{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T22:50:23Z","timestamp":1773269423860,"version":"3.50.1"},"reference-count":127,"publisher":"IOP Publishing","issue":"4","license":[{"start":{"date-parts":[[2023,12,18]],"date-time":"2023-12-18T00:00:00Z","timestamp":1702857600000},"content-version":"vor","delay-in-days":17,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,12,18]],"date-time":"2023-12-18T00:00:00Z","timestamp":1702857600000},"content-version":"tdm","delay-in-days":17,"URL":"https:\/\/iopscience.iop.org\/info\/page\/text-and-data-mining"}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"short-container-title":["Mach. Learn.: Sci. Technol."],"published-print":{"date-parts":[[2023,12,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Rapid access to accurate equation-of-state (EOS) data is crucial in the warm-dense matter (WDM) regime, as it is employed in various applications, such as providing input for hydrodynamic codes to model inertial confinement fusion processes. In this study, we develop neural network models for predicting the EOS based on first-principles data. The first model utilises basic physical properties, while the second model incorporates more sophisticated physical information, using output from average-atom (AA) calculations as features. AA models are often noted for providing a reasonable balance of accuracy and speed; however, our comparison of AA models and higher-fidelity calculations shows that more accurate models are required in the WDM regime. Both the neural network models we propose, particularly the physics-enhanced one, demonstrate significant potential as accurate and efficient methods for computing EOS data in WDM.<\/jats:p>","DOI":"10.1088\/2632-2153\/ad13b9","type":"journal-article","created":{"date-parts":[[2023,12,8]],"date-time":"2023-12-08T22:30:06Z","timestamp":1702074606000},"page":"045055","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["Physics-enhanced neural networks for equation-of-state calculations"],"prefix":"10.1088","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4878-3521","authenticated-orcid":true,"given":"Timothy J","family":"Callow","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0131-0628","authenticated-orcid":true,"given":"Jan","family":"Nikl","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0139-258X","authenticated-orcid":false,"given":"Eli","family":"Kraisler","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9162-262X","authenticated-orcid":false,"given":"Attila","family":"Cangi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"266","published-online":{"date-parts":[[2023,12,18]]},"reference":[{"key":"mlstad13b9bib1","doi-asserted-by":"publisher","first-page":"B441","DOI":"10.1088\/0741-3335\/47\/12B\/S31","article-title":"Progress in the study of warm dense matter","volume":"47","author":"Koenig","year":"2005","journal-title":"Plasma Phys. Control. Fusion"},{"key":"mlstad13b9bib2","author":"U.S. DOE","year":"2009"},{"key":"mlstad13b9bib3","first-page":"pp 123","author":"Brown","year":"2014"},{"key":"mlstad13b9bib4","doi-asserted-by":"publisher","first-page":"e59","DOI":"10.1017\/hpl.2018.53","article-title":"Experimental methods for warm dense matter research","volume":"6","author":"Falk","year":"2018","journal-title":"High Power Laser Sci. Eng."},{"key":"mlstad13b9bib5","doi-asserted-by":"publisher","DOI":"10.1063\/1.5143225","article-title":"Ab initio simulation of warm dense matter","volume":"27","author":"Bonitz","year":"2020","journal-title":"Phys. Plasmas"},{"key":"mlstad13b9bib6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.physrep.2018.04.001","article-title":"The uniform electron gas at warm dense matter conditions","volume":"744","author":"Dornheim","year":"2018","journal-title":"Phys. Rep."},{"key":"mlstad13b9bib7","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.107.015002","article-title":"In-flight measurements of capsule shell adiabats in laser-driven implosions","volume":"107","author":"Kritcher","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"mlstad13b9bib8","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1063\/1.1578638","article-title":"The physics basis for ignition using indirect-drive targets on the National Ignition Facility","volume":"11","author":"Lindl","year":"2004","journal-title":"Phys. Plasmas"},{"key":"mlstad13b9bib9","doi-asserted-by":"publisher","first-page":"435","DOI":"10.1038\/nphys3736","article-title":"Inertial-confinement fusion with lasers","volume":"12","author":"Betti","year":"2016","journal-title":"Nat. Phys."},{"key":"mlstad13b9bib10","doi-asserted-by":"publisher","DOI":"10.1063\/1.4934714","article-title":"Direct-drive inertial confinement fusion: a review","volume":"22","author":"Craxton","year":"2015","journal-title":"Phys. Plasmas"},{"key":"mlstad13b9bib11","doi-asserted-by":"publisher","first-page":"2011","DOI":"10.1063\/1.872570","article-title":"Liquid metallic hydrogen and the structure of brown dwarfs and giant planets","volume":"4","author":"Hubbard","year":"1997","journal-title":"Phys. Plasmas"},{"key":"mlstad13b9bib12","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1038\/s41586-020-2535-y","article-title":"A measurement of the equation of state of carbon envelopes of white dwarfs","volume":"584","author":"Kritcher","year":"2020","journal-title":"Nature"},{"key":"mlstad13b9bib13","doi-asserted-by":"publisher","first-page":"994","DOI":"10.1088\/0004-637X\/703\/1\/994","article-title":"Electron\u2013ion scattering in dense multi-component plasmas: application to the outer crust of an accreting neutron star","volume":"703","author":"Daligault","year":"2009","journal-title":"Astrophys. J."},{"key":"mlstad13b9bib14","doi-asserted-by":"publisher","DOI":"10.1063\/1.3101818","article-title":"Frontiers of the physics of dense plasmas and planetary interiors: experiments, theory and applications","volume":"16","author":"Fortney","year":"2009","journal-title":"Phys. Plasmas"},{"key":"mlstad13b9bib15","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1038\/nature02248","article-title":"Melting of iron at the physical conditions of the Earth\u2019s core","volume":"427","author":"Nguyen","year":"2004","journal-title":"Nature"},{"key":"mlstad13b9bib16","doi-asserted-by":"publisher","first-page":"A191","DOI":"10.1088\/0741-3335\/47\/5A\/014","article-title":"High energy density laboratory astrophysics","volume":"47","author":"Remington","year":"2005","journal-title":"Plasma Phys. Control. Fusion"},{"key":"mlstad13b9bib17","doi-asserted-by":"publisher","DOI":"10.1063\/1.4742317","article-title":"A database for equations of state and resistivities measurements in the warm dense matter regime","volume":"19","author":"Cl\u00e9rouin","year":"2012","journal-title":"Phys. Plasmas"},{"key":"mlstad13b9bib18","doi-asserted-by":"publisher","first-page":"1625","DOI":"10.1103\/RevModPhys.81.1625","article-title":"X-ray Thomson scattering in high energy density plasmas","volume":"81","author":"Glenzer","year":"2009","journal-title":"Rev. Mod. Phys."},{"key":"mlstad13b9bib19","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.125.195001","article-title":"Probing the electronic structure of warm dense nickel via resonant inelastic x-ray scattering","volume":"125","author":"Humphries","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"mlstad13b9bib20","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.109.065002","article-title":"Direct measurements of the ionization potential depression in a dense plasma","volume":"109","author":"Ciricosta","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"mlstad13b9bib21","doi-asserted-by":"publisher","DOI":"10.1063\/1.4920943","article-title":"The complex ion structure of warm dense carbon measured by spectrally resolved x-ray scattering","volume":"22","author":"Kraus","year":"2015","journal-title":"Phys. Plasmas"},{"key":"mlstad13b9bib22","doi-asserted-by":"publisher","DOI":"10.1063\/1.3116505","article-title":"The National Ignition Facility: ushering in a new age for high energy density science","volume":"16","author":"Moses","year":"2009","journal-title":"Phys. Plasmas"},{"key":"mlstad13b9bib23","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.88.015007","article-title":"Linac coherent light source: the first five years","volume":"88","author":"Bostedt","year":"2016","journal-title":"Rev. Mod. Phys."},{"key":"mlstad13b9bib24","doi-asserted-by":"publisher","first-page":"456","DOI":"10.1038\/nphoton.2011.178","article-title":"First light from sacla","volume":"5","author":"Pile","year":"2011","journal-title":"Nat. Photon."},{"key":"mlstad13b9bib25","doi-asserted-by":"publisher","first-page":"592","DOI":"10.3390\/app7060592","article-title":"Photon beam transport and scientific instruments at the European XFEL","volume":"7","author":"Tschentscher","year":"2017","journal-title":"Appl. Sci."},{"key":"mlstad13b9bib26","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.4.023055","article-title":"First-principles derivation and properties of density-functional average-atom models","volume":"4","author":"Callow","year":"2022","journal-title":"Phys. Rev. Res."},{"key":"mlstad13b9bib27","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1016\/j.hedp.2018.08.001","article-title":"A review of equation-of-state models for inertial confinement fusion materials","volume":"28","author":"Gaffney","year":"2018","journal-title":"High Energy Density Phys."},{"key":"mlstad13b9bib28","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.103.013203","article-title":"First-principles equation of state database for warm dense matter computation","volume":"103","author":"Militzer","year":"2021","journal-title":"Phys. Rev. E"},{"key":"mlstad13b9bib29","doi-asserted-by":"publisher","DOI":"10.1063\/1.4984780","article-title":"First-principles equation-of-state table of beryllium based on density-functional theory calculations","volume":"24","author":"Ding","year":"2017","journal-title":"Phys. Plasmas"},{"key":"mlstad13b9bib30","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.84.224109","article-title":"First-principles equation-of-state table of deuterium for inertial confinement fusion applications","volume":"84","author":"Hu","year":"2011","journal-title":"Phys. Rev. B"},{"key":"mlstad13b9bib31","doi-asserted-by":"publisher","first-page":"4981","DOI":"10.1103\/PhysRevB.20.4981","article-title":"Self-consistent field model for condensed matter","volume":"20","author":"David","year":"1979","journal-title":"Phys. Rev. B"},{"key":"mlstad13b9bib32","doi-asserted-by":"publisher","DOI":"10.1063\/1.4764937","article-title":"Average atom transport properties for pure and mixed species in the hot and warm dense matter regimes","volume":"19","author":"Starrett","year":"2012","journal-title":"Phys. Plasmas"},{"key":"mlstad13b9bib33","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.99.053201","article-title":"Pressure in warm and hot dense matter using the average-atom model","volume":"99","author":"Faussurier","year":"2019","journal-title":"Phys. Rev. E"},{"key":"mlstad13b9bib34","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.91.013104","article-title":"Pseudoatom molecular dynamics","volume":"91","author":"Starrett","year":"2015","journal-title":"Phys. Rev. E"},{"key":"mlstad13b9bib35","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.83.026403","article-title":"Variational-average-atom-in-quantum-plasmas (VAAQP) code and virial theorem: equation-of-state and shock-hugoniot calculations for warm dense Al, Fe, Cu and Pb","volume":"83","author":"Piron","year":"2011","journal-title":"Phys. Rev. E"},{"key":"mlstad13b9bib36","doi-asserted-by":"publisher","author":"James","year":"2018","DOI":"10.2172\/1487368"},{"key":"mlstad13b9bib37","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6587\/aadd6c","article-title":"Characterizing the ionization potential depression in dense carbon plasmas with high-precision spectrally resolved x-ray scattering","volume":"61","author":"Kraus","year":"2018","journal-title":"Plasma Phys. Control. Fusion"},{"key":"mlstad13b9bib38","doi-asserted-by":"publisher","first-page":"7911","DOI":"10.1038\/s41467-022-35578-7","article-title":"Accurate temperature diagnostics for matter under extreme conditions","volume":"13","author":"Dornheim","year":"2022","journal-title":"Nat. Commun."},{"key":"mlstad13b9bib39","doi-asserted-by":"publisher","first-page":"1093","DOI":"10.1038\/s41598-022-05034-z","article-title":"Thermal excitation signals in the inhomogeneous warm dense electron gas","volume":"12","author":"Moldabekov","year":"2022","journal-title":"Sci. Rep."},{"key":"mlstad13b9bib40","doi-asserted-by":"publisher","DOI":"10.1063\/5.0139560","article-title":"Imaginary-time correlation function thermometry: a new, high-accuracy and model-free temperature analysis technique for x-ray Thomson scattering data","volume":"30","author":"Dornheim","year":"2023","journal-title":"Phys. Plasmas"},{"key":"mlstad13b9bib41","doi-asserted-by":"publisher","first-page":"A1133","DOI":"10.1103\/PhysRev.140.A1133","article-title":"Self-consistent equations including exchange and correlation effects","volume":"140","author":"Kohn","year":"1965","journal-title":"Phys. Rev."},{"key":"mlstad13b9bib42","first-page":"pp 117","author":"Alexander Wang","year":"2002"},{"key":"mlstad13b9bib43","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1063\/1.1390175","article-title":"Jacob\u2019s ladder of density functional approximations for the exchange-correlation energy","volume":"577","author":"Perdew","year":"2001","journal-title":"AIP Conf. Proc."},{"key":"mlstad13b9bib44","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1021\/cr200107z","article-title":"Challenges for density functional theory","volume":"112","author":"Cohen","year":"2012","journal-title":"Chem. Rev."},{"key":"mlstad13b9bib45","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1146\/annurev-physchem-052516-044835","article-title":"Development of new density functional approximations","volume":"68","author":"Neil Qiang","year":"2017","journal-title":"Annu. Rev. Phys. Chem."},{"key":"mlstad13b9bib46","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.93.063207","article-title":"Importance of finite-temperature exchange correlation for warm dense matter calculations","volume":"93","author":"Valentin","year":"2016","journal-title":"Phys. Rev. E"},{"key":"mlstad13b9bib47","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.101.195129","article-title":"Influence of finite temperature exchange-correlation effects in hydrogen","volume":"101","author":"Ramakrishna","year":"2020","journal-title":"Phys. Rev. B"},{"key":"mlstad13b9bib48","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.93.245131","article-title":"Exact thermal density functional theory for a model system: correlation components and accuracy of the zero-temperature exchange-correlation approximation","volume":"93","author":"Smith","year":"2016","journal-title":"Phys. Rev. B"},{"key":"mlstad13b9bib49","doi-asserted-by":"publisher","DOI":"10.1063\/5.0079695","article-title":"Approximate bounds and temperature dependence of adiabatic connection integrands for the uniform electron gas","volume":"156","author":"Harding","year":"2022","journal-title":"J. Chem. Phys."},{"key":"mlstad13b9bib50","doi-asserted-by":"publisher","DOI":"10.1063\/5.0062325","article-title":"The relevance of electronic perturbations in the warm dense electron gas","volume":"155","author":"Moldabekov","year":"2021","journal-title":"J. Chem. Phys."},{"key":"mlstad13b9bib51","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.105.035134","article-title":"Benchmarking exchange-correlation functionals in the spin-polarized inhomogeneous electron gas under warm dense conditions","volume":"105","author":"Moldabekov","year":"2022","journal-title":"Phys. Rev. B"},{"key":"mlstad13b9bib52","doi-asserted-by":"publisher","DOI":"10.1063\/5.0135729","article-title":"Assessing the accuracy of hybrid exchange-correlation functionals for the density response of warm dense electrons","volume":"158","author":"Moldabekov","year":"2023","journal-title":"J. Chem. Phys."},{"key":"mlstad13b9bib53","doi-asserted-by":"publisher","first-page":"1326","DOI":"10.1021\/acs.jpclett.2c03670","article-title":"Non-empirical mixing coefficient for hybrid XC functionals from analysis of the XC kernel","volume":"14","author":"Moldabekov","year":"2023","journal-title":"J. Phys. Chem. Lett."},{"key":"mlstad13b9bib54","doi-asserted-by":"publisher","first-page":"2519","DOI":"10.1016\/j.cpc.2012.06.016","article-title":"Issues and challenges in orbital-free density functional calculations","volume":"183","author":"Karasiev","year":"2012","journal-title":"Comput. Phys. Commun."},{"key":"mlstad13b9bib55","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.88.161108","article-title":"Nonempirical generalized gradient approximation free-energy functional for orbital-free simulations","volume":"88","author":"Karasiev","year":"2013","journal-title":"Phys. Rev. B"},{"key":"mlstad13b9bib56","doi-asserted-by":"publisher","first-page":"625","DOI":"10.1103\/PhysRevA.38.625","article-title":"Exact properties of the pauli potential for the square root of the electron density and the kinetic energy functional","volume":"38","author":"Levy","year":"1988","journal-title":"Phys. Rev. A"},{"key":"mlstad13b9bib57","doi-asserted-by":"publisher","DOI":"10.1063\/1.5099217","article-title":"The first order atomic fragment approach-an orbital-free implementation of density functional theory","volume":"151","author":"Finzel","year":"2019","journal-title":"J. Chem. Phys."},{"key":"mlstad13b9bib58","doi-asserted-by":"publisher","DOI":"10.1002\/qua.25601","article-title":"The Liu-Parr power series expansion of the Pauli kinetic energy functional with the incorporation of shell-inducing traits: atoms","volume":"118","author":"Lude na","year":"2018","journal-title":"Int. J. Quantum Chem."},{"key":"mlstad13b9bib59","doi-asserted-by":"publisher","first-page":"1237","DOI":"10.1007\/BF01030009","article-title":"Fermion nodes","volume":"63","author":"Ceperley","year":"1991","journal-title":"J. Stat. Phys."},{"key":"mlstad13b9bib60","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.94.170201","article-title":"Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations","volume":"94","author":"Troyer","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"mlstad13b9bib61","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.129.066402","article-title":"Static electronic density response of warm dense hydrogen: ab initio path integral Monte Carlo simulations","volume":"129","author":"B\u00f6hme","year":"2022","journal-title":"Phys. Rev. Lett."},{"key":"mlstad13b9bib62","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1038\/s41524-022-00734-6","article-title":"Recent advances and applications of deep learning methods in materials science","volume":"8","author":"Choudhary","year":"2022","journal-title":"npj Comput. Mater."},{"key":"mlstad13b9bib63","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevMaterials.6.040301","article-title":"Deep dive into machine learning density functional theory for materials science and chemistry","volume":"6","author":"Fiedler","year":"2022","journal-title":"Phys. Rev. Mater."},{"key":"mlstad13b9bib64","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1038\/s42254-022-00470-2","article-title":"Machine learning and density functional theory","volume":"4","author":"Pederson","year":"2022","journal-title":"Nat. Rev. Phys."},{"key":"mlstad13b9bib65","doi-asserted-by":"publisher","first-page":"5223","DOI":"10.1038\/s41467-020-19093-1","article-title":"Quantum chemical accuracy from density functional approximations via machine learning","volume":"11","author":"Bogojeski","year":"2020","journal-title":"Nat. Commun."},{"key":"mlstad13b9bib66","doi-asserted-by":"publisher","first-page":"1385","DOI":"10.1126\/science.abj6511","article-title":"Pushing the frontiers of density functionals by solving the fractional electron problem","volume":"374","author":"Kirkpatrick","year":"2021","journal-title":"Science"},{"key":"mlstad13b9bib67","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.2.033429","article-title":"Ab initio solution of the many-electron Schr\u00f6dinger equation with deep neural networks","volume":"2","author":"Pfau","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"mlstad13b9bib68","doi-asserted-by":"publisher","first-page":"602","DOI":"10.1126\/science.aag2302","article-title":"Solving the quantum many-body problem with artificial neural networks","volume":"355","author":"Carleo","year":"2017","journal-title":"Science"},{"key":"mlstad13b9bib69","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1038\/s41586-021-03382-w","article-title":"The data-driven future of high-energy-density physics","volume":"593","author":"Hatfield","year":"2021","journal-title":"Nature"},{"key":"mlstad13b9bib70","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.125.235001","article-title":"Effective static approximation: a fast and reliable tool for warm-dense matter theory","volume":"125","author":"Dornheim","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"mlstad13b9bib71","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.104.035120","article-title":"Accelerating finite-temperature Kohn-Sham density functional theory with deep neural networks","volume":"104","author":"Ellis","year":"2021","journal-title":"Phys. Rev. B"},{"key":"mlstad13b9bib72","doi-asserted-by":"publisher","DOI":"10.1088\/2632-2153\/ac9956","article-title":"Training-free hyperparameter optimization of neural networks for electronic structures in matter","volume":"3","author":"Fiedler","year":"2022","journal-title":"Mach. Learn.: Sci. Technol."},{"key":"mlstad13b9bib73","author":"Goodfellow","year":"2016"},{"key":"mlstad13b9bib74","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1016\/0893-6080(91)90009-T","article-title":"Approximation capabilities of multilayer feedforward networks","volume":"4","author":"Hornik","year":"1991","journal-title":"Neural Netw."},{"key":"mlstad13b9bib75","doi-asserted-by":"publisher","DOI":"10.1063\/5.0029723","article-title":"Development of uncertainty-aware equation-of-state models: application to copper","volume":"128","author":"Ali","year":"2020","journal-title":"J. Appl. Phys."},{"key":"mlstad13b9bib76","article-title":"Constraining model uncertainty in plasma equation-of-state models with a physics-constrained gaussian process","author":"Gaffney","year":"2022"},{"key":"mlstad13b9bib77","doi-asserted-by":"publisher","DOI":"10.1063\/5.0087210","article-title":"Uncertainty quantification for a multi-phase carbon equation of state model","volume":"131","author":"Lindquist","year":"2022","journal-title":"J. Appl. Phys."},{"key":"mlstad13b9bib78","doi-asserted-by":"publisher","DOI":"10.1063\/5.0126708","article-title":"Neural network surrogate models for equations of state","volume":"30","author":"Mentzer","year":"2023","journal-title":"Phys. Plasmas"},{"key":"mlstad13b9bib79","article-title":"Of course, the performance of the network is evaluated on data that was not seen during training; the point is that the training and test data is drawn from the same database"},{"key":"mlstad13b9bib80","doi-asserted-by":"publisher","first-page":"804","DOI":"10.1103\/PhysRev.43.804","article-title":"On the constitution of metallic sodium","volume":"43","author":"Wigner","year":"1933","journal-title":"Phys. Rev."},{"key":"mlstad13b9bib81","doi-asserted-by":"publisher","first-page":"1561","DOI":"10.1103\/PhysRev.75.1561","article-title":"Equations of state of elements based on the generalized Fermi-Thomas theory","volume":"75","author":"Feynman","year":"1949","journal-title":"Phys. Rev."},{"key":"mlstad13b9bib82","doi-asserted-by":"publisher","first-page":"2096","DOI":"10.1103\/PhysRevA.26.2096","article-title":"Density-functional theory of hydrogen plasmas","volume":"26","author":"Dharma-wardana","year":"1982","journal-title":"Phys. Rev. A"},{"key":"mlstad13b9bib83","doi-asserted-by":"publisher","first-page":"3035","DOI":"10.1103\/PhysRevA.43.3035","article-title":"Photoabsorption in hot plasmas based on the ion-sphere and ion-correlation models","volume":"43","author":"Balazs","year":"1991","journal-title":"Phys. Rev. A"},{"key":"mlstad13b9bib84","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.69.016409","article-title":"Equation of state and transport coefficients for dense plasmas","volume":"69","author":"Blancard","year":"2004","journal-title":"Phys. Rev. E"},{"key":"mlstad13b9bib85","doi-asserted-by":"publisher","first-page":"278","DOI":"10.1016\/j.hedp.2007.02.037","article-title":"Equation of state, occupation probabilities and conductivities in the average atom Purgatorio code","volume":"3","author":"Sterne","year":"2007","journal-title":"High Energy Density Phys."},{"key":"mlstad13b9bib86","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.3.023026","article-title":"Reconciling ionization energies and band gaps of warm dense matter derived with ab initio simulations and average atom models","volume":"3","author":"Massacrier","year":"2021","journal-title":"Phys. Rev. Res."},{"key":"mlstad13b9bib87","doi-asserted-by":"publisher","first-page":"13244","DOI":"10.1103\/PhysRevB.45.13244","article-title":"Accurate and simple analytic representation of the electron-gas correlation energy","volume":"45","author":"Perdew","year":"1992","journal-title":"Phys. Rev. B"},{"key":"mlstad13b9bib88","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.5.013049","article-title":"Improved calculations of mean ionization states with an average-atom model","volume":"5","author":"Callow","year":"2023","journal-title":"Phys. Rev. Res."},{"key":"mlstad13b9bib89","article-title":"ato MEC","author":"Callow","year":"2021"},{"key":"mlstad13b9bib90","first-page":"pp 31","article-title":"atoMEC: an open-source average-atom Python code","author":"Callow","year":"2022"},{"key":"mlstad13b9bib91","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1038\/s41586-020-2649-2","article-title":"Array programming with NumPy","volume":"585","author":"Harris","year":"2020","journal-title":"Nature"},{"key":"mlstad13b9bib92","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1038\/s41592-019-0686-2","article-title":"SciPy 1.0: fundamental algorithms for scientific computing in python","volume":"17","author":"(SciPy 1.0 Contributors)","year":"2020","journal-title":"Nat. Methods"},{"key":"mlstad13b9bib93","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.softx.2017.11.002","article-title":"Recent developments in libxc\u2014a comprehensive library of functionals for density functional theory","volume":"7","author":"Lehtola","year":"2018","journal-title":"SoftwareX"},{"key":"mlstad13b9bib94","article-title":"mendeleev\u2014a python resource for properties of chemical elements, ions and isotopes","author":"Mentel","year":"2014","unstructured":"Mentel L M 2014 mendeleev\u2014a python resource for properties of chemical elements, ions and isotopes (available at: https:\/\/github.com\/lmmentel\/mendeleev ) (Accessed 12 December 2023)"},{"key":"mlstad13b9bib95","article-title":"Joblib: running python functions as pipeline jobs","author":"Joblib Development Team","year":"2020"},{"key":"mlstad13b9bib96","doi-asserted-by":"publisher","first-page":"1234","DOI":"10.1103\/PhysRevA.19.1234","article-title":"Quantum-statistical model for high-density matter","volume":"19","author":"Richard","year":"1979","journal-title":"Phys. Rev. A"},{"key":"mlstad13b9bib97","first-page":"pp 305","author":"More","year":"1985"},{"key":"mlstad13b9bib98","doi-asserted-by":"publisher","first-page":"1553","DOI":"10.1088\/0953-4075\/40\/8\/008","article-title":"A model of dense-plasma atomic structure for equation-of-state calculations","volume":"40","author":"Pain","year":"2007","journal-title":"J. Phys. B: At. Mol. Opt. Phys."},{"key":"mlstad13b9bib99","article-title":"The ideal approximation, as implied by the name, is a known approximation, and cannot be derived from the functional derivative of the free energy"},{"key":"mlstad13b9bib100","article-title":"We note that, in equation (12), we have written the xc-free energy Fxc[n] as part of the internal energy. However, in principle, it also contains an entropic contribution, because the entropy S[n] is approximated by the non-interacting entropy functional in KS-DFT. We direct readers to [26] for a more detailed discussion of the xc energy term in ground-state and finite-temperature KS-DFT"},{"key":"mlstad13b9bib101","doi-asserted-by":"publisher","first-page":"4869","DOI":"10.1103\/PhysRevE.51.4869","article-title":"Pressure ionization in the spherical ion-cell model of dense plasmas and a pressure formula in the relativistic Pauli approximation","volume":"51","author":"Blenski","year":"1995","journal-title":"Phys. Rev. E"},{"key":"mlstad13b9bib102","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.98.033205","article-title":"D\u2019yakov-Kontorovitch instability of shock waves in hot plasmas","volume":"98","author":"Wetta","year":"2018","journal-title":"Phys. Rev. E"},{"key":"mlstad13b9bib103","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.104.025209","article-title":"Carbon ionization from a quantum average-atom model up to gigabar pressures","volume":"104","author":"Faussurier","year":"2021","journal-title":"Phys. Rev. E"},{"key":"mlstad13b9bib104","doi-asserted-by":"publisher","first-page":"2389","DOI":"10.1063\/1.1678599","article-title":"Hellmann-Feynman and virial theorems in the x\u03b1 method","volume":"57","author":"John","year":"1972","journal-title":"J. Chem. Phys."},{"key":"mlstad13b9bib105","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1119\/1.1987655","article-title":"Virial theorem generalized","volume":"42","author":"McLellan","year":"1974","journal-title":"Am. J. Phys."},{"key":"mlstad13b9bib106","doi-asserted-by":"publisher","first-page":"8167","DOI":"10.1103\/PhysRevB.37.8167","article-title":"Quantum-mechanical stress and a generalized virial theorem for clusters and solids","volume":"37","author":"Ziesche","year":"1988","journal-title":"Phys. Rev. B"},{"key":"mlstad13b9bib107","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1088\/0953-8984\/13\/2\/306","article-title":"Virial theorem and pressure calculations in the GGA","volume":"13","author":"Legrand","year":"2001","journal-title":"J. Phys.: Condens. Matter"},{"key":"mlstad13b9bib108","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1002\/qua.989","article-title":"Local kinetic energy and local temperature in the density-functional theory of electronic structure","volume":"90","author":"Ayers","year":"2002","journal-title":"Int. J. Quantum Chem."},{"key":"mlstad13b9bib109","doi-asserted-by":"publisher","first-page":"788","DOI":"10.1063\/1.437511","article-title":"Local kinetic energy in quantum mechanics","volume":"70","author":"Cohen","year":"1979","journal-title":"J. Chem. Phys."},{"key":"mlstad13b9bib110","doi-asserted-by":"publisher","first-page":"875","DOI":"10.1002\/(SICI)1097-461X(1996)60:43.0.CO;2-4","article-title":"Atomic shell structure and electron numbers","volume":"60","author":"Kohout","year":"1996","journal-title":"Int. J. Quantum Chem."},{"key":"mlstad13b9bib111","article-title":"Atoms at finite temperatures","author":"Johnson","year":"2000"},{"key":"mlstad13b9bib112","doi-asserted-by":"publisher","first-page":"2278","DOI":"10.1109\/5.726791","article-title":"Gradient-based learning applied to document recognition","volume":"86","author":"Lecun","year":"1998","journal-title":"Proc. IEEE"},{"key":"mlstad13b9bib113","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1145\/3065386","article-title":"ImageNet classification with deep convolutional neural networks","volume":"60","author":"Krizhevsky","year":"2017","journal-title":"Commun. ACM"},{"key":"mlstad13b9bib114","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.120.145301","article-title":"Crystal graph convolutional neural networks for an accurate and interpretable prediction of material properties","volume":"120","author":"Xie","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"mlstad13b9bib115","doi-asserted-by":"publisher","first-page":"1735","DOI":"10.1162\/neco.1997.9.8.1735","article-title":"Long short-term memory","volume":"9","author":"Hochreiter","year":"1997","journal-title":"Neural Comput."},{"key":"mlstad13b9bib116","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1145\/3422622","article-title":"Generative adversarial networks","volume":"63","author":"Goodfellow","year":"2020","journal-title":"Commun. ACM"},{"key":"mlstad13b9bib117","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1038\/323533a0","article-title":"Learning representations by back-propagating errors","volume":"323","author":"Rumelhart","year":"1986","journal-title":"Nature"},{"key":"mlstad13b9bib118","author":"Hastie","year":"2009","edition":"2nd edn"},{"key":"mlstad13b9bib119","article-title":"The reason we use two different notations is because Pref is used to denote the reference pressure in general, and is used later when evaluating the raw AA results, as well as the neural network results. Y 0 is only used in the context of the neural network training procedure, and is used to denote a specific reference pressure for the given stage\/subset of the training workflow. Furthermore, as will be discussed in section 4.1 Y 0 is a transformation of the original pressure via a scaling relation"},{"key":"mlstad13b9bib120","doi-asserted-by":"publisher","first-page":"18","DOI":"10.1007\/BF01238023","article-title":"Combining classifiers: a theoretical framework","volume":"1","author":"Kittler","year":"1998","journal-title":"Pattern Anal. Appl."},{"key":"mlstad13b9bib121","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1093\/biomet\/30.1-2.81","article-title":"A new measure of rank correlation","volume":"30","author":"Kendall","year":"1938","journal-title":"Biometrika"},{"key":"mlstad13b9bib122","doi-asserted-by":"publisher","first-page":"B864","DOI":"10.1103\/PhysRev.136.B864","article-title":"Inhomogeneous electron gas","volume":"136","author":"Hohenberg","year":"1964","journal-title":"Phys. Rev."},{"key":"mlstad13b9bib123","article-title":"Adam: a method for stochastic optimization","author":"Kingma","year":"2014"},{"key":"mlstad13b9bib124","article-title":"TensorFlow: large-scale machine learning on heterogeneous systems","author":"Abadi","year":"2015"},{"key":"mlstad13b9bib125","doi-asserted-by":"crossref","DOI":"10.1145\/3292500.3330701","article-title":"Optuna: a next-generation hyperparameter optimization framework","author":"Akiba","year":"2019"},{"key":"mlstad13b9bib126","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1016\/j.hedp.2009.07.003","article-title":"Advances in NLTE modeling for integrated simulations","volume":"6","author":"Scott","year":"2010","journal-title":"High Energy Density Phys."},{"key":"mlstad13b9bib127","doi-asserted-by":"publisher","DOI":"10.1016\/j.hedp.2020.100746","article-title":"An accelerated approach to inline non-lte modeling","volume":"34","author":"Holladay","year":"2020","journal-title":"High Energy Density Phys."}],"container-title":["Machine Learning: Science and Technology"],"original-title":[],"link":[{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad13b9","content-type":"text\/html","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad13b9\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad13b9","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad13b9\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad13b9\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad13b9\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad13b9\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad13b9\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,18]],"date-time":"2023-12-18T07:53:37Z","timestamp":1702886017000},"score":1,"resource":{"primary":{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad13b9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,1]]},"references-count":127,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,12,18]]},"published-print":{"date-parts":[[2023,12,1]]}},"URL":"https:\/\/doi.org\/10.1088\/2632-2153\/ad13b9","relation":{},"ISSN":["2632-2153"],"issn-type":[{"value":"2632-2153","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,12,1]]},"assertion":[{"value":"Physics-enhanced neural networks for equation-of-state calculations","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 2023 The Author(s). Published by IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2023-05-05","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2023-12-08","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2023-12-18","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}]}}