{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T18:40:32Z","timestamp":1778092832413,"version":"3.51.4"},"reference-count":67,"publisher":"Public Library of Science (PLoS)","issue":"3","license":[{"start":{"date-parts":[[2024,3,12]],"date-time":"2024-03-12T00:00:00Z","timestamp":1710201600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["R01HL154150"],"award-info":[{"award-number":["R01HL154150"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000006","name":"Office of Naval Research","doi-asserted-by":"publisher","award":["N00014-22-1-2795"],"award-info":[{"award-number":["N00014-22-1-2795"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.ploscompbiol.org"],"crossmark-restriction":false},"short-container-title":["PLoS Comput Biol"],"abstract":"<jats:p>Discovering mathematical equations that govern physical and biological systems from observed data is a fundamental challenge in scientific research. We present a new physics-informed framework for parameter estimation and missing physics identification (gray-box) in the field of Systems Biology. The proposed framework\u2014named AI-Aristotle\u2014combines the eXtreme Theory of Functional Connections (X-TFC) domain-decomposition and Physics-Informed Neural Networks (PINNs) with symbolic regression (SR) techniques for parameter discovery and gray-box identification. We test the accuracy, speed, flexibility, and robustness of AI-Aristotle based on two benchmark problems in Systems Biology: a pharmacokinetics drug absorption model and an ultradian endocrine model for glucose-insulin interactions. We compare the two machine learning methods (X-TFC and PINNs), and moreover, we employ two different symbolic regression techniques to cross-verify our results. To test the performance of AI-Aristotle, we use sparse synthetic data perturbed by uniformly distributed noise. More broadly, our work provides insights into the accuracy, cost, scalability, and robustness of integrating neural networks with symbolic regressors, offering a comprehensive guide for researchers tackling gray-box identification challenges in complex dynamical systems in biomedicine and beyond.<\/jats:p>","DOI":"10.1371\/journal.pcbi.1011916","type":"journal-article","created":{"date-parts":[[2024,3,12]],"date-time":"2024-03-12T17:21:23Z","timestamp":1710264083000},"page":"e1011916","update-policy":"https:\/\/doi.org\/10.1371\/journal.pcbi.corrections_policy","source":"Crossref","is-referenced-by-count":44,"title":["AI-Aristotle: A physics-informed framework for systems biology gray-box identification"],"prefix":"10.1371","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2485-8987","authenticated-orcid":true,"given":"Nazanin","family":"Ahmadi Daryakenari","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2285-3074","authenticated-orcid":true,"given":"Mario","family":"De Florio","sequence":"additional","affiliation":[]},{"given":"Khemraj","family":"Shukla","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9713-7120","authenticated-orcid":true,"given":"George Em","family":"Karniadakis","sequence":"additional","affiliation":[]}],"member":"340","published-online":{"date-parts":[[2024,3,12]]},"reference":[{"key":"pcbi.1011916.ref001","doi-asserted-by":"crossref","unstructured":"A. Tarantola, Inverse problem theory and methods for model parameter estimation, SIAM, 2005.","DOI":"10.1137\/1.9780898717921"},{"key":"pcbi.1011916.ref002","doi-asserted-by":"crossref","unstructured":"Rico-Martinez R, Anderson J, Kevrekidis I. Continuous-time nonlinear signal processing: a neural network based approach for gray box identification. In: Proceedings of IEEE Workshop on Neural Networks for Signal Processing. IEEE; 1994. p. 596\u2013605.","DOI":"10.1109\/NNSP.1994.366006"},{"issue":"15","key":"pcbi.1011916.ref003","doi-asserted-by":"crossref","first-page":"3932","DOI":"10.1073\/pnas.1517384113","article-title":"Discovering governing equations from data by sparse identification of nonlinear dynamical systems","volume":"113","author":"SL Brunton","year":"2016","journal-title":"Proceedings of the national academy of sciences"},{"issue":"1","key":"pcbi.1011916.ref004","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","article-title":"Regression shrinkage and selection via the lasso","volume":"58","author":"R Tibshirani","year":"1996","journal-title":"Journal of the Royal Statistical Society: Series B (Methodological)"},{"issue":"4","key":"pcbi.1011916.ref005","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1109\/TIT.2006.871582","article-title":"Compressed sensing","volume":"52","author":"DL Donoho","year":"2006","journal-title":"IEEE Transactions on information theory"},{"issue":"16","key":"pcbi.1011916.ref006","doi-asserted-by":"crossref","first-page":"eaay2631","DOI":"10.1126\/sciadv.aay2631","article-title":"AI Feynman: A physics-inspired method for symbolic regression","volume":"6","author":"SM Udrescu","year":"2020","journal-title":"Science Advances"},{"issue":"1","key":"pcbi.1011916.ref007","doi-asserted-by":"crossref","first-page":"1777","DOI":"10.1038\/s41467-023-37236-y","article-title":"Combining data and theory for derivable scientific discovery with AI-Descartes","volume":"14","author":"C Cornelio","year":"2023","journal-title":"Nature Communications"},{"key":"pcbi.1011916.ref008","doi-asserted-by":"crossref","DOI":"10.7551\/mitpress\/14179.001.0001","volume-title":"The art of abduction","author":"I Douven","year":"2022"},{"key":"pcbi.1011916.ref009","unstructured":"Brol\u00f8s KR, Machado MV, Cave C, Kasak J, Stentoft-Hansen V, Batanero VG, et al. An approach to symbolic regression using feyn. arXiv preprint arXiv:210405417. 2021;."},{"key":"pcbi.1011916.ref010","unstructured":"Wilstrup C, Kasak J. Symbolic regression outperforms other models for small data sets. arXiv preprint arXiv:210315147. 2021;."},{"issue":"15","key":"pcbi.1011916.ref011","doi-asserted-by":"crossref","first-page":"3749","DOI":"10.1093\/bioinformatics\/btac405","article-title":"Identifying interactions in omics data for clinical biomarker discovery using symbolic regression","volume":"38","author":"NJ Christensen","year":"2022","journal-title":"Bioinformatics"},{"key":"pcbi.1011916.ref012","doi-asserted-by":"crossref","unstructured":"Andras P. Random projection neural network approximation. In: 2018 International Joint Conference on Neural Networks (IJCNN). IEEE; 2018. p. 1\u20138.","DOI":"10.1109\/IJCNN.2018.8489215"},{"key":"pcbi.1011916.ref013","doi-asserted-by":"crossref","unstructured":"Wouter F. Schmidt, Martin A. Kraaijveld, Robert PW. Duin, and others, Feed forward neural networks with random weights, in International conference on pattern recognition, pages 1\u20131, 1992, organization = IEEE Computer Society Press.","DOI":"10.1109\/ICPR.1992.201708"},{"key":"pcbi.1011916.ref014","doi-asserted-by":"crossref","unstructured":"Boris Igelnik and Yoh-Han Pao, Stochastic choice of basis functions in adaptive function approximation and the functional-link net, IEEE transactions on Neural Networks, volume 6, number 6, pages 1320\u20131329, 1995, publisher = IEEE.","DOI":"10.1109\/72.471375"},{"issue":"6","key":"pcbi.1011916.ref015","doi-asserted-by":"crossref","DOI":"10.1063\/5.0086649","article-title":"Physics-informed neural networks and functional interpolation for stiff chemical kinetics","volume":"32","author":"M De Florio","year":"2022","journal-title":"Chaos: An Interdisciplinary Journal of Nonlinear Science"},{"key":"pcbi.1011916.ref016","doi-asserted-by":"crossref","unstructured":"Gianluca Fabiani, Evangelos Galaris, Lucia Russo, and Constantinos Siettos, Parsimonious physics-informed random projection neural networks for initial value problems of ODEs and index-1 DAEs, Chaos: An Interdisciplinary Journal of Nonlinear Science, volume 33, number 4, 2023, publisher = AIP Publishing.","DOI":"10.1063\/5.0135903"},{"key":"pcbi.1011916.ref017","doi-asserted-by":"crossref","unstructured":"Evangelos Galaris, Gianluca Fabiani, Ioannis Gallos, Ioannis Kevrekidis, and Constantinos Siettos, Numerical bifurcation analysis of PDEs from lattice Boltzmann model simulations: a parsimonious machine learning approach, Journal of Scientific Computing, volume 92, number 2, pages 34, 2022, publisher = Springer.","DOI":"10.1007\/s10915-022-01883-y"},{"key":"pcbi.1011916.ref018","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1016\/j.neucom.2020.10.062","article-title":"Interaction-transformation symbolic regression with extreme learning machine","volume":"423","author":"FO de Franca","year":"2021","journal-title":"Neurocomputing"},{"key":"pcbi.1011916.ref019","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.ins.2018.02.040","article-title":"A greedy search tree heuristic for symbolic regression","volume":"442","author":"FO de Fran\u00e7a","year":"2018","journal-title":"Information Sciences"},{"issue":"2","key":"pcbi.1011916.ref020","doi-asserted-by":"crossref","first-page":"1565","DOI":"10.1007\/s11063-021-10465-2","article-title":"Symbolic regression based extreme learning machine models for system identification","volume":"53","author":"BE K\u00f6kt\u00fcrk-G\u00fczel","year":"2021","journal-title":"Neural Processing Letters"},{"key":"pcbi.1011916.ref021","unstructured":"Mario De Florio, Ioannis G Kevrekidis, George Em Karniadakis, AI-Lorenz: A physics-data-driven framework for black-box and gray-box identification of chaotic systems with symbolic regression, arXiv preprint arXiv:2312.14237, 2023."},{"issue":"2","key":"pcbi.1011916.ref022","doi-asserted-by":"crossref","first-page":"025305","DOI":"10.1103\/PhysRevE.107.025305","article-title":"Black and gray box learning of amplitude equations: Application to phase field systems","volume":"107","author":"FP Kemeth","year":"2023","journal-title":"Physical Review E"},{"issue":"6","key":"pcbi.1011916.ref023","doi-asserted-by":"crossref","first-page":"2611","DOI":"10.1021\/acs.iecr.9b04507","article-title":"Partial observations and conservation laws: Gray-box modeling in biotechnology and optogenetics","volume":"59","author":"RJ Lovelett","year":"2019","journal-title":"Industrial & Engineering Chemistry Research"},{"issue":"23","key":"pcbi.1011916.ref024","doi-asserted-by":"crossref","first-page":"3209","DOI":"10.1093\/bioinformatics\/btm510","article-title":"Estimating parameters and hidden variables in non-linear state-space models based on ODEs for biological networks inference","volume":"23","author":"M Quach","year":"2007","journal-title":"Bioinformatics"},{"issue":"13","key":"pcbi.1011916.ref025","doi-asserted-by":"crossref","first-page":"2314","DOI":"10.1093\/bioinformatics\/bty089","article-title":"ShinyKGode: an interactive application for ODE parameter inference using gradient matching","volume":"34","author":"J Wandy","year":"2018","journal-title":"Bioinformatics"},{"issue":"24","key":"pcbi.1011916.ref026","doi-asserted-by":"crossref","first-page":"4266","DOI":"10.1093\/bioinformatics\/bty514","article-title":"Hierarchical optimization for the efficient parametrization of ODE models","volume":"34","author":"C Loos","year":"2018","journal-title":"Bioinformatics"},{"key":"pcbi.1011916.ref027","doi-asserted-by":"crossref","unstructured":"Seungjoon Lee, Yorgos M. Psarellis, Constantinos I. Siettos, and Ioannis G. Kevrekidis, Learning black-and gray-box chemotactic PDEs\/closures from agent-based Monte Carlo simulation data, Journal of Mathematical Biology, volume 87, number 1, pages 15, 2023, publisher = Springer.","DOI":"10.1007\/s00285-023-01946-0"},{"key":"pcbi.1011916.ref028","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":"M Raissi","year":"2019","journal-title":"Journal of Computational physics"},{"issue":"11","key":"pcbi.1011916.ref029","doi-asserted-by":"crossref","first-page":"e1007575","DOI":"10.1371\/journal.pcbi.1007575","article-title":"Systems biology informed deep learning for inferring parameters and hidden dynamics","volume":"16","author":"A Yazdani","year":"2020","journal-title":"PLoS computational biology"},{"key":"pcbi.1011916.ref030","doi-asserted-by":"crossref","unstructured":"Daneker M, Zhang Z, Karniadakis GE, Lu L. Systems biology: Identifiability analysis and parameter identification via systems-biology-informed neural networks. In: Computational Modeling of Signaling Networks. Springer; 2023. p. 87\u2013105.","DOI":"10.1007\/978-1-0716-3008-2_4"},{"key":"pcbi.1011916.ref031","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1016\/j.neucom.2021.06.015","article-title":"Extreme theory of functional connections: A fast physics-informed neural network method for solving ordinary and partial differential equations","volume":"457","author":"E Schiassi","year":"2021","journal-title":"Neurocomputing"},{"key":"pcbi.1011916.ref032","unstructured":"Virgolin M, Pissis SP. Symbolic regression is np-hard. arXiv preprint arXiv:220701018. 2022;."},{"key":"pcbi.1011916.ref033","unstructured":"Cranmer M. Interpretable machine learning for science with PySR and SymbolicRegression. jl. arXiv preprint arXiv:230501582. 2023;."},{"key":"pcbi.1011916.ref034","unstructured":"Stephens T. gplearn: Genetic programming in python, with a scikitlearn inspired api. [Online]. Available: https:\/\/github.com\/trevorstephens\/gplearn; 2015."},{"key":"pcbi.1011916.ref035","doi-asserted-by":"crossref","unstructured":"Kiyani E, Shukla K, Karniadakis GE, Karttunen M. A Framework Based on Symbolic Regression Coupled with eXtended Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion from Data. arXiv preprint arXiv:230510706. 2023;.","DOI":"10.1016\/j.cma.2023.116258"},{"key":"pcbi.1011916.ref036","doi-asserted-by":"crossref","first-page":"116647","DOI":"10.1016\/j.cma.2023.116647","article-title":"Discovering a reaction\u2013diffusion model for Alzheimer\u2019s disease by combining PINNs with symbolic regression","volume":"419","author":"Zhen Zhang","year":"2024","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"pcbi.1011916.ref037","unstructured":"Barnes B, Fulford GR. Mathematical modelling with case studies: a differential equations approach using Maple and MATLAB. vol. 25. CRC Press; 2011."},{"issue":"5","key":"pcbi.1011916.ref038","doi-asserted-by":"crossref","first-page":"E801","DOI":"10.1152\/ajpendo.1991.260.5.E801","article-title":"Computer model for mechanisms underlying ultradian oscillations of insulin and glucose","volume":"260","author":"J Sturis","year":"1991","journal-title":"American Journal of Physiology-Endocrinology And Metabolism"},{"issue":"6","key":"pcbi.1011916.ref039","doi-asserted-by":"crossref","first-page":"e96443","DOI":"10.1371\/journal.pone.0096443","article-title":"Dynamical phenotyping: using temporal analysis of clinically collected physiologic data to stratify populations","volume":"9","author":"DJ Albers","year":"2014","journal-title":"PloS one"},{"issue":"4","key":"pcbi.1011916.ref040","doi-asserted-by":"crossref","first-page":"57","DOI":"10.3390\/math5040057","article-title":"The theory of connections: Connecting points","volume":"5","author":"D Mortari","year":"2017","journal-title":"Mathematics"},{"issue":"3","key":"pcbi.1011916.ref041","doi-asserted-by":"crossref","first-page":"65","DOI":"10.3390\/mca26030065","article-title":"Theory of functional connections applied to linear ODEs subject to integral constraints and linear ordinary integro-differential equations","volume":"26","author":"M De Florio","year":"2021","journal-title":"Mathematical and Computational Applications"},{"issue":"4","key":"pcbi.1011916.ref042","doi-asserted-by":"crossref","first-page":"48","DOI":"10.3390\/math5040048","article-title":"Least-squares solution of linear differential equations","volume":"5","author":"D Mortari","year":"2017","journal-title":"Mathematics"},{"issue":"1-3","key":"pcbi.1011916.ref043","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1016\/j.neucom.2005.12.126","article-title":"Extreme learning machine: theory and applications","volume":"70","author":"GB Huang","year":"2006","journal-title":"Neurocomputing"},{"key":"pcbi.1011916.ref044","doi-asserted-by":"crossref","first-page":"108833","DOI":"10.1016\/j.anucene.2021.108833","article-title":"Physics-informed neural networks for the point kinetics equations for nuclear reactor dynamics","volume":"vol. 167","author":"Schiassi Enrico","year":"2022","journal-title":"Annals of Nuclear Energy"},{"key":"pcbi.1011916.ref045","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.neucom.2022.05.015","article-title":"Self-adaptive loss balanced Physics-informed neural networks","volume":"496","author":"Z Xiang","year":"2022","journal-title":"Neurocomputing (Amsterdam)"},{"key":"pcbi.1011916.ref046","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2022.111722","article-title":"Self-adaptive physics-informed neural networks","volume":"474","author":"LD McClenny","year":"2023","journal-title":"Journal of Computational Physics"},{"key":"pcbi.1011916.ref047","unstructured":"Kingma DP, Ba J. Adam: A method for stochastic optimization. arXiv preprint arXiv:14126980. 2014;."},{"issue":"1","key":"pcbi.1011916.ref048","first-page":"503","article-title":"On the limited memory BFGS method for large scale optimization","volume":"45","author":"DC Liu","year":"1989","journal-title":"Mathematical programming"},{"issue":"no. 1","key":"pcbi.1011916.ref049","article-title":"Physics-informed neural networks for predicting gas flow dynamics and unknown parameters in diesel engines","volume":"vol. 13","author":"Kamaljyoti Nath","journal-title":"Scientific Reports"},{"issue":"no. 22","key":"pcbi.1011916.ref050","doi-asserted-by":"crossref","first-page":"21117","DOI":"10.1007\/s11071-023-08933-6","article-title":"Parameter estimation and modeling of nonlinear dynamical systems based on Runge\u2013Kutta physics-informed neural network","volume":"vol. 111","author":"Weida Zhai","year":"2023","journal-title":"Nonlinear Dynamics"},{"key":"pcbi.1011916.ref051","doi-asserted-by":"crossref","unstructured":"Jochen Stiasny, Samuel Chevalier, Spyros Chatzivasileiadis, Learning without data: Physics-informed neural networks for fast time-domain simulation, in 2021 IEEE International Conference on Communications, Control, and Computing Technologies for Smart Grids (SmartGridComm), pp. 438\u2013443, 2021, IEEE.","DOI":"10.1109\/SmartGridComm51999.2021.9631995"},{"key":"pcbi.1011916.ref052","unstructured":"Enrico Schiassi, Andrea D\u2019Ambrosio, Hunter Johnston, Mario De Florio, Kristofer Drozd, Roberto Furfaro, Fabio Curti, and Daniele Mortari, Physics-informed extreme theory of functional connections applied to optimal orbit transfer, in Proceedings of the AAS\/AIAA Astrodynamics Specialist Conference, Lake Tahoe, CA, USA, pages = 9\u201313, 2020."},{"issue":"no. 4","key":"pcbi.1011916.ref053","article-title":"Physics-informed neural networks for rarefied-gas dynamics: Thermal creep flow in the Bhatnagar\u2013Gross\u2013Krook approximation","volume":"vol. 33","author":"Florio Mario De","year":"2021","journal-title":"Physics of Fluids"},{"issue":"no. 3","key":"pcbi.1011916.ref054","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1007\/s00033-022-01767-z","article-title":"Physics-informed neural networks for rarefied-gas dynamics: Poiseuille flow in the BGK approximation","volume":"vol. 73","author":"Florio Mario De","year":"2022","journal-title":"Zeitschrift f\u00fcr angewandte Mathematik und Physik"},{"key":"pcbi.1011916.ref055","doi-asserted-by":"crossref","first-page":"107384","DOI":"10.1016\/j.jqsrt.2020.107384","article-title":"Solutions of Chandrasekhar\u2019s basic problem in radiative transfer via theory of functional connections","volume":"vol. 259","author":"Florio Mario De","year":"2021","journal-title":"Journal of Quantitative Spectroscopy and Radiative Transfer"},{"key":"pcbi.1011916.ref056","doi-asserted-by":"crossref","first-page":"115396","DOI":"10.1016\/j.cam.2023.115396","article-title":"Physics-Informed Neural Networks for 2nd order ODEs with sharp gradients","volume":"vol. 436","author":"Florio Mario De","year":"2024","journal-title":"Journal of Computational and Applied Mathematics"},{"issue":"no. 17","key":"pcbi.1011916.ref057","doi-asserted-by":"crossref","first-page":"2069","DOI":"10.3390\/math9172069","article-title":"Physics-informed neural networks and functional interpolation for data-driven parameters discovery of epidemiological compartmental models","volume":"vol. 9","author":"Schiassi Enrico","year":"2021","journal-title":"Mathematics"},{"key":"pcbi.1011916.ref058","doi-asserted-by":"crossref","first-page":"111731","DOI":"10.1016\/j.jcp.2022.111731","article-title":"Optimal control of PDEs using physics-informed neural networks","volume":"vol. 473","author":"Saviz Mowlavi","year":"2023","journal-title":"Journal of Computational Physics"},{"key":"pcbi.1011916.ref059","doi-asserted-by":"crossref","first-page":"113547","DOI":"10.1016\/j.cma.2020.113547","article-title":"hp-VPINNs: Variational physics-informed neural networks with domain decomposition","volume":"vol. 374","author":"Kharazmi Ehsan","year":"2021","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"pcbi.1011916.ref060","doi-asserted-by":"crossref","first-page":"110676","DOI":"10.1016\/j.jcp.2021.110676","article-title":"Physics-informed neural networks for solving forward and inverse flow problems via the Boltzmann-BGK formulation","volume":"vol. 447","author":"Lou Qin","year":"2021","journal-title":"Journal of Computational Physics"},{"issue":"no. 3","key":"pcbi.1011916.ref061","doi-asserted-by":"crossref","first-page":"B811","DOI":"10.1137\/20M1360153","article-title":"Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker\u2013Planck Equation and Physics-Informed Neural Networks","volume":"vol. 43","author":"Chen Xiaoli","year":"2021","journal-title":"SIAM Journal on Scientific Computing"},{"key":"pcbi.1011916.ref062","unstructured":"Enrico Schiassi, Andrea D\u2019Ambrosio, and Roberto Furfaro, Bellman Neural Networks for the Class of Optimal Control Problems With Integral Quadratic Cost, IEEE Transactions on Artificial Intelligence, 2022, IEEE."},{"key":"pcbi.1011916.ref063","doi-asserted-by":"crossref","first-page":"116042","DOI":"10.1016\/j.cma.2023.116042","article-title":"Physics-informed deep learning for simultaneous surrogate modeling and PDE-constrained optimization of an airfoil geometry","volume":"vol. 411","author":"Sun Yubiao","year":"2023","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"pcbi.1011916.ref064","unstructured":"S. Shekarpaz, F. Zeng, and G. Karniadakis, Splitting physics-informed neural networks for inferring the dynamics of integer-and fractional-order neuron models, arXiv preprint arXiv:2304.13205, Apr 26, 2023."},{"issue":"no. 3","key":"pcbi.1011916.ref065","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1137\/0705041","article-title":"On the construction and comparison of difference schemes","volume":"vol. 5","author":"Gilbert Strang","year":"1968","journal-title":"SIAM journal on numerical analysis"},{"issue":"no 3","key":"pcbi.1011916.ref066","first-page":"271","article-title":"Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics","volume":"vol. 47","author":"Sergei K Godunov","year":"1959","journal-title":"Matemati\u010deskij sbornik"},{"key":"pcbi.1011916.ref067","doi-asserted-by":"crossref","unstructured":"Ismail Alaoui Abdellaoui, Siamak Mehrkanoon, Symbolic regression for scientific discovery: an application to wind speed forecasting, 2021 IEEE Symposium Series on Computational Intelligence (SSCI), pages = 01\u201308, 2021, IEEE.","DOI":"10.1109\/SSCI50451.2021.9659860"}],"container-title":["PLOS Computational Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pcbi.1011916","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,14]],"date-time":"2024-11-14T05:01:34Z","timestamp":1731560494000},"score":1,"resource":{"primary":{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pcbi.1011916"}},"subtitle":[],"editor":[{"given":"Piero","family":"Fariselli","sequence":"first","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2024,3,12]]},"references-count":67,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3,12]]}},"URL":"https:\/\/doi.org\/10.1371\/journal.pcbi.1011916","relation":{},"ISSN":["1553-7358"],"issn-type":[{"value":"1553-7358","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,3,12]]}}}