{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T20:25:38Z","timestamp":1783628738570,"version":"3.55.0"},"reference-count":74,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Computer Physics Communications"],"published-print":{"date-parts":[[2026,10]]},"DOI":"10.1016\/j.cpc.2026.110255","type":"journal-article","created":{"date-parts":[[2026,6,6]],"date-time":"2026-06-06T04:23:47Z","timestamp":1780719827000},"page":"110255","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":1,"special_numbering":"C","title":["PINNIES: An efficient physics-informed neural network framework for integral operator problems"],"prefix":"10.1016","volume":"327","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9505-819X","authenticated-orcid":false,"given":"Alireza Afzal","family":"Aghaei","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mahdi Movahedian","family":"Moghaddam","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kourosh","family":"Parand","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.cpc.2026.110255_bib0001","series-title":"Fractional Calculus and Waves in Linear Viscoelasticity: an Introduction to Mathematical Models","author":"Mainardi","year":"2022"},{"key":"10.1016\/j.cpc.2026.110255_bib0002","series-title":"Mathematical Modeling","author":"Meerschaert","year":"2013"},{"key":"10.1016\/j.cpc.2026.110255_bib0003","series-title":"Linear and Nonlinear Integral Equations","volume":"vol. 639","author":"Wazwaz","year":"2011"},{"key":"10.1016\/j.cpc.2026.110255_bib0004","series-title":"Optimal Control","author":"Lewis","year":"2012"},{"key":"10.1016\/j.cpc.2026.110255_bib0005","series-title":"Inverse Problems: Tikhonov Theory and Algorithms","volume":"vol. 22","author":"Ito","year":"2014"},{"key":"10.1016\/j.cpc.2026.110255_bib0006","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."},{"issue":"153","key":"10.1016\/j.cpc.2026.110255_bib0007","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.cpc.2026.110255_bib0008","series-title":"Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications","author":"Podlubny","year":"1998"},{"key":"10.1016\/j.cpc.2026.110255_bib0009","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1007\/s00521-010-0489-y","article-title":"A neural network approach for solving Fredholm integral equations of the second kind","volume":"21","author":"Effati","year":"2012","journal-title":"Neural Comput. Appl."},{"key":"10.1016\/j.cpc.2026.110255_bib0010","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1016\/j.asoc.2014.10.036","article-title":"Artificial neural networks based modeling for solving Volterra integral equations system","volume":"27","author":"Jafarian","year":"2015","journal-title":"Appl. Soft Comput."},{"key":"10.1016\/j.cpc.2026.110255_bib0011","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1016\/j.ins.2022.09.017","article-title":"Extended artificial neural networks approach for solving two-dimensional fractional-order Volterra-type integro-differential equations","volume":"612","author":"Saneifard","year":"2022","journal-title":"Inf. Sci."},{"issue":"1","key":"10.1016\/j.cpc.2026.110255_bib0012","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1186\/s13661-023-01762-x","article-title":"An application of artificial neural networks for solving fractional higher-order linear integro-differential equations","volume":"2023","author":"Allahviranloo","year":"2023","journal-title":"Bound. Value Probl."},{"key":"10.1016\/j.cpc.2026.110255_bib0013","doi-asserted-by":"crossref","unstructured":"M. Saadat, D. Mangal, S. Jamali, UniFIDES: universal fractional integro-differential equation solvers,(2024) arXiv prepr. arXiv: 2407.01848. 10.2139\/ssrn.4907999.","DOI":"10.2139\/ssrn.4907999"},{"issue":"1","key":"10.1016\/j.cpc.2026.110255_bib0014","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1216\/jie.2024.36.23","article-title":"Deep neural network solutions for oscillatory Fredholm integral equations","volume":"36","author":"Jiang","year":"2024","journal-title":"J. Integral Equ. Appl."},{"issue":"1","key":"10.1016\/j.cpc.2026.110255_bib0015","first-page":"91","article-title":"Numerical solution of fuzzy linear Fredholm integro-differential equation by fuzzy neural network","volume":"11","author":"Mosleh","year":"2014","journal-title":"Iran. J. Fuzzy Syst."},{"key":"10.1016\/j.cpc.2026.110255_bib0016","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40096-014-0117-6","article-title":"Utilizing artificial neural network approach for solving two-dimensional integral equations","volume":"8","author":"Asady","year":"2014","journal-title":"Math. Sci."},{"key":"10.1016\/j.cpc.2026.110255_bib0017","series-title":"Fractional Volterra Integral Equations: A Neural Network Approach","volume":"vol. 1","author":"Martire","year":"2022"},{"key":"10.1016\/j.cpc.2026.110255_bib0018","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2023.116012","article-title":"BINN: a deep learning approach for computational mechanics problems based on boundary integral equations","volume":"410","author":"Sun","year":"2023","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"7","key":"10.1016\/j.cpc.2026.110255_bib0019","doi-asserted-by":"crossref","first-page":"732","DOI":"10.1080\/00207160.2024.2374820","article-title":"E-PINN: extended physics informed neural network for the forward and inverse problems of high-order nonlinear integro-differential equations","volume":"101","author":"Zhang","year":"2024","journal-title":"Int. J. Comput. Math."},{"key":"10.1016\/j.cpc.2026.110255_bib0020","doi-asserted-by":"crossref","DOI":"10.3389\/fncom.2023.1120516","article-title":"Approximate solutions to several classes of Volterra and Fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine","volume":"17","author":"Lu","year":"2023","journal-title":"Front. Comput. Neurosci."},{"key":"10.1016\/j.cpc.2026.110255_bib0021","unstructured":"Z. Liu, P. Ma, Y. Wang, W. Matusik, M. Tegmark, KAN 2.0: Kolmogorov-Arnold networks meet science, 2024. arXiv: 2408.10205 [cs.LG]."},{"key":"10.1016\/j.cpc.2026.110255_bib0022","doi-asserted-by":"crossref","first-page":"2669","DOI":"10.1007\/s11071-017-3616-9","article-title":"Nonlinear fractional optimal control problems with neural network and dynamic optimization schemes","volume":"89","author":"Ghasemi","year":"2017","journal-title":"Nonlinear Dyn."},{"issue":"1\u20134","key":"10.1016\/j.cpc.2026.110255_bib0023","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1080\/0954898X.2019.1688878","article-title":"Fractional infinite-horizon optimal control problems with a feed forward neural network scheme","volume":"30","author":"Yavari","year":"2019","journal-title":"Netw.: Comput. Neural Syst."},{"issue":"9","key":"10.1016\/j.cpc.2026.110255_bib0024","first-page":"620","article-title":"Deep learning as optimal control problems","volume":"54","author":"Benning","year":"2021","journal-title":"IFAC-Pap."},{"key":"10.1016\/j.cpc.2026.110255_bib0025","unstructured":"K.-M. Na, C.-H. Lee, Physics-informed deep learning approach to solve optimal control problem, 2024, 10.2514\/6.2024-0945."},{"key":"10.1016\/j.cpc.2026.110255_bib0026","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/s11063-016-9510-5","article-title":"A neural network approach for solving a class of fractional optimal control problems","volume":"45","author":"Sabouri","year":"2017","journal-title":"Neural Process. Lett."},{"issue":"1\u20132","key":"10.1016\/j.cpc.2026.110255_bib0027","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1080\/0954898X.2023.2173817","article-title":"Solving time delay fractional optimal control problems via a Gudermannian neural network and convergence results","volume":"34","author":"Kheyrinataj","year":"2023","journal-title":"Netw.: Comput. Neural Syst."},{"key":"10.1016\/j.cpc.2026.110255_bib0028","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2022.111731","article-title":"Optimal control of PDEs using physics-informed neural networks","volume":"473","author":"Mowlavi","year":"2023","journal-title":"J. Comput. Phys."},{"issue":"1","key":"10.1016\/j.cpc.2026.110255_bib0029","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1080\/09540091.2019.1605498","article-title":"Fractional power series neural network for solving delay fractional optimal control problems","volume":"32","author":"Kheyrinataj","year":"2020","journal-title":"Connect. Sci."},{"issue":"1","key":"10.1016\/j.cpc.2026.110255_bib0030","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.cpc.2026.110255_bib0031","unstructured":"K. Zubov, Z. McCarthy, Y. Ma, F. Calisto, V. Pagliarino, S. Azeglio, L. Bottero, E. Luj\u00e1n, V. Sulzer, A. Bharambe, et al., NeuralPDE: automating physics-informed neural networks (pinns) with error approximations,(2021). arXiv prepr. arXiv: 2107.09443."},{"issue":"1","key":"10.1016\/j.cpc.2026.110255_bib0032","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3844\/jmssp.2017.1.13","article-title":"Study of fractional order integro-differential equations by using Chebyshev neural network","volume":"13","author":"Chaharborj","year":"2017","journal-title":"J. Math. Stat."},{"issue":"8","key":"10.1016\/j.cpc.2026.110255_bib0033","doi-asserted-by":"crossref","first-page":"6135","DOI":"10.1109\/TAP.2021.3070152","article-title":"Physics embedded deep neural network for solving volume integral equation: 2-D case","volume":"70","author":"Guo","year":"2021","journal-title":"IEEE Trans. Antennas Propag."},{"issue":"2","key":"10.1016\/j.cpc.2026.110255_bib0034","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s40819-022-01288-3","article-title":"Solving Fredholm integral equations using deep learning","volume":"8","author":"Guan","year":"2022","journal-title":"Int. J. Appl. Comput. Math."},{"key":"10.1016\/j.cpc.2026.110255_bib0035","article-title":"Physics-informed neural networks for the solution of electromagnetic scattering by integral equations","author":"Zhang","year":"2022","journal-title":"2022 International Applied Computational Electromagnetics Society Symposium, ACES-China 2022"},{"key":"10.1016\/j.cpc.2026.110255_bib0036","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2022.111260","article-title":"A-PINN: auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations","volume":"462","author":"Yuan","year":"2022","journal-title":"J. Comput. Phys."},{"issue":"7","key":"10.1016\/j.cpc.2026.110255_bib0037","doi-asserted-by":"crossref","first-page":"1480","DOI":"10.1080\/00207160.2023.2191746","article-title":"A feedforward neural network based on Legendre polynomial for solving linear Fredholm integro-differential equations","volume":"100","author":"Shao","year":"2023","journal-title":"Int. J. Comput. Math."},{"key":"10.1016\/j.cpc.2026.110255_bib0038","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1016\/j.enganabound.2023.08.020","article-title":"An efficient artificial neural network algorithm for solving boundary integral equations in elasticity","volume":"156","author":"Ruocco","year":"2023","journal-title":"Eng. Anal. Bound. Elem."},{"key":"10.1016\/j.cpc.2026.110255_bib0039","article-title":"A Novel Neural Network Architecture for Solving Fractional Differential Equations","author":"Firoozsalari","year":"2025","journal-title":"Comput. Methods Differ. Equ."},{"key":"10.1016\/j.cpc.2026.110255_bib0040","article-title":"Solution of integral equations by physics-informed neural networks for electromagnetic scattering","author":"Zhang","year":"2024","journal-title":"Authorea Prepr."},{"key":"10.1016\/j.cpc.2026.110255_bib0041","doi-asserted-by":"crossref","DOI":"10.1016\/j.cnsns.2024.108242","article-title":"Machine learning for nonlinear integro-differential equations with degenerate kernel scheme","volume":"138","author":"Li","year":"2024","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"10.1016\/j.cpc.2026.110255_bib0042","unstructured":"K. Georgiou, C. Siettos, A.N. Yannacopoulos, Fredholm neural networks, (2024). arXiv prepr. arXiv: 2408.09484."},{"issue":"6","key":"10.1016\/j.cpc.2026.110255_bib0043","doi-asserted-by":"crossref","first-page":"4111","DOI":"10.1007\/s00521-024-10752-8","article-title":"Non-local physics informed neural networks for forward and inverse problems containing non-local operators","volume":"37","author":"Singh","year":"2025","journal-title":"Neural Comput. Appl."},{"issue":"2","key":"10.1016\/j.cpc.2026.110255_bib0044","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/S0893-6080(02)00232-0","article-title":"Intelligent optimal control with dynamic neural networks","volume":"16","author":"Becerikli","year":"2003","journal-title":"Neural Netw."},{"issue":"3","key":"10.1016\/j.cpc.2026.110255_bib0045","doi-asserted-by":"crossref","first-page":"628","DOI":"10.1109\/TASE.2012.2198057","article-title":"Neural-network-based optimal control for a class of unknown discrete-time nonlinear systems using globalized dual heuristic programming","volume":"9","author":"Liu","year":"2012","journal-title":"IEEE Trans. Autom. Sci. Eng."},{"key":"10.1016\/j.cpc.2026.110255_bib0046","doi-asserted-by":"crossref","first-page":"2093","DOI":"10.1007\/s00521-012-1156-2","article-title":"Optimal control problem via neural networks","volume":"23","author":"Effati","year":"2013","journal-title":"Neural Comput. Appl."},{"issue":"1","key":"10.1016\/j.cpc.2026.110255_bib0047","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1007\/s11063-019-10099-5","article-title":"On infinite horizon optimal control problems with a feed forward neural network scheme","volume":"51","author":"Mortezaee","year":"2020","journal-title":"Neural Process. Lett."},{"issue":"1","key":"10.1016\/j.cpc.2026.110255_bib0048","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1038\/s41467-021-27590-0","article-title":"AI Pontryagin or how artificial neural networks learn to control dynamical systems","volume":"13","author":"B\u00f6ttcher","year":"2022","journal-title":"Nat. Commun."},{"key":"10.1016\/j.cpc.2026.110255_bib0049","article-title":"Physics-informed neural networks for PDE-constrained optimization and control","author":"Barry-Straume","year":"2022","journal-title":"Commun. Appl. Math. Comput."},{"key":"10.1016\/j.cpc.2026.110255_bib0050","unstructured":"Y. Chen, Y. Shi, B. Zhang, Optimal control via neural networks: a convex approach, International Conference on Learning Representations (2018). https:\/\/openreview.net\/forum?id=H1MW72AcK7."},{"issue":"5","key":"10.1016\/j.cpc.2026.110255_bib0051","doi-asserted-by":"crossref","first-page":"1122","DOI":"10.2514\/1.G002357","article-title":"Real-time optimal control via deep neural networks: study on landing problems","volume":"41","author":"S\u00e1nchez-S\u00e1nchez","year":"2018","journal-title":"J. Guid. Control Dyn."},{"issue":"3","key":"10.1016\/j.cpc.2026.110255_bib0052","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1007\/s00285-023-01873-0","article-title":"Optimal control by deep learning techniques and its applications on epidemic models","volume":"86","author":"Yin","year":"2023","journal-title":"J. Math. Biol."},{"key":"10.1016\/j.cpc.2026.110255_bib0053","unstructured":"Y. Dai, B. Jin, R. Sau, Z. Zhou, Solving elliptic optimal control problems via neural networks and optimality system, arXiv e-prints (2023) arXiv\u20132308. 10.1007\/s10444-025-10241-z."},{"key":"10.1016\/j.cpc.2026.110255_bib0054","doi-asserted-by":"crossref","unstructured":"R.D. Nzoyem Ngueguin, D.A.W. Barton, T. Deakin, A comparison of mesh-free differentiable programming and data-driven strategies for optimal control under PDE constraints, 2023, 10.1145\/3624062.3626078.","DOI":"10.1145\/3624062.3626078"},{"issue":"1","key":"10.1016\/j.cpc.2026.110255_bib0055","doi-asserted-by":"crossref","first-page":"C127","DOI":"10.1137\/22M154209X","article-title":"AONN: an adjoint-oriented neural network method for all-at-once solutions of parametric optimal control problems","volume":"46","author":"Yin","year":"2024","journal-title":"SIAM J. Sci. Comput."},{"key":"10.1016\/j.cpc.2026.110255_bib0056","first-page":"1","article-title":"Fractional calculus: history, definitions and applications for the engineer","author":"Loverro","year":"2004","journal-title":"Rapp. Tech. Univeristy Notre Dame: Dep. Aerosp. Mech. Eng."},{"key":"10.1016\/j.cpc.2026.110255_bib0057","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/j.cnsns.2018.04.019","article-title":"A new collection of real world applications of fractional calculus in science and engineering","volume":"64","author":"Sun","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"21","key":"10.1016\/j.cpc.2026.110255_bib0058","first-page":"1021","article-title":"Applications of fractional calculus","volume":"4","author":"Dalir","year":"2010","journal-title":"Appl. Math. Sci."},{"key":"10.1016\/j.cpc.2026.110255_bib0059","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/j.neucom.2021.10.122","article-title":"Applications of fractional calculus in computer vision: a survey","volume":"489","author":"Arora","year":"2022","journal-title":"Neurocomputing"},{"key":"10.1016\/j.cpc.2026.110255_bib0060","series-title":"Fractional Calculus with its Applications in Engineering and Technology","author":"Yang","year":"2022"},{"issue":"8","key":"10.1016\/j.cpc.2026.110255_bib0061","doi-asserted-by":"crossref","first-page":"1827","DOI":"10.1140\/epjst\/e2013-01967-y","article-title":"Fractional calculus: a survey of useful formulas","volume":"222","author":"Val\u00e9rio","year":"2013","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"10.1016\/j.cpc.2026.110255_bib0062","unstructured":"T. Taheri, A.A. Aghaei, K. Parand, Accelerating fractional PINNs using operational matrices of derivative, (2024). arXiv prepr. arXiv: 2401.14081."},{"key":"10.1016\/j.cpc.2026.110255_bib0063","article-title":"Hyperparameter optimization of orthogonal functions in the numerical solution of differential equations","author":"Afzal Aghaei","year":"2024","journal-title":"Math. Methods Appl. Sci."},{"issue":"3","key":"10.1016\/j.cpc.2026.110255_bib0064","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1137\/S0036144595294850","article-title":"Classroom note: numerical and analytical solutions of Volterra\u2019s population model","volume":"39","author":"TeBeest","year":"1997","journal-title":"SIAM Rev."},{"key":"10.1016\/j.cpc.2026.110255_bib0065","doi-asserted-by":"crossref","DOI":"10.1016\/j.asoc.2025.114512","article-title":"Symbolic solution of fractional Volterra population models via physics-informed neural networks and distributed particle swarm optimization","volume":"189","author":"Movahedian Moghaddam","year":"2026","journal-title":"Appl. Soft Comput."},{"issue":"3","key":"10.1016\/j.cpc.2026.110255_bib0066","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1016\/j.scient.2011.06.012","article-title":"Application of Chebyshev polynomials to derive efficient algorithms for the solution of optimal control problems","volume":"19","author":"Kafash","year":"2012","journal-title":"Sci. Iran."},{"issue":"7","key":"10.1016\/j.cpc.2026.110255_bib0067","doi-asserted-by":"crossref","first-page":"6469","DOI":"10.1007\/s11071-022-08177-w","article-title":"Hybrid of block-pulse functions and generalized Mott polynomials and their applications in solving delay fractional optimal control problems","volume":"111","author":"Rabiei","year":"2023","journal-title":"Nonlinear Dyn."},{"key":"10.1016\/j.cpc.2026.110255_bib0068","series-title":"Proceedings of the 32nd International Conference on Neural Information Processing Systems","first-page":"8580","article-title":"Neural tangent kernel: convergence and generalization in neural networks","author":"Jacot","year":"2018"},{"key":"10.1016\/j.cpc.2026.110255_bib0069","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2024.116805","article-title":"Residual-based attention in physics-informed neural networks","volume":"421","author":"Anagnostopoulos","year":"2024","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"7\u20138","key":"10.1016\/j.cpc.2026.110255_bib0070","doi-asserted-by":"crossref","first-page":"1806","DOI":"10.1177\/10775463211070902","article-title":"An approach to solve fractional optimal control problems via fractional-order Boubaker wavelets","volume":"29","author":"Rabiei","year":"2023","journal-title":"J. Vib. Control"},{"issue":"5\u20136","key":"10.1016\/j.cpc.2026.110255_bib0071","doi-asserted-by":"crossref","first-page":"1164","DOI":"10.1177\/10775463211059364","article-title":"Orthonormal piecewise Bernoulli functions: application for optimal control problems generated using fractional integro-differential equations","volume":"29","author":"Heydari","year":"2023","journal-title":"J. Vib. Control"},{"key":"10.1016\/j.cpc.2026.110255_bib0072","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10915-020-01213-0","article-title":"Generalized Bernoulli polynomials: solving nonlinear 2D fractional optimal control problems","volume":"83","author":"Hassani","year":"2020","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.cpc.2026.110255_bib0073","series-title":"Orthogonal Polynomials: Computation and Approximation","author":"Gautschi","year":"2004"},{"key":"10.1016\/j.cpc.2026.110255_bib0074","article-title":"Approximation Theory and Approximation Practice","author":"Trefethen","year":"2013"}],"container-title":["Computer Physics Communications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0010465526002377?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0010465526002377?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T19:58:26Z","timestamp":1783627106000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0010465526002377"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10]]},"references-count":74,"alternative-id":["S0010465526002377"],"URL":"https:\/\/doi.org\/10.1016\/j.cpc.2026.110255","relation":{},"ISSN":["0010-4655"],"issn-type":[{"value":"0010-4655","type":"print"}],"subject":[],"published":{"date-parts":[[2026,10]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"PINNIES: An efficient physics-informed neural network framework for integral operator problems","name":"articletitle","label":"Article Title"},{"value":"Computer Physics Communications","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cpc.2026.110255","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":"110255"}}