{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T17:27:05Z","timestamp":1781890025586,"version":"3.54.5"},"reference-count":115,"publisher":"IOP Publishing","issue":"4","license":[{"start":{"date-parts":[[2025,11,28]],"date-time":"2025-11-28T00:00:00Z","timestamp":1764288000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2025,11,28]],"date-time":"2025-11-28T00:00:00Z","timestamp":1764288000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/iopscience.iop.org\/info\/page\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100025019","name":"Support for Pioneering Research Initiated by the Next Generation","doi-asserted-by":"crossref","award":["JPMJSP2136"],"award-info":[{"award-number":["JPMJSP2136"]}],"id":[{"id":"10.13039\/501100025019","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"crossref","award":["JP23K17807"],"award-info":[{"award-number":["JP23K17807"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100004298","name":"Secom Science and Technology Foundation","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004298","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"short-container-title":["Mach. Learn.: Sci. Technol."],"published-print":{"date-parts":[[2025,12,30]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Physics-informed neural networks (PINNs) have attracted significant attention in scientific machine learning for their capability to solve forward and inverse problems governed by partial differential equations. However, the accuracy of PINN solutions is often limited by the treatment of boundary conditions. Conventional penalty-based methods, which incorporate boundary conditions as penalty terms in the loss function, cannot guarantee exact satisfaction of the given boundary conditions and are highly sensitive to the choice of penalty parameters. This paper demonstrates that distance functions, specifically R-functions, can be leveraged to enforce boundary conditions, overcoming these limitations. R-functions provide normalized distance fields, enabling flexible representation of boundary geometries, including non-convex domains, and facilitating various types of boundary conditions. Nevertheless, distance functions alone are insufficient for accurate inverse analysis in PINNs. To address this, we propose an integrated framework that combines the normalized distance field with bias-corrected adaptive weight tuning to improve both accuracy and efficiency. Numerical results show that the proposed method provides more accurate and efficient solutions to various inverse problems than penalty-based approaches, even in the presence of non-convex geometries with complex boundary conditions. This approach offers a reliable and efficient framework for inverse analysis using PINNs, with potential applications across a wide range of engineering problems.<\/jats:p>","DOI":"10.1088\/2632-2153\/ae1b71","type":"journal-article","created":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T22:55:43Z","timestamp":1762296943000},"page":"045055","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":5,"title":["Reliable and efficient inverse analysis using physics-informed neural networks with normalized distance functions and adaptive weight tuning"],"prefix":"10.1088","volume":"6","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9538-8663","authenticated-orcid":true,"given":"Shota","family":"Deguchi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1124-2895","authenticated-orcid":true,"given":"Mitsuteru","family":"Asai","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"266","published-online":{"date-parts":[[2025,11,28]]},"reference":[{"key":"mlstae1b71bib1","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1002\/cnm.1640100303","type":"journal-article","article-title":"Neural-network-based approximations for solving partial differential equations","volume":"10","author":"Dissanayake","year":"1994","journal-title":"Commun. Numer. Methods Eng."},{"key":"mlstae1b71bib2","doi-asserted-by":"publisher","first-page":"987","DOI":"10.1109\/72.712178","type":"journal-article","article-title":"Artificial neural networks for solving ordinary and partial differential equations","volume":"9","author":"Lagaris","year":"1998","journal-title":"IEEE Trans. Neural Netw."},{"key":"mlstae1b71bib3","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1016\/S0377-0427(02)00397-7","type":"journal-article","article-title":"Novel determination of differential-equation solutions: universal approximation method","volume":"146","author":"Leephakpreeda","year":"2002","journal-title":"J. Comput. Appl. Math."},{"key":"mlstae1b71bib4","doi-asserted-by":"publisher","first-page":"1221","DOI":"10.1109\/TNN.2009.2020735","type":"journal-article","article-title":"Artificial neural network method for solution of boundary value problems with exact satisfaction of arbitrary boundary conditions","volume":"20","author":"McFall","year":"2009","journal-title":"IEEE Trans. Neural Netw."},{"key":"mlstae1b71bib5","doi-asserted-by":"publisher","first-page":"82","DOI":"10.1016\/j.inffus.2019.12.012","type":"journal-article","article-title":"Explainable artificial intelligence (XAI): concepts, taxonomies, opportunities and challenges toward responsible AI","volume":"58","author":"Barredo Arrieta","year":"2020","journal-title":"Inf. Fusion"},{"key":"mlstae1b71bib6","doi-asserted-by":"publisher","first-page":"42200","DOI":"10.1109\/ACCESS.2020.2976199","type":"journal-article","article-title":"Explainable machine learning for scientific insights and discoveries","volume":"8","author":"Roscher","year":"2020","journal-title":"IEEE Access"},{"key":"mlstae1b71bib7","doi-asserted-by":"publisher","first-page":"614","DOI":"10.1109\/TKDE.2021.3079836","type":"journal-article","article-title":"Informed machine learning\u2014a taxonomy and survey of integrating prior knowledge into learning systems","volume":"35","author":"von Rueden","year":"2023","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"mlstae1b71bib8","article-title":"Universal differential equations for scientific machine learning","author":"Rackauckas","year":"2021","type":"preprint"},{"key":"mlstae1b71bib9","doi-asserted-by":"publisher","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","type":"journal-article","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."},{"key":"mlstae1b71bib10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40304-018-0127-z","type":"journal-article","article-title":"The deep ritz method: a deep learning-based numerical algorithm for solving variational problems","volume":"6","author":"Weinan","year":"2018","journal-title":"Commun. Math. Stat."},{"key":"mlstae1b71bib11","doi-asserted-by":"publisher","first-page":"1365","DOI":"10.4208\/cicp.OA-2020-0219","type":"journal-article","article-title":"Deep nitsche method: deep ritz method with essential boundary conditions","volume":"29","author":"Liao","year":"2021","journal-title":"Commun. Comput. Phys."},{"key":"mlstae1b71bib12","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2019.112790","type":"journal-article","article-title":"An energy approach to the solution of partial differential equations in computational mechanics via machine learning: concepts, implementation and applications","volume":"362","author":"Samaniego","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"mlstae1b71bib13","article-title":"Variational physics-informed neural networks for solving partial differential equations","author":"Kharazmi","year":"2019","type":"preprint"},{"key":"mlstae1b71bib14","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2020.113547","type":"journal-article","article-title":"hp-VPINNs: variational physics-informed neural networks with domain decomposition","volume":"374","author":"Kharazmi","year":"2021","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"mlstae1b71bib15","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109409","type":"journal-article","article-title":"Weak adversarial networks for high-dimensional partial differential equations","volume":"411","author":"Zang","year":"2020","journal-title":"J. Comput. Phys."},{"key":"mlstae1b71bib16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110930","type":"journal-article","article-title":"MIM: a deep mixed residual method for solving high-order partial differential equations","volume":"452","author":"Lyu","year":"2022","journal-title":"J. Comput. Phys."},{"key":"mlstae1b71bib17","doi-asserted-by":"publisher","first-page":"22","DOI":"10.2208\/jscejj.22-15011","type":"journal-article","article-title":"Efficiency improvement of PINNs inverse analysis by extracting spatial features of data","volume":"79","author":"Deguchi","year":"2023","journal-title":"Japan. J. JSCE"},{"key":"mlstae1b71bib18","article-title":"Robustness of physics-informed neural networks to noise in sensor data","author":"Wong","year":"2022","type":"preprint"},{"key":"mlstae1b71bib19","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6501\/ad3fd3","type":"journal-article","article-title":"Physics-informed deep-learning applications to experimental fluid mechanics","volume":"35","author":"Eivazi","year":"2024","journal-title":"Meas. Sci. Technol."},{"key":"mlstae1b71bib20","doi-asserted-by":"publisher","DOI":"10.1029\/2019WR026731","type":"journal-article","article-title":"Physics-informed deep neural networks for learning parameters and constitutive relationships in subsurface flow problems","volume":"56","author":"Tartakovsky","year":"2020","journal-title":"Water Resour. Res."},{"key":"mlstae1b71bib21","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2019.112623","type":"journal-article","article-title":"Machine learning in cardiovascular flows modeling: predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks","volume":"358","author":"Kissas","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"mlstae1b71bib22","doi-asserted-by":"publisher","DOI":"10.1063\/5.0055600","type":"journal-article","article-title":"Uncovering near-wall blood flow from sparse data with physics-informed neural networks","volume":"33","author":"Arzani","year":"2021","journal-title":"Phys. Fluids"},{"key":"mlstae1b71bib23","type":"conference-proceedings","article-title":"Sobolev training for neural networks","volume":"vol 30","author":"Czarnecki","year":"2017"},{"key":"mlstae1b71bib24","doi-asserted-by":"publisher","DOI":"10.1088\/2632-2153\/ac3712","type":"journal-article","article-title":"Inverse Dirichlet weighting enables reliable training of physics informed neural networks","volume":"3","author":"Maddu","year":"2022","journal-title":"Mach. Learn.: Sci. Technol."},{"key":"mlstae1b71bib25","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.114823","type":"journal-article","article-title":"Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems","volume":"393","author":"Yu","year":"2022","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"mlstae1b71bib26","doi-asserted-by":"publisher","first-page":"1679","DOI":"10.4310\/CMS.2023.v21.n6.a11","type":"journal-article","article-title":"Sobolev training for physics-informed neural networks","volume":"21","author":"Son","year":"2023","journal-title":"Commun. Math. Sci."},{"key":"mlstae1b71bib27","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.113695","type":"journal-article","article-title":"Sobolev training of thermodynamic-informed neural networks for interpretable elasto-plasticity models with level set hardening","volume":"377","author":"Vlassis","year":"2021","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"mlstae1b71bib28","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2023.116042","type":"journal-article","article-title":"Physics-informed deep learning for simultaneous surrogate modeling and PDE-constrained optimization of an airfoil geometry","volume":"411","author":"Sun","year":"2023","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"mlstae1b71bib29","doi-asserted-by":"publisher","first-page":"1339","DOI":"10.1016\/j.jcp.2018.08.029","type":"journal-article","article-title":"DGM: a deep learning algorithm for solving partial differential equations","volume":"375","author":"Sirignano","year":"2018","journal-title":"J. Comput. Phys."},{"key":"mlstae1b71bib30","doi-asserted-by":"publisher","first-page":"A3055","DOI":"10.1137\/20M1318043","type":"journal-article","article-title":"Understanding and mitigating gradient flow pathologies in physics-informed neural networks","volume":"43","author":"Wang","year":"2021","journal-title":"SIAM J. Sci. Comput."},{"key":"mlstae1b71bib31","article-title":"An expert\u2019s guide to training physics-informed neural networks","author":"Wang","year":"2023","type":"preprint"},{"key":"mlstae1b71bib32","article-title":"Fourier features let networks learn high frequency functions in low dimensional domains","author":"Tancik","year":"2020","type":"conference-proceedings"},{"key":"mlstae1b71bib33","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.113938","type":"journal-article","article-title":"On the eigenvector bias of fourier feature networks: from regression to solving multi-scale pdes with physics-informed neural networks","volume":"384","author":"Wang","year":"2021","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"mlstae1b71bib34","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1016\/j.camwa.2022.12.008","type":"journal-article","article-title":"Physics-informed neural networks combined with polynomial interpolation to solve nonlinear partial differential equations","volume":"132","author":"Tang","year":"2023","journal-title":"Comput. Math. Appl."},{"key":"mlstae1b71bib35","doi-asserted-by":"publisher","first-page":"2002","DOI":"10.4208\/cicp.OA-2020-0164","type":"journal-article","article-title":"Extended physics-informed neural networks (XPINNs): a generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations","volume":"28","author":"Ameya Jagtap","year":"2020","journal-title":"Commun. Comput. Phys."},{"key":"mlstae1b71bib36","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2022.114963","type":"journal-article","article-title":"A deep domain decomposition method based on Fourier features","volume":"423","author":"Li","year":"2023","journal-title":"J. Comput. Appl. Math."},{"key":"mlstae1b71bib37","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10444-023-10065-9","type":"journal-article","article-title":"Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations","volume":"49","author":"Moseley","year":"2023","journal-title":"Adv. Comput. Math."},{"key":"mlstae1b71bib38","doi-asserted-by":"publisher","DOI":"10.1016\/j.engappai.2023.107183","type":"journal-article","article-title":"Augmented physics-informed neural networks (APINNs): a gating network-based soft domain decomposition methodology","volume":"126","author":"Hu","year":"2023","journal-title":"Eng. Appl. Artif. Intell."},{"key":"mlstae1b71bib39","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2019.112732","type":"journal-article","article-title":"Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data","volume":"361","author":"Sun","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"mlstae1b71bib40","doi-asserted-by":"publisher","first-page":"354","DOI":"10.4208\/cmr.2020-0051","type":"journal-article","article-title":"A comparison study of Deep Galerkin Method and Deep Ritz Method for elliptic problems with different boundary conditions","volume":"36","author":"Chen","year":"2020","journal-title":"Commun. Math. Res."},{"key":"mlstae1b71bib41","doi-asserted-by":"publisher","first-page":"930","DOI":"10.4208\/cicp.OA-2020-0086","type":"journal-article","article-title":"Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks","volume":"29","author":"Wight","year":"2021","journal-title":"Commun. Comput. Phys."},{"key":"mlstae1b71bib42","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1016\/j.neucom.2018.06.056","type":"journal-article","article-title":"A unified deep artificial neural network approach to partial differential equations in complex geometries","volume":"317","author":"Berg","year":"2018","journal-title":"Neurocomputing"},{"key":"mlstae1b71bib43","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)EM.1943-7889.0001947","type":"journal-article","article-title":"Physics-informed deep learning for computational elastodynamics without labeled data","volume":"147","author":"Rao","year":"2021","journal-title":"J. Eng. Mech."},{"key":"mlstae1b71bib44","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.110085","type":"journal-article","article-title":"PFNN: a penalty-free neural network method for solving a class of second-order boundary-value problems on complex geometries","volume":"428","author":"Sheng","year":"2021","journal-title":"J. Comput. Phys."},{"key":"mlstae1b71bib45","doi-asserted-by":"publisher","first-page":"B1105","DOI":"10.1137\/21M1397908","type":"journal-article","article-title":"Physics-informed neural networks with hard constraints for inverse design","volume":"43","author":"Lu","year":"2021","journal-title":"SIAM J. Sci. Comput."},{"key":"mlstae1b71bib46","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.114333","type":"journal-article","article-title":"Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks","volume":"389","author":"Sukumar","year":"2022","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"mlstae1b71bib47","doi-asserted-by":"publisher","DOI":"10.1088\/2399-6528\/ace416","type":"journal-article","article-title":"Dynamic & norm-based weights to normalize imbalance in back-propagated gradients of physics-informed neural networks","volume":"7","author":"Deguchi","year":"2023","journal-title":"J. Phys. Commun."},{"key":"mlstae1b71bib48","type":"conference-proceedings","article-title":"Neural tangent kernel: convergence and generalization in neural networks","volume":"vol 31","author":"Jacot","year":"2018"},{"key":"mlstae1b71bib49","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110768","type":"journal-article","article-title":"When and why PINNs fail to train: a neural tangent kernel perspective","volume":"449","author":"Wang","year":"2022","journal-title":"J. Comput. Phys."},{"key":"mlstae1b71bib50","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1007\/BF00883366","type":"journal-article","article-title":"Taking account of singularities at angular points and juncture points of the boundary conditions in the method of r-functions","volume":"18","author":"Sheiko","year":"1982","journal-title":"Sov. Appl. Mech."},{"key":"mlstae1b71bib51","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/S0167-8396(01)00015-2","type":"journal-article","article-title":"Transfinite interpolation over implicitly defined sets","volume":"18","author":"Rvachev","year":"2001","journal-title":"Comput. Aided Geom. Design"},{"key":"mlstae1b71bib52","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1017\/S096249290631001X","type":"journal-article","article-title":"Semi-analytic geometry with R-functions","volume":"16","author":"Shapiro","year":"2007","journal-title":"Acta Numer."},{"key":"mlstae1b71bib53","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1016\/0167-8396(94)90030-2","type":"journal-article","article-title":"Real functions for representation of rigid solids","volume":"11","author":"Shapiro","year":"1994","journal-title":"Comput. Aided Geom. Design"},{"key":"mlstae1b71bib54","first-page":"pp 258","type":"conference-proceedings","article-title":"Implicit functions with guaranteed differential properties","author":"Shapiro","year":"1999"},{"key":"mlstae1b71bib55","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1007\/s003660200027","type":"journal-article","article-title":"The architecture of SAGE - a meshfree system based on RFM","volume":"18","author":"Shapiro","year":"2002","journal-title":"Eng. Comput."},{"key":"mlstae1b71bib56","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1002\/nme.760","type":"journal-article","article-title":"A meshfree method for incompressible fluid dynamics problems","volume":"58","author":"Tsukanov","year":"2003","journal-title":"Int. J. Numer. Methods Eng."},{"key":"mlstae1b71bib57","doi-asserted-by":"publisher","first-page":"712","DOI":"10.1016\/j.cma.2014.10.012","type":"journal-article","article-title":"Cell-based maximum-entropy approximants","volume":"284","author":"Daniel Mill\u00e1n","year":"2015","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"mlstae1b71bib58","doi-asserted-by":"publisher","DOI":"10.1016\/j.heliyon.2023.e18820","type":"journal-article","article-title":"Enforcing Dirichlet boundary conditions in physics-informed neural networks and variational physics-informed neural networks","volume":"9","author":"Berrone","year":"2023","journal-title":"Heliyon"},{"key":"mlstae1b71bib59","doi-asserted-by":"publisher","DOI":"10.1016\/j.enganabound.2025.106161","type":"journal-article","article-title":"Fo-pinn: a first-order formulation for physics-informed neural networks","volume":"174","author":"Gladstone","year":"2025","journal-title":"Eng. Anal. Bound. Elem."},{"key":"mlstae1b71bib60","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1115\/1.3005099","type":"journal-article","article-title":"R-Functions in Boundary Value Problems in Mechanics","volume":"48","author":"Rvachev","year":"1995","journal-title":"Appl. Mech. Rev."},{"key":"mlstae1b71bib61","doi-asserted-by":"publisher","first-page":"60","DOI":"10.1016\/j.camwa.2024.01.021","type":"journal-article","article-title":"Physical informed neural networks with soft and hard boundary constraints for solving advection-diffusion equations using Fourier expansions","volume":"159","author":"Li","year":"2024","journal-title":"Comput. Math. Appl."},{"key":"mlstae1b71bib62","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1016\/0893-6080(89)90020-8","type":"journal-article","article-title":"Multilayer feedforward networks are universal approximators","volume":"2","author":"Hornik","year":"1989","journal-title":"Neural Netw."},{"key":"mlstae1b71bib63","doi-asserted-by":"publisher","first-page":"861","DOI":"10.1016\/S0893-6080(05)80131-5","type":"journal-article","article-title":"Multilayer feedforward networks with a nonpolynomial activation function can approximate any function","volume":"6","author":"Leshno","year":"1993","journal-title":"Neural Netw."},{"key":"mlstae1b71bib64","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1609\/aaai.v39i1.32040","type":"conference-proceedings","article-title":"Number Theoretic Accelerated Learning of Physics-Informed Neural Networks","volume":"vol 39","author":"Matsubara","year":"2025"},{"key":"mlstae1b71bib65","doi-asserted-by":"publisher","first-page":"86252","DOI":"10.1109\/ACCESS.2023.3302892","type":"journal-article","article-title":"Data vs. physics: The apparent Pareto front of physics-informed neural networks","volume":"11","author":"Rohrhofer","year":"2023","journal-title":"IEEE Access"},{"key":"mlstae1b71bib66","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1016\/j.gmod.2004.01.003","type":"journal-article","article-title":"Approximate distance fields with non-vanishing gradients","volume":"66","author":"Biswas","year":"2004","journal-title":"Graph. Models"},{"key":"mlstae1b71bib67","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1007\/s004660050479","type":"journal-article","article-title":"On completeness of RFM solution structures","volume":"25","author":"Rvachev","year":"2000","journal-title":"Comput. Mech."},{"key":"mlstae1b71bib68","first-page":"pp 26548","type":"conference-proceedings","article-title":"Characterizing possible failure modes in physics-informed neural networks","volume":"vol 34","author":"Krishnapriyan","year":"2021"},{"key":"mlstae1b71bib69","article-title":"Mitigating propagation failures in physics-informed neural networks using retain-resample-release (r3) sampling","author":"Daw","year":"2023","type":"conference-proceedings"},{"key":"mlstae1b71bib70","article-title":"On the Pareto front of physics-informed neural networks","author":"Rohrhofer","year":"2021","type":"other"},{"key":"mlstae1b71bib71","first-page":"pp 5824","type":"conference-proceedings","article-title":"Gradient surgery for multi-task learning","volume":"vol 33","author":"Yu","year":"2020"},{"key":"mlstae1b71bib72","doi-asserted-by":"publisher","DOI":"10.1063\/5.0151244","type":"journal-article","article-title":"Physics-informed neural network based on a new adaptive gradient descent algorithm for solving partial differential equations of flow problems","volume":"35","author":"Li","year":"2023","journal-title":"Phys. Fluids"},{"key":"mlstae1b71bib73","article-title":"Adam: a method for stochastic optimization","author":"Kingma","year":"2015","type":"conference-proceedings"},{"key":"mlstae1b71bib74","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109951","type":"journal-article","article-title":"NSFnets (Navier-Stokes flow nets): physics-informed neural networks for the incompressible Navier-Stokes equations","volume":"426","author":"Jin","year":"2021","journal-title":"J. Comput. Phys."},{"key":"mlstae1b71bib75","first-page":"pp 176","type":"conference-proceedings","article-title":"Balancing multiple back-propagated gradients with dynamic weight tuning for accurate training and inference of physics-informed neural network","volume":"vol 28","author":"Deguchi","year":"2023"},{"key":"mlstae1b71bib76","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2020.113028","type":"journal-article","article-title":"Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems","volume":"365","author":"Jagtap","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"mlstae1b71bib77","doi-asserted-by":"publisher","first-page":"208","DOI":"10.1137\/19M1274067","type":"journal-article","article-title":"DeepXDE: a deep learning library for solving differential equations","volume":"63","author":"Lu","year":"2021","journal-title":"SIAM Rev."},{"key":"mlstae1b71bib78","first-page":"pp 249","type":"conference-proceedings","article-title":"Understanding the difficulty of training deep feedforward neural networks","volume":"vol 9)","author":"Glorot","year":"2010"},{"key":"mlstae1b71bib79","first-page":"p 3","type":"conference-proceedings","article-title":"Rectifier nonlinearities improve neural network acoustic models","volume":"vol 30","author":"Maas","year":"2013"},{"key":"mlstae1b71bib80","first-page":"pp 1026","type":"conference-proceedings","article-title":"Delving deep into rectifiers: surpassing human-level performance on imagenet classification","author":"He","year":"2015"},{"key":"mlstae1b71bib81","article-title":"Fast and accurate deep network learning by exponential linear units (ELUs)","author":"Clevert","year":"2016","type":"conference-proceedings"},{"key":"mlstae1b71bib82","doi-asserted-by":"publisher","first-page":"I_35","DOI":"10.2208\/jscejam.77.2_I_35","type":"journal-article","article-title":"Unknown parameter estimation using physics-informed neural networks with noised observation data","volume":"77","author":"Deguchi","year":"2021","journal-title":"J. Japan Soc. Civil Eng."},{"key":"mlstae1b71bib83","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2019.109136","type":"journal-article","article-title":"Adaptive activation functions accelerate convergence in deep and physics-informed neural networks","volume":"404","author":"Jagtap","year":"2020","journal-title":"J. Comput. Phys."},{"key":"mlstae1b71bib84","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.113741","type":"journal-article","article-title":"A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics","volume":"379","author":"Haghighat","year":"2021","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"mlstae1b71bib85","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1016\/j.neunet.2017.12.012","type":"journal-article","article-title":"Sigmoid-weighted linear units for neural network function approximation in reinforcement learning","volume":"107","author":"Elfwing","year":"2018","journal-title":"Neural Netw."},{"key":"mlstae1b71bib86","article-title":"Searching for activation functions","author":"Ramachandran","year":"2017","type":"preprint"},{"key":"mlstae1b71bib87","article-title":"Gaussian error linear units (GELUs)","author":"Hendrycks","year":"2016","type":"preprint"},{"key":"mlstae1b71bib88","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2020.0334","type":"journal-article","article-title":"Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks","volume":"476","author":"Jagtap","year":"2020","journal-title":"Proc. R. Soc. A"},{"key":"mlstae1b71bib89","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111722","type":"journal-article","article-title":"Self-adaptive physics-informed neural networks","volume":"474","author":"McClenny","year":"2023","journal-title":"J. Comput. Phys."},{"key":"mlstae1b71bib90","article-title":"TensorFlow: large-scale machine learning on heterogeneous systems","author":"Abadi","year":"2015","type":"other"},{"key":"mlstae1b71bib91","first-page":"pp 173","type":"book","article-title":"Computational design of the basic dynamical processes of the ucla general circulation model","volume":"vol 17)","author":"Arakawa","year":"1977"},{"key":"mlstae1b71bib92","doi-asserted-by":"publisher","DOI":"10.2514\/6.1984-340","type":"conference-proceedings","article-title":"Computation of high Reynolds number flow around a circular cylinder with surface roughness","author":"Kawamura","year":"1984"},{"key":"mlstae1b71bib93","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1016\/0169-5983(86)90014-6","type":"journal-article","article-title":"Computation of high Reynolds number flow around a circular cylinder with surface roughness","volume":"1","author":"Kawamura","year":"1986","journal-title":"Fluid Dyn. Res."},{"key":"mlstae1b71bib94","doi-asserted-by":"publisher","first-page":"745","DOI":"10.1090\/S0025-5718-1968-0242392-2","type":"journal-article","article-title":"Numerical solution of the Navier-Stokes equations","volume":"22","author":"Chorin","year":"1968","journal-title":"Math. Comput."},{"key":"mlstae1b71bib95","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1147\/rd.112.0215","type":"journal-article","article-title":"On the partial difference equations of mathematical physics","volume":"11","author":"Courant","year":"1967","journal-title":"IBM J. Res. Dev."},{"key":"mlstae1b71bib96","doi-asserted-by":"publisher","first-page":"232","DOI":"10.1063\/1.1699639","type":"journal-article","article-title":"A method for the numerical calculation of hydrodynamic shocks","volume":"21","author":"von Neumann","year":"1950","journal-title":"J. Appl. Phys."},{"key":"mlstae1b71bib97","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1016\/0021-9991(82)90058-4","type":"journal-article","article-title":"High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method","volume":"48","author":"Ghia","year":"1982","journal-title":"J. Comput. Phys."},{"key":"mlstae1b71bib98","doi-asserted-by":"publisher","first-page":"747","DOI":"10.1002\/fld.953","type":"journal-article","article-title":"Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers","volume":"48","author":"Erturk","year":"2005","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"mlstae1b71bib99","author":"Chorin","year":"1993","type":"book"},{"key":"mlstae1b71bib100","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1515\/jnum-2012-0013","type":"journal-article","article-title":"New development in FreeFem++","volume":"20","author":"Hecht","year":"2012","journal-title":"J. Numer. Math."},{"key":"mlstae1b71bib101","author":"Brenner","year":"2008","type":"book"},{"key":"mlstae1b71bib102","doi-asserted-by":"publisher","DOI":"10.1007\/s10409-024-24140-x","type":"journal-article","article-title":"VW-PINNs: a volume weighting method for PDE residuals in physics-informed neural networks","volume":"41","author":"Song","year":"2024","journal-title":"Acta Mech. Sin."},{"key":"mlstae1b71bib103","doi-asserted-by":"publisher","DOI":"10.25080\/majora-212e5952-005","type":"conference-proceedings","article-title":"Experience report of physics-informed neural networks in fluid simulations: pitfalls and frustration","author":"Chuang","year":"2022"},{"key":"mlstae1b71bib104","doi-asserted-by":"publisher","first-page":"278","DOI":"10.1109\/TVCG.2014.2356196","type":"journal-article","article-title":"Space-time transfinite interpolation of volumetric material properties","volume":"21","author":"Sanchez","year":"2015","journal-title":"IEEE Trans. Vis. Comput. Graph."},{"key":"mlstae1b71bib105","doi-asserted-by":"publisher","first-page":"8598","DOI":"10.1109\/LRA.2022.3188894","type":"journal-article","article-title":"Using R-functions to control the shape of soft robots","volume":"7","author":"Mulroy","year":"2022","journal-title":"IEEE Robot. Autom. Lett."},{"key":"mlstae1b71bib106","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2024.116813","type":"journal-article","article-title":"Respecting causality for training physics-informed neural networks","volume":"421","author":"Wang","year":"2024","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"mlstae1b71bib107","first-page":"1","type":"journal-article","article-title":"PirateNets: physics-informed deep learning with residual adaptive networks","volume":"25","author":"Wang","year":"2024","journal-title":"J. Mach. Learn. Res."},{"key":"mlstae1b71bib108","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1016\/j.taml.2020.01.039","type":"journal-article","article-title":"Physics-informed deep learning for incompressible laminar flows","volume":"10","author":"Rao","year":"2020","journal-title":"Theor. Appl. Mech. Lett."},{"key":"mlstae1b71bib109","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.110079","type":"journal-article","article-title":"PhyGeoNet: physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state pdes on irregular domain","volume":"428","author":"Gao","year":"2021","journal-title":"J. Comput. Phys."},{"key":"mlstae1b71bib110","type":"conference-proceedings","article-title":"Attention is all you need","volume":"vol 30","author":"Vaswani","year":"2017"},{"key":"mlstae1b71bib111","doi-asserted-by":"publisher","first-page":"627","DOI":"10.1016\/0893-6080(96)00006-8","type":"journal-article","article-title":"A feedforward neural network with function shape autotuning","volume":"9","author":"Chen","year":"1996","journal-title":"Neural Netw."},{"key":"mlstae1b71bib112","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpgr.2023.109551","type":"journal-article","article-title":"Gradient-enhanced physics-informed neural networks for power systems operational support","volume":"223","author":"Mohammadian","year":"2023","journal-title":"Electr. Power Syst. Res."},{"key":"mlstae1b71bib113","doi-asserted-by":"publisher","DOI":"10.1016\/j.engappai.2023.106908","type":"journal-article","article-title":"A gradient-enhanced physics-informed neural network (gPINN) scheme for the coupled non-fickian\/non-fourierian diffusion-thermoelasticity analysis: a novel gPINN structure","volume":"126","author":"Eshkofti","year":"2023","journal-title":"Eng. Appl. Artif. Intell."},{"key":"mlstae1b71bib114","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115346","type":"journal-article","article-title":"Bayesian physics informed neural networks for real-world nonlinear dynamical systems","volume":"402","author":"Linka","year":"2022","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"mlstae1b71bib115","doi-asserted-by":"publisher","DOI":"10.1016\/j.engappai.2023.106425","type":"journal-article","article-title":"Self-adaptive physics-driven deep learning for seismic wave modeling in complex topography","volume":"123","author":"Ding","year":"2023","journal-title":"Eng. Appl. Artif. Intell."}],"container-title":["Machine Learning: Science and Technology"],"original-title":[],"link":[{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ae1b71","content-type":"text\/html","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ae1b71\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ae1b71","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ae1b71\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ae1b71\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ae1b71\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ae1b71\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ae1b71\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,28]],"date-time":"2025-11-28T08:15:50Z","timestamp":1764317750000},"score":1,"resource":{"primary":{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ae1b71"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,28]]},"references-count":115,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,11,28]]},"published-print":{"date-parts":[[2025,12,30]]}},"URL":"https:\/\/doi.org\/10.1088\/2632-2153\/ae1b71","relation":{},"ISSN":["2632-2153"],"issn-type":[{"value":"2632-2153","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,28]]},"assertion":[{"value":"Reliable and efficient inverse analysis using physics-informed neural networks with normalized distance functions and adaptive weight tuning","name":"article_title","label":"Article Title"},{"value":"Machine Learning: Science and Technology","name":"journal_title","label":"Journal Title"},{"value":"paper","name":"article_type","label":"Article Type"},{"value":"\u00a9 2025 The Author(s). Published by IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2025-03-18","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2025-11-04","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2025-11-28","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}]}}