3D code for MAgneto-Thermal evolution in Isolated Neutron Stars, MATINS: The Magnetic Field Formalism

The long-term evolution of the internal, strong magnetic fields of neutron stars needs a specific numerical modeling. The diversity of the observed phenomenology of neutron stars indicates that their magnetic topology is rather complex and three-dimensional simulations are required, for example, to explain the observed bursting mechanisms and the creation of surface hotspots. We present MATINS, a new three dimensions numerical code for magneto-thermal evolution in neutron stars, based on a finite-volume scheme that employs the cubed-sphere system of coordinates. In this first work, we focus on the crustal magnetic evolution, with the inclusion of realistic calculations for the neutron star structure, composition and electrical conductivity assuming a simple temperature evolution profile. MATINS follows the evolution of strong fields (1014 − 1015 Gauss) with complex non-axisymmetric topologies and dominant Hall-drift terms, and it is suitable for handling sharp current sheets. After introducing the technical description of our approach and some tests, we present long-term simulations of the non-linear field evolution in realistic neutron star crusts. The results show how the non-axisymmetric Hall cascade redistributes the energy over different spatial scales. Following the exploration of different initial topologies, we conclude that during a few tens of kyr, an equipartition of energy between the poloidal and toroidal components happens at small-scales. However, the magnetic field keeps a strong memory of the initial large-scales, which are much harder to be restructured or created. This indicates that large-scale configuration attained during the neutron star formation is crucial to determine the field topology at any evolution stage.CD and NR are supported by the ERC Consolidator Grant “MAGNESIA” No. 817661 (PI: Rea) and this work has been carried out within the framework of the doctoral program in Physics of the Universitat Autònoma de Barcelona. This work was also partially supported by the program Unidad de Excelencia María de Maeztu CEX2020-001058-M. DV is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC Starting Grant "IMAGINE" No. 948582, PI: DV). JAP acknowledges support from the Generalitat Valenciana (PROMETEO/2019/071) and the AEI grant PID2021-127495NB-I00

How bright can old magnetars be? Assessing the impact of magnetized envelopes and field topology on neutron star cooling

Neutron stars cool down during their lifetime through the combination of neutrino emission from the interior and photon cooling from the surface. Strongly magnetised neutron stars, called magnetars, are no exception, but the effect of their strong fields adds further complexities to the cooling theory. Besides other factors, modelling the outermost hundred meters (the envelope) plays a crucial role in predicting their surface temperatures. In this letter, we revisit the influence of envelopes on the cooling properties of neutron stars, with special focus on the critical effects of the magnetic field. We explore how our understanding of the relation between the internal and surface temperatures has evolved over the past two decades, and how different assumptions about the neutron star envelope and field topology lead to radically different conclusions on the surface temperature and its cooling with age. In particular, we find that relatively old magnetars with core-threading magnetic fields are actually much cooler than arotation-powered pulsar of the same age. This is at variance with what is typically observed in crustal-confined models. Our results have important implications for the estimates of the X-ray luminosities of aged magnetars, and the subsequent population study of the different neutron star classes.JAP acknowledges support from the Generalitat Valenciana grants PROMETEO/2019/071 and ASFAE/2022/026 (with funding from NextGenerationEU PRTR-C17.I1) and the AEI grant PID2021-127495NB-I00. CD and NR are supported by the ERC Consolidator Grant “MAGNESIA” No. 817661 (PI: Rea) and this work has been carried out within the framework of the doctoral program in Physics of the Universitat Autònoma de Barcelona and it is partially supported by the program Unidad de Excelencia María de Maeztu CEX2020-001058-M. DV is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC Starting Grant "IMAGINE" No. 948582, PI: DV)

Modelling Force-Free Neutron Star Magnetospheres using Physics-Informed Neural Networks

Using Physics-Informed Neural Networks (PINNs) to solve a specific boundary value problem is becoming more popular as an alternative to traditional methods. However, depending on the specific problem, they could be computationally expensive and potentially less accurate. The functionality of PINNs for real-world physical problems can significantly improve if they become more flexible and adaptable. To address this, our work explores the idea of training a PINN for general boundary conditions and source terms expressed through a limited number of coefficients, introduced as additional inputs in the network. Although this process increases the dimensionality and is computationally costly, using the trained network to evaluate new general solutions is much faster. Our results indicate that PINN solutions are relatively accurate, reliable, and well-behaved. We applied this idea to the astrophysical scenario of the magnetic field evolution in the interior of a neutron star connected to a force-free magnetosphere. Solving this problem through a global simulation in the entire domain is expensive due to the elliptic solver's needs for the exterior solution. The computational cost with a PINN was more than an order of magnitude lower than the similar case solved with classical methods. These results pave the way for the future extension to 3D of this (or a similar) problem, where generalised boundary conditions are very costly to implement.Comment: 11 pages, 10 figures, submitted for publication in MNRA

Thermal luminosity degeneracy of magnetized neutron stars with and without hyperon cores

The dissipation of intense crustal electric currents produces high Joule heating rates in cooling neutron stars. Here it is shown that Joule heating can counterbalance fast cooling, making it difficult to infer the presence of hyperons (which accelerate cooling) from measurements of the observed thermal luminosity Lγ. Models with and without hyperon cores match Lγ of young magnetars (with poloidal-dipolar field Bdip ≳ 1014 G at the polar surface and Lγ ≳ 1034 erg s−1 at t ≲ 105 yr) as well as mature, moderately magnetized stars (with Bdip ≲ 1014 G and 1031 erg s−1 ≲ Lγ ≲ 1032 erg s−1 at t ≳ 105 yr). In magnetars, the crustal temperature is almost independent of hyperon direct Urca cooling in the core, regardless of whether the latter is suppressed or not by hyperon superfluidity. The thermal luminosities of light magnetars without hyperons and heavy magnetars with hyperons have Lγ in the same range and are almost indistinguishable. Likewise, Lγ data of neutron stars with Bdip ≲ 1014 G but with strong internal fields are not suitable to extract information about the equation of state as long as hyperons are superfluid, with maximum amplitude of the energy gaps of the order ≈1 MeV.FA is supported by The University of Melbourne through a Melbourne Research Scholarship. AM acknowledges funding from an Australian Research Council Discovery Project grant (DP170103625) and the Australian Research Council Centre of Excellence for Gravitational Wave Discovery (OzGrav) (CE170100004). DV is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC Starting Grant "IMAGINE" No. 948582, PI DV). CD is supported by the ERC Consolidator Grant “MAGNESIA” (No. 817661, PI Nanda Rea) and this work has been carried out within the framework of the doctoral program in Physics of the Universitat Autònoma de Barcelona. JAP acknowledges support by the Generalitat Valenciana (PROMETEO/2019/071), AEI grant PGC2018-095984-B-I00 and the Alexander von Humboldt Stiftung through a Humboldt Research Award

On the Rate of Crustal Failures in Young Magnetars

The activity of magnetars is powered by their intense and dynamic magnetic fields and has been proposed as the trigger to extragalactic fast radio bursts. Here we estimate the frequency of crustal failures in young magnetars, by computing the magnetic stresses in detailed magnetothermal simulations including Hall drift and ohmic dissipation. The initial internal topology at birth is poorly known but is likely to be much more complex than a dipole. Thus, we explore a wide range of initial configurations, finding that the expected rate of crustal failures varies by orders of magnitude depending on the initial magnetic configuration. Our results show that this rate scales with the crustal magnetic energy, rather than with the often used surface value of the dipolar component related to the spin-down torque. The estimated frequency of crustal failures for a given dipolar component can vary by orders of magnitude for different initial conditions, depending on how much magnetic energy is distributed in the crustal nondipolar components, likely dominant in newborn magnetars. The quantitative reliability of the expected event rate could be improved by a better treatment of the magnetic evolution in the core and the elastic/plastic crustal response, not included here. Regardless of that, our results are useful inputs in modeling the outburst rate of young Galactic magnetars, and their relation with the fast radio bursts in our and other galaxies.C.D., D.V., N.R., and A.G.G. are supported by the ERC Consolidator Grant “MAGNESIA” (No. 817661) and acknowledge funding from grants SGR2017-1383 and PGC2018-095512-BI00. J.A.P. acknowledges support by the Generalitat Valenciana (PROMETEO/2019/071) and by AEI grant PGC2018-095984-BI00. R.P. acknowledges support from NSF award AST-1616157. We acknowledge support from the PHAROS COST Action (CA16214)

Constraining the Nature of the 18 min Periodic Radio Transient GLEAM-X J162759.5-523504.3 via Multiwavelength Observations and Magneto-thermal Simulations

We observed the periodic radio transient GLEAM-X J162759.5-523504.3 (GLEAM-X J1627) using the Chandra X-ray Observatory for about 30 ks on 2022 January 22–23, simultaneously with radio observations from the Murchison Widefield Array, MeerKAT, and the Australia Telescope Compact Array. Its radio emission and 18 min periodicity led the source to be tentatively interpreted as an extreme magnetar or a peculiar highly magnetic white dwarf. The source was not detected in the 0.3–8 keV energy range with a 3σ upper limit on the count rate of 3 × 10−4 counts s−1. No radio emission was detected during our X-ray observations either. Furthermore, we studied the field around GLEAM-X J1627 using archival European Southern Observatory and DECam Plane Survey data, as well as recent Southern African Large Telescope observations. Many sources are present close to the position of GLEAM-X J1627, but only two within the 2'' radio position uncertainty. Depending on the assumed spectral distribution, the upper limits converted to an X-ray luminosity of LX < 6.5 × 1029 erg s−1 for a blackbody with temperature kT = 0.3 keV, or LX < 9 × 1029 erg s−1 for a power law with photon index Γ = 2 (assuming a 1.3 kpc distance). Furthermore, we performed magneto-thermal simulations for neutron stars considering crust- and core-dominated field configurations. Based on our multiband limits, we conclude that (i) in the magnetar scenario, the X-ray upper limits suggest that GLEAM-X J1627 should be older than ∼1 Myr, unless it has a core-dominated magnetic field or has experienced fast cooling; (ii) in the white dwarf scenario, we can rule out most binary systems, a hot sub-dwarf, and a hot magnetic isolated white dwarf (T ≳ 10.000 K), while a cold isolated white dwarf is still compatible with our limits.N.R., F.C.Z., C.D., M.R., V.G., C.P., A.B., and E.P. are supported by the ERC Consolidator Grant "MAGNESIA" under grant agreement No. 817661, and National Spanish grant No. PGC2018-095512-BI00. F.C.Z., A.B., and V.G. are also supported by Juan de la Cierva Fellowships. C.D., M.R., and C.A.'s work has been carried out within the framework of the doctoral program in Physics of the Universitat Autónoma de Barcelona. N.H.W. is supported by an Australian Research Council Future Fellowship (project number FT190100231) funded by the Australian Government. D.d.M. acknowledges financial support from the Italian Space Agency (ASI) and National Institute for Astrophysics (INAF) under agreements ASI-INAF I/037/12/0 and ASI-INAF n.2017-14-H.0 and from INAF "Sostegno alla ricerca scientifica main streams dell'INAF," Presidential Decree 43/2018 and from INAF "SKA/CTA projects," Presidential Decree 70/2016. D.B. acknowledges support from the South African National Research Foundation. D.V. is supported by the ERC Starting Grant "IMAGINE" under grant agreement No. 948582. This work was also partially supported by the program Unidad de Excelencia Maria de Maetzu de Maeztu CEX2020-001058-M and by the PHAROS COST Action (grant No. CA16214)

Unveiling the Physics of Neutron Stars : A 3D expedition into MAgneto-Thermal evolution in Isolated Neutron Stars with 'MATINS'

Aquesta tesi doctoral investiga a fons l'evolució a llarg termini dels camps magnètics interns i forts que es troben dins de les estrelles de neutrons aïllades, les quals són els objectes magnètics més poderosos de l'univers. Aquestes estrelles no només tenen un impacte magnètic més enllà de la seva superfície, creant una magnetosfera que afecta les seves característiques observables, sinó que també experimenten canvis magnètics i tèrmics al llarg de milions d'anys. Aquesta complexa topologia magnètica i la seva interacció amb la temperatura requereixen simulacions numèriques avançades. En el nucli d'aquesta tesi es troba l'exploració de models d'evolució magneto-tèrmica tridimensionals d'última generació. Aquesta investigació representa un avenç pioner, ja que realitza la simulació 3D més realista fins a la data, abastant el primer milió d'anys de vida d'una estrella de neutrons. Aquestes simulacions tenen en compte detalls com la dissipació òhmica i la deriva de Hall a la crosta de l'estrella, i utilitzen dades reals sobre la temperatura i l'estructura de l'estrella. A més, l'ús de coordenades de tipus cub tetraèdric aborda les singularitats axials en les simulacions 3D. També s'incorporen factors relativistes i un model d'envoltori de la literatura, juntament amb una topologia de camp magnètic inicial derivada de simulacions de dinàmica de protoestrelles de neutrons. A través d'aquestes simulacions, s'estudia la lluminositat tèrmica, les propietats de temporització i l'evolució del camp magnètic. Dels diversos estudis realitzats durant aquesta tesi, es poden destacar algunes conclusions clau. Primer, es va descobrir que la ubicació de les corrents elèctriques dins de l'estrella determina si els magnetars de mitjana edat mostren una brillantor relativa o una lluminositat molt baixa, amb implicacions importants per a estudis de població de pulsars i magnetars. També es van imposar restriccions a les equacions d'estat nuclear mitjançant comparacions entre dades observacionals i simulacions teòriques. Aquestes equacions d'estat han de ser capaces d'explicar objectes tant brillants com magnetars com objectes molt febles a edats joves. Finalment, es va trobar que el component dipolar, important en molts models de magnetars-FRB, no és el principal desencadenant de les erupcions de la magnetosfera durant les fallades de la crosta. En canvi, l'energia magnètica de la crosta és un indicador fiable del nombre d'esdeveniments esperats. A més, es va observar que el camp magnètic conserva una memòria de les estructures a gran escala inicials, destacant la importància de la configuració inicial durant la formació de l'estrella de neutrons. En resum, aquesta tesi aprofundeix en l'evolució dels camps magnètics a l'interior d'estrelles de neutrons aïllades, mitjançant simulacions avançades, i aporta noves comprensions sobre els magnetars i les seves característiques observables.Esta tesis doctoral aborda una investigación exhaustiva sobre la evolución a largo plazo de los intensos campos magnéticos internos de las estrellas de neutrones aisladas, consideradas los objetos magnéticos más potentes del universo. Estos campos magnéticos extienden su influencia más allá de la superficie, alcanzando el plasma magnetizado cercano conocido como magnetosfera. Esta configuración magnética afecta significativamente las características observables de las estrellas de neutrones altamente magnetizadas, llamadas magnetars. Los campos magnéticos internos evolucionan durante miles a millones de años, en conexión con la evolución térmica. La diversidad de fenómenos observables en las estrellas de neutrones resalta la complejidad tridimensional de su topología magnética, lo que requiere simulaciones numéricas avanzadas. La tesis se enfoca en modelos tridimensionales avanzados de evolución magneto-térmica. Realizamos la simulación 3D más realista hasta la fecha, abarcando el primer millón de años de vida de una estrella de neutrones, utilizando el código MATINS recién desarrollado. Este código considera la disipación ohmica y la deriva de Hall en la corteza de la estrella. Las simulaciones incorporan cálculos microfísicos precisos dependientes de la temperatura y la estructura estelar derivada de una ecuación de estado realista. Además, utilizamos coordenadas de esfera cúbica para abordar las singularidades axiales en 3D. Las simulaciones consideran factores relativistas y un modelo de envoltura avanzado, junto con una topología inicial del campo magnético derivada de simulaciones de dinamo de protoestrella de neutrones. Cuantitativamente, simulamos la luminosidad térmica, las propiedades de temporización y la evolución del campo magnético, ampliando los límites de la modelización numérica. Entre las conclusiones clave de esta tesis, destacamos que la ubicación de las corrientes eléctricas en la estrella determina si los magnetars de mediana edad son brillantes o de baja luminosidad. Además, mediante comparaciones entre datos observacionales y simulaciones teóricas, establecimos restricciones en las ecuaciones de estado nucleares. Un modelo adecuado debe explicar tanto objetos brillantes como los magnetars y objetos tenues en edades tempranas. Considerando un enfoque simplificado, estimamos que el 75% de las ecuaciones de estado propuestas son válidas. Otro hallazgo importante es que el componente dipolar, central en muchos modelos de magnetars de FRB, desempeña un papel secundario en la desencadenación de explosiones en la magnetosfera durante fallos en la corteza. En cambio, la energía magnética de la corteza sirve como indicador confiable del número esperado de eventos. Además, en las simulaciones magneto-térmicas en 3D, encontramos que el campo magnético retiene una fuerte memoria de las estructuras iniciales a gran escala, lo que subraya la importancia de la configuración a gran escala en la evolución del campo. Adoptamos una topología inicial compleja derivada de simulaciones de dinamo de protoestrella de neutrones. Esta simulación no explica la luminosidad de rayos-X en magnetars, pero puede explicarla en magnetars de campo bajo y objetos compactos centrales. El componente dipolar en la superficie, responsable del par electromagnético en la disminución de la rotación, no aumenta con el tiempo a partir de esta topología inicial.This doctoral thesis undertakes a comprehensive investigation of the long-term evolution of the internal, strong magnetic fields found within isolated neutron stars. These astronomical entities stand as the most potent magnetic objects in the universe. Within the context of neutron stars, their magnetic influence extends beyond their surface, encompassing the magnetized plasma in their vicinity, referred to as the magnetosphere. This overarching magnetic configuration significantly impacts the observable characteristics of the highly magnetized neutron stars, commonly known as magnetars. Conversely, the magnetic fields within their interiors undergo prolonged evolution spanning from thousands to millions of years. This magnetic evolution is intrinsically linked to the concurrent thermal evolution. The diverse range of observable phenomena associated with neutron star underscores the complex and three-dimensional nature of their magnetic topology, thereby requiring sophisticated numerical simulations. These simulations are crucial not only for explaining observed burst mechanisms and the creation of surface hotspots, but also for comprehending the intricate interplay between magnetic fields and temperature under the extreme conditions inherent in these cosmic objects. A central focus of this thesis involves a thorough exploration of state-of-the-art three-dimensional coupled magneto-thermal evolution models. This marks a pioneering achievement as we conduct, for the first time, the most realistic 3D simulation to date, spanning the first million years of a neutron star's life, using the newly developed code MATINS. This code adeptly accounts for both Ohmic dissipation and Hall drift within the neutron star's crust. Our simulations incorporate highly accurate temperature-dependent microphysical calculations and adopt the star's structure derived from a realistic equation of state. Furthermore, the use of cubed-sphere coordinates addresses the challenge posed by axial singularities in 3D simulations, providing a deeper understanding of neutron star evolution. Additionally, our simulations consider the relativistic factors in the evolution equations and utilize the state-of-the-art envelope model from existing literature, along with an initial magnetic field topology derived from proto-neutron star dynamo simulations. Within this framework, we quantitatively simulate the thermal luminosity, timing properties, and magnetic field evolution, pushing the boundaries of numerical modeling capabilities. Several key conclusions can be drawn from the diverse studies conducted during this doctorate. To begin with, we found that the location of bulk electrical currents within the star determines whether middle-aged magnetars appear relatively bright or exhibit very low luminosities. These insights carry significant implications for population synthesis studies of pulsars and magnetars. Furthermore, through a comparison between observational data of young and faint sources and theoretical simulations, we imposed constraints on the nuclear equation of states. A suitable equation of state should elucidate both bright objects like magnetars and extremely faint objects at young ages. Considering a simplified meta-modeling approach, we estimate that 75% of the proposed equation of states fall within the excluded range. Another notable thesis discovery is that the dipolar component, central to many FRB-magnetar models, plays a minor role in triggering magnetosphere bursts during crustal failures. Instead, the crustal magnetic energy serves as a reliable tracer of the expected number of events. Moreover, performing 3D magneto-thermal simulations, we found that the magnetic field retains a strong memory of the initial large scales structures. This indicates that the large-scale configuration attained during neutron star formation is crucial in determining the field topology at any evolution stage. Therefore, we adopted a complex initial topology derived from proto-neutron star dynamo simulations. This simulation does not account for the X-ray luminosity observed in Magnetars but can explain the luminosity in low-field magnetars and Central Compact Objects. Moreover, the surface dipolar component, responsible for the dominant electromagnetic spin-down torque, does not exhibit any increase over time when starting from this initial topology

Fast cooling and internal heating in hyperon stars

Neutron star models with maximum mass close to 2 M⊙ reach high central densities, which may activate nucleonic and hyperon direct Urca neutrino emission. To alleviate the tension between fast theoretical cooling rates and thermal luminosity observations of moderately magnetized, isolated thermally emitting stars (with Lγ ≳ 1031 erg s−1 at t ≳ 105.3 yr), some internal heating source is required. The power supplied by the internal heater is estimated for both a phenomenological source in the inner crust and Joule heating due to magnetic field decay, assuming different superfluidity models and compositions of the outer stellar envelope. It is found that a thermal power of W(t) ≈ 1034 erg s−1 allows neutron star models to match observations of moderately magnetized, isolated stars with ages t ≳ 105.3 yr. The requisite W(t) can be supplied by Joule heating due to crust-confined initial magnetic configurations with (i) mixed poloidal–toroidal fields, with surface strength Bdip = 1013 G at the pole of the dipolar poloidal component and ∼90 per cent of the magnetic energy stored in the toroidal component; and (ii) poloidal-only configurations with Bdip = 1014 G.FA is supported by the University of Melbourne through a Melbourne Research Scholarship. AM acknowledges funding from an Australian Research Council Discovery Project grant (DP170103625). DV is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC Starting Grant ‘IMAGINE’ No. 948582, PI DV). CD is supported by the ERC Consolidator Grant ‘MAGNESIA’ (No. 817661, PI Nanda Rea) and this work has been carried out within the framework of the doctoral program in Physics of the Universitat Autònoma de Barcelona. JAP acknowledges support by the Generalitat Valenciana (PROMETEO/2019/071), AEI grant PGC2018-095984-B-I00, and the Alexander von Humboldt Stiftung through a Humboldt Research Award

Magneto-thermal evolution of neutron stars with coupled Ohmic, Hall and ambipolar effects via accurate finite-volume simulations

Simulating the long-term evolution of temperature and magnetic fields in neutron stars is a major effort in astrophysics, having significant impact in several topics. A detailed evolutionary model requires, at the same time, the numerical solution of the heat diffusion equation, the use of appropriate numerical methods to control non-linear terms in the induction equation, and the local calculation of realistic microphysics coefficients. Here we present the latest extension of the magneto-thermal 2D code in which we have coupled the crustal evolution to the core evolution, including ambipolar diffusion. It has also gained in modularity, accuracy, and efficiency. We revise the most suitable numerical methods to accurately simulate magnetar-like magnetic fields, reproducing the Hall-driven magnetic discontinuities. From the point of view of computational performance, most of the load falls on the calculation of microphysics coefficients. To a lesser extent, the thermal evolution part is also computationally expensive because it requires large matrix inversions due to the use of an implicit method. We show two representative case studies: (i) a non-trivial multipolar configuration confined to the crust, displaying long-lived small-scale structures and discontinuities; and (ii) a preliminary study of ambipolar diffusion in normal matter. The latter acts on timescales that are too long to have relevant effects on the timescales of interest but sets the stage for future works where superfluid and superconductivity need to be included.DV, AGG, CD and VG are supported by the ERC Consolidator Grant “MAGNESIA” (nr. 817661, PI Nanda Rea) and acknowledge funding from grants SGR2017-1383 and PGC2018-095512-BI00. JAP acknowledges support by the Generalitat Valenciana (PROMETEO/2019/071), AEI grant PGC2018-095984-B-I00 and the Alexander von Humboldt Stiftung through a Humboldt Research Award. DV acknowledges his Short Term Scientific Mission in Durham (UK) funded by the COST Action PHAROS (CA16214)

Modelling force-free neutron star magnetospheres using physics-informed neural networks

Using physics-informed neural networks (PINNs) to solve a specific boundary value problem is becoming more popular as an alternative to traditional methods. However, depending on the specific problem, they could be computationally expensive and potentially less accurate. The functionality of PINNs for real-world physical problems can significantly improve if they become more flexible and adaptable. To address this, our work explores the idea of training a PINN for general boundary conditions and source terms expressed through a limited number of coefficients, introduced as additional inputs in the network. Although this process increases the dimensionality and is computationally costly, using the trained network to evaluate new general solutions is much faster. Our results indicate that PINN solutions are relatively accurate, reliable, and well behaved. We applied this idea to the astrophysical scenario of the magnetic field evolution in the interior of a neutron star connected to a force-free magnetosphere. Solving this problem through a global simulation in the entire domain is expensive due to the elliptic solver’s needs for the exterior solution. The computational cost with a PINN was more than an order of magnitude lower than the similar case solved with a finite difference scheme, arguably at the cost of accuracy. These results pave the way for the future extension to three-dimensional of this (or a similar) problem, where generalized boundary conditions are very costly to implement.We acknowledge the support through the grant PID2021-127495NB-I00 funded by MCIN/AEI/10.13039/501100011033 and by the European Union, and the Astrophysics and High Energy Physics programme of the Generalitat Valenciana ASFAE/2022/026 funded by MCIN and the European Union NextGenerationEU (PRTR-C17.I1). JFU is supported by the predoctoral fellowship UAFPU21-103 funded by the University of Alicante. CD is supported by the ERC Consolidator Grant ‘MAGNESIA’ no. 817661 (PI: N. Rea) and has the partial support of NORDITA. This work has been carried out within the framework of the doctoral program in Physics of the Universitat Autònoma de Barcelona and it is partially supported by the programme Unidad de Excelencia María de Maeztu CEX2020-001058-M.Peer reviewe