{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,5]],"date-time":"2025-07-05T15:40:07Z","timestamp":1751730007502,"version":"3.41.0"},"publisher-location":"Cham","reference-count":41,"publisher":"Springer Nature Switzerland","isbn-type":[{"type":"print","value":"9783031975561"},{"type":"electronic","value":"9783031975578"}],"license":[{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"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":[],"published-print":{"date-parts":[[2025]]},"DOI":"10.1007\/978-3-031-97557-8_24","type":"book-chapter","created":{"date-parts":[[2025,7,5]],"date-time":"2025-07-05T14:58:50Z","timestamp":1751727530000},"page":"332-346","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Physics Informed Neural Networks for\u00a0Non-stationary Material Science Problems"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5111-6981","authenticated-orcid":false,"given":"Pawe\u0142","family":"Maczuga","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6217-4274","authenticated-orcid":false,"given":"Tomasz","family":"S\u0142u\u017calec","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4872-406X","authenticated-orcid":false,"given":"\u0141ukasz","family":"Sztangret","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2915-8317","authenticated-orcid":false,"given":"Danuta","family":"Szeliga","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8426-6345","authenticated-orcid":false,"given":"Marcin","family":"\u0141o\u015b","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7766-6052","authenticated-orcid":false,"given":"Maciej","family":"Paszy\u0144ski","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,7,6]]},"reference":[{"key":"24_CR1","doi-asserted-by":"publisher","unstructured":"DPM: A novel training method for physics-informed neural networks in extrapolation. In: The Thirty-Fifth AAAI Conference on Artificial Intelligence, vol. 35. https:\/\/doi.org\/10.1609\/aaai.v35i9.16992. https:\/\/ojs.aaai.org\/index.php\/AAAI\/article\/view\/16992","DOI":"10.1609\/aaai.v35i9.16992"},{"key":"24_CR2","doi-asserted-by":"publisher","unstructured":"Alber, M., et al.: Integrating machine learning and multiscale modeling-perspectives, challenges, and opportunities in the biologica biomedical, and behavioral sciences. NPJ Digit. Med. 2 (2019). https:\/\/doi.org\/10.1038\/s41746-019-0193-y","DOI":"10.1038\/s41746-019-0193-y"},{"issue":"12","key":"24_CR3","doi-asserted-by":"publisher","first-page":"1727","DOI":"10.1007\/s10409-021-01148-1","volume":"37","author":"S Cai","year":"2021","unstructured":"Cai, S., Mao, Z., Wang, Z., Yin, M., Karniadakis, G.E.: Physics-informed neural networks (PINNs) for fluid mechanics: a review. Acta. Mech. Sin. 37(12), 1727\u20131738 (2021)","journal-title":"Acta. Mech. Sin."},{"issue":"8","key":"24_CR4","doi-asserted-by":"publisher","first-page":"11618","DOI":"10.1364\/OE.384875","volume":"28","author":"Y Chen","year":"2020","unstructured":"Chen, Y., Lu, L., Karniadakis, G.E., Dal Negro, L.: Physics-informed neural networks for inverse problems in nano-optics and metamaterials. Opt. Express 28(8), 11618\u201311633 (2020)","journal-title":"Opt. Express"},{"key":"24_CR5","doi-asserted-by":"crossref","unstructured":"Demkowicz, L.: Computing with $$hp$$-Adaptive Finite Elements, vol.\u00a01. Wiley (2006)","DOI":"10.1201\/9781420011685"},{"key":"24_CR6","doi-asserted-by":"crossref","unstructured":"Demkowicz, L., Kurtz, J., Pardo, D., Paszynski, M., Rachowicz, W., Zdunek, A.: Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume II Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications, 1st edn. Chapman and Hall\/CRC (2007)","DOI":"10.1201\/9781420011692"},{"key":"24_CR7","doi-asserted-by":"publisher","unstructured":"Geneva, N., Zabaras, N.: Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks. J. Comput. Phys. 403 (2020). https:\/\/doi.org\/10.1016\/j.jcp.2019.109056","DOI":"10.1016\/j.jcp.2019.109056"},{"key":"24_CR8","doi-asserted-by":"publisher","unstructured":"Goik, D., Jopek, K., Paszy\u0144ski, M., Lenharth, A., Nguyen, D., Pingali, K.: Graph grammar based multi-thread multi-frontal direct solver with Galois scheduler. Procedia Comput. Sci. 29, 960\u2013969 (2014). https:\/\/doi.org\/10.1016\/j.procs.2014.05.086. https:\/\/www.sciencedirect.com\/science\/article\/pii\/S1877050914002634, 2014 International Conference on Computational Science","DOI":"10.1016\/j.procs.2014.05.086"},{"key":"24_CR9","doi-asserted-by":"publisher","unstructured":"Goswami, S., Anitescu, C., Chakraborty, S., Rabczuk, T.: Transfer learning enhanced physics informed neural network for phase-field modeling of fracture. Theoret. Appl. Fract. Mach. 106 (2020). https:\/\/doi.org\/10.1016\/j.tafmec.2019.102447","DOI":"10.1016\/j.tafmec.2019.102447"},{"key":"24_CR10","doi-asserted-by":"publisher","unstructured":"Guermond, J., Minev, P.: A new class of massively parallel direction splitting for the incompressible Navier\u2013Stokes equations. Comput. Methods Appl. Mech. Eng. 200(23), 2083\u20132093 (2011). https:\/\/doi.org\/10.1016\/j.cma.2011.02.007. https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0045782511000429","DOI":"10.1016\/j.cma.2011.02.007"},{"key":"24_CR11","doi-asserted-by":"crossref","unstructured":"Huang, X., et al.: A universal PINNs method for solving Partial Differential Equations with a point source. In: Proceedings of the Fourteen International Joint Conference on Artificial Intelligence (IJCAI-2022), pp. 3839\u20133846 (2022)","DOI":"10.24963\/ijcai.2022\/533"},{"key":"24_CR12","doi-asserted-by":"publisher","unstructured":"Jin, H., Mattheakis, M., Protopapas, P.: Physics-informed neural networks for quantum eigenvalue problems. In: 2022 International Joint Conference on Neural Networks (IJCNN), pp.\u00a01\u20138 (2022). https:\/\/doi.org\/10.1109\/IJCNN55064.2022.9891944","DOI":"10.1109\/IJCNN55064.2022.9891944"},{"key":"24_CR13","doi-asserted-by":"publisher","unstructured":"Kharazmi, E., Zhang, Z., Karniadakis, G.E.: hp-VPINNs: variational physics-informed neural networks with domain decomposition. Comput. Methods Appl. Mech. Eng. 374, 113547 (2021). https:\/\/doi.org\/10.1016\/j.cma.2020.113547. https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0045782520307325","DOI":"10.1016\/j.cma.2020.113547"},{"key":"24_CR14","unstructured":"Kingma, D.P., Ba, J.: Adam: a method for stochastic optimization. CoRR abs\/1412.6980 (2014). https:\/\/api.semanticscholar.org\/CorpusID:6628106"},{"key":"24_CR15","doi-asserted-by":"publisher","unstructured":"Kissas, G., Yang, Y., Hwuang, E., Witschey, W.R., Detre, J.A., Perdikaris, P.: Machine learning in cardiovascular flows modeling: predicting arterial blood pressure from non-invasive 4d flow MRI data using physics-informed neural networks. Comput. Methods Appl. Mech. Eng. 358 (2020). https:\/\/doi.org\/10.1016\/j.cma.2019.112623","DOI":"10.1016\/j.cma.2019.112623"},{"key":"24_CR16","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1017\/jfm.2016.615","volume":"807","author":"J Ling","year":"2016","unstructured":"Ling, J., Kurzawski, A., Templeton, J.: Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fuild Mech. 807, 155\u2013166 (2016). https:\/\/doi.org\/10.1017\/jfm.2016.615","journal-title":"J. Fuild Mech."},{"key":"24_CR17","doi-asserted-by":"publisher","unstructured":"Lu, L., Meng, X., Mao, Z., Karniadakis, G.E.: DeepXDE: a deep learning library for solving differential equations. SIAM Rev. 63(1), 208\u2013228 (2021). https:\/\/doi.org\/10.1137\/19M1274067","DOI":"10.1137\/19M1274067"},{"issue":"6","key":"24_CR18","doi-asserted-by":"publisher","first-page":"B1105","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. SIAM J. Sci. Comput. 43(6), B1105\u2013B1132 (2021). https:\/\/doi.org\/10.1137\/21M1397908","journal-title":"SIAM J. Sci. Comput."},{"key":"24_CR19","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1007\/978-3-031-35995-8_6","volume-title":"Computational Science - ICCS 2023","author":"P Maczuga","year":"2023","unstructured":"Maczuga, P., Paszy\u0144ski, M.: Influence of activation functions on the convergence of physics-informed neural networks for 1d wave equation. In: Miky\u0161ka, J., de Mulatier, C., Paszynski, M., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M. (eds.) Computational Science - ICCS 2023, pp. 74\u201388. Springer, Cham (2023)"},{"key":"24_CR20","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2019.112789","volume":"360","author":"Z Mao","year":"2020","unstructured":"Mao, Z., Jagtap, A.D., Karniadakis, G.E.: Physics-informed neural networks for high-speed flows. Comput. Methods Appl. Mech. Eng. 360, 112789 (2020)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"24_CR21","unstructured":"Marsden, J., Hughes, T.: Mathematical Foundations of Elasticity. Dover Publications, Inc. (1983)"},{"issue":"2","key":"24_CR22","doi-asserted-by":"publisher","first-page":"981","DOI":"10.1093\/imanum\/drab032","volume":"42","author":"S Mishra","year":"2022","unstructured":"Mishra, S., Molinaro, R.: Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs. IMA J. Numer. Anal. 42(2), 981\u20131022 (2022)","journal-title":"IMA J. Numer. Anal."},{"key":"24_CR23","doi-asserted-by":"publisher","unstructured":"Nellikkath, R., Chatzivasileiadis, S.: Physics-informed neural networks for minimising worst-case violations in DC optimal power flow. In: 2021 IEEE International Conference on Communications, Control, and Computing Technologies for Smart Grids (SmartGridComm), pp. 419\u2013424 (2021). https:\/\/doi.org\/10.1109\/SmartGridComm51999.2021.9632308","DOI":"10.1109\/SmartGridComm51999.2021.9632308"},{"key":"24_CR24","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"604","DOI":"10.1007\/978-3-540-69389-5_68","volume-title":"Computational Science \u2013 ICCS 2008","author":"A Paszy\u0144ska","year":"2008","unstructured":"Paszy\u0144ska, A., Paszy\u0144ski, M., Grabska, E.: Graph transformations for modeling hp-adaptive finite element method with triangular elements. In: Bubak, M., van Albada, G.D., Dongarra, J., Sloot, P. (eds.) ICCS 2008. LNCS, vol. 5103, pp. 604\u2013613. Springer, Heidelberg (2008). https:\/\/doi.org\/10.1007\/978-3-540-69389-5_68"},{"key":"24_CR25","doi-asserted-by":"crossref","unstructured":"Paszy\u0144ska, A., et al.: Quasi-optimal elimination trees for 2d grids with singularities. Sci. Program. (1), 303024 (2015)","DOI":"10.1155\/2015\/303024"},{"key":"24_CR26","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"114","DOI":"10.1007\/978-3-030-77961-0_11","volume-title":"Computational Science \u2013 ICCS 2021","author":"M Paszy\u0144ski","year":"2021","unstructured":"Paszy\u0144ski, M., Grzeszczuk, R., Pardo, D., Demkowicz, L.: Deep learning driven self-adaptive Hp finite element method. In: Paszynski, M., Kranzlm\u00fcller, D., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P. (eds.) ICCS 2021. LNCS, vol. 12742, pp. 114\u2013121. Springer, Cham (2021). https:\/\/doi.org\/10.1007\/978-3-030-77961-0_11"},{"key":"24_CR27","doi-asserted-by":"publisher","first-page":"1313","DOI":"10.1007\/978-3-540-68111-3_139","volume-title":"Parallel Processing and Applied Mathematics","author":"M Paszy\u0144ski","year":"2008","unstructured":"Paszy\u0144ski, M., Paszy\u0144ska, A.: Graph transformations for modeling parallel hp-adaptive finite element method. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Wasniewski, J. (eds.) Parallel Processing and Applied Mathematics, pp. 1313\u20131322. Springer, Heidelberg (2008)"},{"key":"24_CR28","unstructured":"Peng, W., Zhang, J., Zhou, W., Zhao, X., Yao, W., Chen, X.: IDRLNet: a physics-informed neural network library (2021). https:\/\/arxiv.org\/abs\/2107.04320"},{"key":"24_CR29","unstructured":"Qin, S., Li, M., Xu, T., Dong, S.: RAR-PINN algorithm for the data-driven vector-soliton solutions and parameter discovery of coupled nonlinear equations. ArXiv abs\/2205.10230 (2022). https:\/\/api.semanticscholar.org\/CorpusID:248965018"},{"key":"24_CR30","doi-asserted-by":"publisher","unstructured":"Raissi, M., Perdikaris, P., Karniadakis, G.: 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. https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0021999118307125","DOI":"10.1016\/j.jcp.2018.10.045"},{"key":"24_CR31","doi-asserted-by":"crossref","unstructured":"Rasht-Behesht, M., Huber, C., Shukla, K., Karniadakis, G.E.: Physics-informed neural networks (PINNs) for wave propagation and full waveform inversions. J. Geophys. Res. Solid Earth 127(5), e2021JB023120 (2022)","DOI":"10.1029\/2021JB023120"},{"key":"24_CR32","unstructured":"Shin, Y., Darbon, J., Karniadakis, G.E.: On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs. Commun. Comput. Phys. (2020). https:\/\/api.semanticscholar.org\/CorpusID:225054225"},{"key":"24_CR33","doi-asserted-by":"crossref","unstructured":"Sun, F., Liu, Y., Sun, H.: Physics-informed spline learning for nonlinear dynamics discovery. In: Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence (IJCAI 2021), pp. 2054\u20132061 (2021)","DOI":"10.24963\/ijcai.2021\/283"},{"key":"24_CR34","doi-asserted-by":"publisher","unstructured":"Sun, L., Gao, H., Pan, S., Wang, J.X.: Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data. Comput. Methods Appl. Mech. Eng. 361 (2020). https:\/\/doi.org\/10.1016\/j.cma.2019.112732","DOI":"10.1016\/j.cma.2019.112732"},{"key":"24_CR35","unstructured":"Taylor, C., Hughes, T.: Finite Element Programming of the Navier-Stokes Equations. Fluid Mechanics Its Applications. Pineridge Press (1981). https:\/\/books.google.pl\/books?id=wo0eAQAAIAAJ"},{"key":"24_CR36","doi-asserted-by":"publisher","unstructured":"Wandel, N., Weinmann, M., Neidlin, M., Klein, R.: Spline-PINN: approaching PDEs without data using fast, physics-informed Hermite-spline CNNs. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 36, no. 8, pp. 8529\u20138538 (2022). https:\/\/doi.org\/10.1609\/aaai.v36i8.20830. https:\/\/ojs.aaai.org\/index.php\/AAAI\/article\/view\/20830","DOI":"10.1609\/aaai.v36i8.20830"},{"key":"24_CR37","doi-asserted-by":"publisher","unstructured":"Wang, S., Yu, X., Perdikaris, P.: When and why PINNs fail to train: a neural tangent kernel perspective. J. Comput. Phys. 449, 110768 (2022). https:\/\/doi.org\/10.1016\/j.jcp.2021.110768. https:\/\/www.sciencedirect.com\/science\/article\/pii\/S002199912100663X","DOI":"10.1016\/j.jcp.2021.110768"},{"key":"24_CR38","doi-asserted-by":"publisher","first-page":"136","DOI":"10.1016\/j.jcp.2019.05.027","volume":"394","author":"Y Yang","year":"2019","unstructured":"Yang, Y., Perdikaris, P.: Adversarial uncertainty quantification in physics-informed neural networks. J. Comput. Phys. 394, 136\u2013152 (2019). https:\/\/doi.org\/10.1016\/j.jcp.2019.05.027","journal-title":"J. Comput. Phys."},{"key":"24_CR39","doi-asserted-by":"publisher","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. https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0021999122003229","DOI":"10.1016\/j.jcp.2022.111260"},{"key":"24_CR40","doi-asserted-by":"publisher","unstructured":"\u0141o\u015b, M., Muga, I., Mu\u00f1oz-Matute, J., Paszy\u0144ski, M.: Isogeometric residual minimization (IGRM) for non-stationary stokes and Navier\u2013Stokes problems. Comput. Math. Appl. 95, 200\u2013214 (2021). https:\/\/doi.org\/10.1016\/j.camwa.2020.11.013. https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0898122120304417, recent Advances in Least-Squares and Discontinuous Petrov\u2013Galerkin Finite Element Methods","DOI":"10.1016\/j.camwa.2020.11.013"},{"key":"24_CR41","doi-asserted-by":"publisher","unstructured":"\u0141o\u015b, M., Paszy\u0144ski, M.: Parallel shared-memory open-source code for simulations of transient problems using isogeometric analysis, implicit direction splitting and residual minimization (IGA-ADS-RM). Adv. Eng. Softw. 196, 103723 (2024). https:\/\/doi.org\/10.1016\/j.advengsoft.2024.103723. https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0965997824001303","DOI":"10.1016\/j.advengsoft.2024.103723"}],"container-title":["Lecture Notes in Computer Science","Computational Science \u2013 ICCS 2025 Workshops"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-031-97557-8_24","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,5]],"date-time":"2025-07-05T14:58:53Z","timestamp":1751727533000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-031-97557-8_24"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025]]},"ISBN":["9783031975561","9783031975578"],"references-count":41,"URL":"https:\/\/doi.org\/10.1007\/978-3-031-97557-8_24","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"type":"print","value":"0302-9743"},{"type":"electronic","value":"1611-3349"}],"subject":[],"published":{"date-parts":[[2025]]},"assertion":[{"value":"6 July 2025","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"ICCS","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"International Conference on Computational Science","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Singapore","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Singapore","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2025","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"7 July 2025","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"9 July 2025","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"25","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"iccs-computsci2025","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/www.iccs-meeting.org\/iccs2025\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}