochev","year":"2016","unstructured":"Bochev, P., Gunzburger, M.: Chapter 12 - least-squares methods for hyperbolic problems. In: Handbook of Numerical Methods for Hyperbolic Problems, vol. 17, 289\u2013317. Elsevier, Amsterdam (2016). https:\/\/doi.org\/10.1016\/bs.hna.2016.07.002"},{"issue":"5","key":"2588_CR32","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1016\/0893-6080(89)90020-8","volume":"2","author":"K. Hornik","year":"1989","unstructured":"Hornik, K., Stinchcombe, M., White, H.: Multilayer feedforward networks are universal approximators. Neural Netw 2(5), 359\u2013366 (1989). https:\/\/doi.org\/10.1016\/0893-6080(89)90020-8","journal-title":"Neural Netw"},{"issue":"5","key":"2588_CR33","doi-asserted-by":"publisher","first-page":"551","DOI":"10.1016\/0893-6080(90)90005-6","volume":"3","author":"K. Hornik","year":"1990","unstructured":"Hornik, K., Stinchcombe, M., White, H.: Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks. Neural Netw 3(5), 551\u2013560 (1990). https:\/\/doi.org\/10.1016\/0893-6080(90)90005-6","journal-title":"Neural Netw"},{"key":"2588_CR34","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40304-018-0127-z","volume":"6","author":"W. E","year":"2018","unstructured":"E, W., Yu, B.: The deep ritz method: a deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat. 6, 1\u201312 (2018). https:\/\/doi.org\/10.1007\/s40304-018-0127-z","journal-title":"Commun. Math. Stat."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02588-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-025-02588-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02588-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T15:10:47Z","timestamp":1759331447000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-025-02588-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,4]]},"references-count":34,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["2588"],"URL":"https:\/\/doi.org\/10.1007\/s12190-025-02588-9","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,4]]},"assertion":[{"value":"22 March 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 June 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 June 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 July 2025","order":4,"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 no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}