Physics-informed neural networks (PINNs) provide a novel approach for data-driven numerical simulations, tackling challenges of discretization and enabling seamless integration of noisy data and physical models (e.g., partial differential equations). In this presentation, we discuss the results of our recent studies where we apply PINNs for coronal magnetic field extrapolations of the solar atmosphere, which are essential to understand the genesis and initiation of solar eruptions and to predict the occurrence of high-energy events from our Sun.We utilize our PINN to estimate the 3D coronal magnetic fields based on photospheric vector magnetograms and the force-free physical model. This approach provides state-of-the-art coronal magnetic field extrapolations in quasi real-time. We simulate the evolution of Active Region NOAA 11158 over 5 continuous days, where the derived time profile of the free magnetic energy unambiguously relates to the observed flare activity.We extend this approach by utilizing multi-height magnetic field measurements and combine them in a single magnetic field model. Our evaluation shows that the additional chromospheric field information leads to a more realistic approximation of the solar coronal magnetic field. In addition, our method intrinsically provides an estimate of the height corrugation of the observed magnetograms.We provide an outlook on our ongoing work where we use PINNs for global force-free magnetic field extrapolations. This approach enables a novel understanding of the global magnetic topology with a realistic treatment of current carrying fields.In summary, PINNs have the potential to greatly advance the field of numerical simulations, accelerate scientific research, and enable advanced space weather monitoring.

We aim to better characterize the conditions of the solar corona, especially with respect to the occurrence of confined and eruptive flares. In this work, we model the coronal evolution around 231 large flares observed during solar cycle 24. Using Helioseismic and Magnetic Imager vector magnetic field data around each event, we employed nonlinear force-free field extrapolations to approximate the coronal energy and helicity budgets of the solar source regions. A superposed epoch analysis and dynamical time warping applied to the time series of selected photospheric and coronal quantities were used to pin down the characteristics of the pre- and postflare time evolution, as well as to assess flare-related changes. During the 24 hours leading up to a major flare, the total magnetic energy and unsigned magnetic flux were seen to evolve closely with respect to each other, irrespective of the flare type. Prior to confined flares, the free energy evolves in a way that exhibits more of a similarity with the unsigned flux than the helicity of the current-carrying field, while the opposite trend is seen prior to eruptive flares. Furthermore, the flare type can be predicted correctly in more than 90% of major flares when combining measures of the active regions nonpotentiality and local stability. The coronal energy and helicity budgets return to preflare levels within ≈,6 to 12 hours after eruptive major M-class flares, while the impact of eruptive X-flares lasts considerably longer. Finally, the postflare replenishment times of ≳12 hours after eruptive X-class flares may serve as a partial explanation for the rare observation of eruptive X-class flares within a time frame of a few hours.

Context. The eruption of large-scale solar filaments that extend beyond the core of an active region (AR) into weak-field regions presents a valuable opportunity to investigate flare and eruption dynamics across various magnetic environments. Aims. We performed a detailed analysis of the magnetic structure and asymmetric eruption of a large (similar to 500 Mm in extent) inverse S-shaped filament partially located in AR 13229 on February 24, 2023. Our primary goal was to relate the filament's pre-eruptive magnetic configuration to the observed dynamics of its eruption and the formation of a large-scale coronal dimming in a weak-field region. Methods. The evolution of the eruption and associated coronal dimming were analyzed using multiwavelength observations from the Atmospheric Imaging Assembly (AIA) and H alpha observations from Kanzelh & ouml;he Observatory. A detailed dimming analysis was performed based on an AIA 211 & Aring; logarithmic base-ratio image sequence. To reconstruct the coronal magnetic field, we applied a physics-informed neural network (PINN)-based nonlinear force-free field (NLFFF) extrapolation method to a large computational volume (similar to 730 Mm x 550 Mm), using a pre-eruption photospheric vector magnetogram from the Helioseismic and Magnetic Imager (HMI) as the lower boundary. Results. Our results show a highly asymmetric eruption. The eastern part of the filament erupts freely, creating a coronal dimming associated with its footprint, which subsequently expands (total area of similar to 9 x 10(9) km(2)) together with an inverse J-shaped flare ribbon. The eastern dimming area covers a weak magnetic field region with a mean unsigned flux of only similar to 5 G. The NLFFF extrapolation shows the presence of a large-scale magnetic flux rope (MFR) of similar to 500 Mm in length, consistent with the observed filament. We identified an extended MFR footprint to the east in the NLFFF extrapolation that connects to an inverse J-shaped flare ribbon (hook) observed during the eruption, outlining the area from which the coronal dimming originated. Overlying strapping fields connect to the region into which the coronal dimming and flare ribbon subsequently expand. This configuration offers a plausible explanation for the formation of the dimming as a stationary flux rope and strapping flux dimming. The subsequent expansion of the stationary flux rope dimming is caused by the growth of the MFR footprint through strapping-strapping reconnection. In contrast, the western filament leg shows multiple anchor points along a narrow channel and potentially strong overlying magnetic fields, which could have resulted in the suppressed dimming and partial confinement by overlying loops observed on this side of the filament during the eruption. Conclusions. The reconstructed pre-eruptive NLFFF configuration provides a clear physical explanation for the observed asymmetries in the eruption, flare geometry, and coronal dimming. This successful application shows that PINN-based NLFFF extrapolation can be effective for modeling large-scale filaments extending into weak-field regions, and that combining this method with detailed observational analysis can greatly improve our understanding of complex solar eruptions.

We performed a detailed analysis of the magnetic structure and asymmetric eruption of a large (about 500 Mm) inverse S-shaped filament partially located in AR 13229 on February 24, 2023. We linked the filament's pre-eruptive magnetic configuration to its highly asymmetric eruption dynamics and the formation of a large-scale coronal dimming in a weak-field region (mean unsigned flux of about 5 G). To reconstruct the coronal magnetic field, we applied a physics-informed neural network (PINN)-based nonlinear force-free field (NLFFF) extrapolation method to a pre-eruption HMI vector magnetogram. The NLFFF extrapolation reveals a large-scale magnetic flux rope (MFR) of about 500 Mm in length, consistent with the filament. We identified an extended MFR footprint to the east that connects to the J-shaped flare ribbon, outlining where the coronal dimming began. Overlying strapping fields connect to the area into which the dimming and flare ribbon later expand. This configuration explains the formation of the dimming as a stationary flux rope and strapping flux dimming, with subsequent expansion driven by the growth of the MFR footprint through strapping-strapping reconnection. Conversely, the western filament leg shows multiple anchor points and strong overlying magnetic fields, which suppressed the dimming and partially confined the eruption on that side. The reconstructed pre-eruptive NLFFF configuration offers a clear physical explanation for the asymmetries seen in the eruption, flare geometry, and coronal dimming. This demonstrates that PINN-based NLFFF extrapolation can effectively model large-scale filaments extending into weak-field regions, enhancing our understanding of complex solar eruptions.

In order to improve our understanding on the pre-requisites of eruptive solar flares, we study and compare different measures that characterize the eruptive potential of solar active regions - the critical height for torus instability as a local measure and the helicity ratio as a global measure - with the structural properties of the underlying magnetic field, namely the altitude of the center of the current-carrying magnetic structure. Using time series of 3D optimization-based nonlinear force-free magnetic field models for 10 different active regions (ARs) around the time of large solar flares, we determine the altitudes of the current-weighted centers of the non-potential model structures. Based on the potential magnetic field, we inspect the decay index, n, in multiple vertical planes oriented along of or perpendicular to the flare-relevant polarity inversion line, and estimate the critical height (h_crit) for torus instability (TI) using different thresholds of n. The critical heights are interpreted with respect to the altitudes of the current-weighted centers of the associated non-potential structures, as well as the eruptive character of the associated flares, and the eruptive potential of the host AR, as characterized by the helicity ratio. Our most important findings are that (i) h_crit is more segregated in terms of flare type than the helicity ratio, and that (ii) coronal field configurations with a higher eruptive potential (in terms of the helicity ratio) also appear to be more prone to TI. Furthermore, we find no pronounced differences in the altitudes of the non-potential structures prior to confined and eruptive flares.

Context. Relative field line helicity (RFLH) is a recently developed quantity that can approximate the density of relative magnetic helicity. Aims. This paper aims to determine whether RFLH can be used as an indicator of solar eruptivity. Methods. Starting from magnetographic observations from the Helioseismic and Magnetic Imager instrument on board the Solar Dynamic Observatory of a sample of seven solar active regions (ARs), that comprises over 2000 individual snapshots, we reconstruct the AR's coronal magnetic field with a widely used non-linear force-free method. This enables us to compute the RFLH using two independent gauge conditions for the vector potentials. We focus our study on the times of strong flares in the ARs, above the M class, and in regions around the polarity inversion lines (PILs) of the magnetic field, and of RFLH. Results. We find that the temporal profiles of the relative helicity that is contained in the magnetic PIL follow those of the relative helicity that is computed by the accurate volume method for the whole AR. Additionally, the PIL relative helicity can be used to define a parameter similar to the well-known parameter R, whose high values are related with increased flaring probability. This helicity-based R-parameter correlates closely with the original parameter, showing in some cases even higher values. Additionally, it experiences more pronounced decreases during flares. This means that there exists at least one parameter deduced from RFLH that is important as a solar eruptivity indicator.

Non-linear force-free extrapolations are a common approach to estimate the 3D topology of coronal magnetic fields based on photospheric vector magnetograms. The force-free assumption is a valid approximation at coronal heights, but for the dense plasma conditions in the lower atmosphere, this assumption is not satisfied. In this study, we utilize multi-height magnetic field measurements in combination with physics-informed neural networks to advance solar magnetic field extrapolations. We include a flexible height-mapping, which allows us to account for the different formation heights of the observed magnetic field measurements. The comparison to analytical and simulated magnetic fields demonstrates that including chromospheric magnetic field measurements leads to a significant improvement of our magnetic field extrapolations. We also apply our method to chromospheric line-of-sight magnetograms, from the Vector Spectromagnetograph (VSM) on the Synoptic Optical Long-term Investigations of the Sun (SOLIS) observatory, in combination with photospheric vector magnetograms, from the Helioseismic Magnetic Imager (HMI) onboard the Solar Dynamic Observatory (SDO). The comparison to observations in extreme ultraviolet wavelengths shows that the additional chromospheric information leads to a better agreement with the observed coronal structures. In addition, our method intrinsically provides an estimate of the corrugation of the observed magnetograms. With this new approach, we make efficient use of multi-height magnetic field measurements and advance the realism of coronal magnetic field simulations.

We analyze the complete chain of effects caused by a solar eruptive event in order to better understand the dynamic evolution of magnetic-field related quantities in interplanetary space, in particular that of magnetic flux and helicity. We study a series of connected events (a confined C4.5 flare, a flare-less filament eruption and a double-peak M-class flare) that originated in NOAA active region (AR) 12891 on 2021 November 1 and November 2. We deduce the magnetic structure of AR 12891 using stereoscopy and nonlinear force-free (NLFF) magnetic field modeling, allowing us to identify a coronal flux rope and to estimate its axial flux and helicity. Additionally, we compute reconnection fluxes based on flare ribbon and coronal dimming signatures from remote sensing imagery. Comparison to corresponding quantities of the associated magnetic cloud (MC), deduced from in-situ measurements from Solar Orbiter and near-Earth spacecraft, allows us to draw conclusions on the evolution of the associated interplanetary coronal mass ejection (ICME). The latter are aided through the application of geometric fitting techniques (graduated cylindrical shell modeling; GCS) and interplanetary propagation models (drag based ensemble modeling; DBEM) to the ICME. NLFF modeling suggests the host AR's magnetic structure in the form of a left-handed (negative-helicity) sheared arcade/flux rope reaching to altitudes of 8-10 Mm above photospheric levels, in close agreement with the corresponding stereoscopic estimate. Revealed from GCS and DBEM modeling, the ejected flux rope propagated in a self-similar expanding manner through interplanetary space. Comparison of magnetic fluxes and helicities processed by magnetic reconnection in the solar source region and the respective budgets of the MC indicate a considerable contribution from the eruptive process, though the pre-eruptive budgets appear of relevance too.

Magnetic free energy powers solar flares and coronal mass ejections, and the buildup of magnetic helicity might play a role in the development of unstable structures that subsequently erupt. To better understand the roles of energy and helicity in large flares and eruptions, we have characterized the evolution of magnetic energy and helicity associated with 21 X-class flares from 2010 to 2017. Our sample includes both confined and eruptive events, with 6 and 15 in each category, respectively. Using the Helioseismic and Magnetic Imager vector magnetic field observations from several hours before to several hours after each event, we employ (a) the Differential Affine Velocity Estimator for Vector Magnetograms to determine the photospheric fluxes of energy and helicity, and (b) nonlinear force-free field extrapolations to estimate the coronal content of energy and helicity in source-region fields. Using superposed epoch analysis, we find, on average, the following: (1) decreases in both magnetic energy and helicity, in both photospheric fluxes and coronal content, that persist for a few hours after eruptions, but no clear changes, notably in relative helicity, for confined events; (2) significant increases in the twist of photospheric fields in eruptive events, with twist uncertainties too large in confined events to constrain twist changes (and lower overall twist in confined events); and (3) on longer timescales (event time +12 hr), replenishment of free magnetic energy and helicity content to near preevent levels for eruptive events. For eruptive events, magnetic helicity and free energy in coronal models clearly decrease after flares, with the amounts of decrease proportional to each region’s pre-flare content.

While the photospheric magnetic field of our Sun is routinely measured, its extent into the upper atmosphere is typically not accessible by direct observations. Here we present an approach for coronal magnetic-field extrapolation, using a neural network that integrates observational data and the physical force-free magnetic-field model. Our method flexibly finds a trade-off between the observation and force-free magnetic-field assumption, improving the understanding of the connection between the observation and the underlying physics. We utilize meta-learning concepts to simulate the evolution of active region NOAA 11158. Our simulation of 5 days of observations at full cadence (12 minutes) requires less than 12 hours of total computation time, allowing for real-time force-free magnetic-field extrapolations. The application to an analytical magnetic-field solution, a systematic analysis of the time evolution of free magnetic energy and magnetic helicity in the coronal volume, as well as comparison with extreme-ultraviolet observations, demonstrates the validity of our approach. The obtained temporal and spatial depletion of free magnetic energy unambiguously relates to the observed flare activity. The application of physics-informed neural networks enables an estimation of the solar coronal magnetic field in quasi real time. A comparison with extreme-ultraviolet observations reveals that the model provides a realistic approximation and the modelled coronal field has a clear relationship with flaring activity.

Solar flares and coronal mass ejections (CMEs) are the main drivers for severe space weather disturbances on Earth and other planets. While the geo-effects of CMEs give us a lead time of about 1 to 4 days, the effects of flares and flare-accelerated solar energetic particles (SEPs) are very immediate, 8 minutes for the enhanced radiation and as short as about 20 minutes for the highest energy SEPs arriving at Earth. Thus, predictions of solar flare occurrence at least several hours ahead are of high importance for the mitigation of severe space weather effects.Observations and simulations of solar flares suggest that the structure and evolution of the active region’s magnetic field is a key component for energetic eruptions. The recent advances in deep learning provide tools to directly learn complex relations from multi-dimensional data. Here, we present a novel deep learning method for short-term solar flare prediction. The algorithm is based on the HMI photospheric line-of-sight magnetic field and its temporal evolution together with the coronal evolution as observed by multi-wavelengths EUV filtergrams from the AIA instrument onboard the Solar Dynamics Observatory. We train a neural network to independently identify features in the imaging data based on the dynamic evolution of the coronal structure and the photospheric magnetic field evolution, which may hint at flare occurrence in the near future.We show that our method can reliably predict flares six hours ahead, with 73% correct flaring predictions (89% when considering only M- and X-class flares), and 83% correct quiet active region predictions.In order to overcome the “black box problem” of machine-learning algorithms, and thus to allow for physical interpretation of the network findings, we employ a spatio-temporal attention mechanism. This allows us to extract the emphasized regions, which reveal the neural network interpretation of the flare onset conditions. Our comparison shows that predicted precursors are associated with the position of flare occurrence, respond to dynamic changes, and align with characteristics within the active region.

Coronal mass ejections are among the Sun's most energetic activity events yet the physical mechanisms that lead to their occurrence are not yet fully understood. They can drive major space weather impacts at Earth, so knowing why and when these ejections will occur is required for accurate space weather forecasts. In this study we use a 4 day time series of a quantity known as the helicity ratio, divide H ( J ) divide / divide H ( V ) divide (helicity of the current-carrying part of the active region field to the total relative magnetic helicity within the volume), which has been computed from nonlinear force-free field extrapolations of NOAA active region 11158. We compare the evolution of divide H ( J ) divide / divide H ( V ) divide with the activity produced in the corona of the active region and show this ratio can be used to indicate when the active region is prone to eruption. This occurs when divide H ( J ) divide / divide H ( V ) divide exceeds a value of 0.1, as suggested by previous studies. We find the helicity ratio variations to be more pronounced during times of strong flux emergence, collision and reconnection between fields of different bipoles, shearing motions, and reconfiguration of the corona through failed and successful eruptions. When flux emergence, collision, and shearing motions have lessened, the changes in helicity ratio are somewhat subtle despite the occurrence of significant eruptive activity during this time.

Context. Nonlinear force-free (NLFF) modeling is regularly used to indirectly infer the 3D geometry of the coronal magnetic field, which is not otherwise accessible on a regular basis by means of direct measurements. Aims. We study the effect of binning in time-series NLFF modeling of individual active regions (ARs) in order to quantify the effect of a different underlying spatial sampling on the quality of modeling as well as on the derived physical parameters. Methods. We apply an optimization method to sequences of Solar Dynamics Observatory (SDO) Helioseismic and Magnetic Imager (HMI) vector magnetogram data at three different plate scales for three solar active regions to obtain nine NLFF model time series. From the NLFF models, we deduce active-region magnetic fluxes, electric currents, magnetic energies, and relative helicities, and analyze those with respect to the underlying spatial sampling. We calculate various metrics to quantify the quality of the derived NLFF models and apply a Helmholtz decomposition to characterize solenoidal errors. Results. At a given spatial sampling, the quality of NLFF modeling is different for different ARs, and the quality varies along the individual model time series. For a given AR, modeling at a certain spatial sampling is not necessarily of superior quality compared to that performed with a different plate scale. Generally, the NLFF model quality tends to be higher for larger pixel sizes with the solenoidal quality being the ultimate cause for systematic variations in model-deduced physical quantities. Conclusions. Optimization-based modeling using SDO/HMI vector data binned to larger pixel sizes yields variations in magnetic energy and helicity estimates of ≲30% on overall, given that concise checks ensure the physical plausibility and high solenoidal quality of the tested model. Spatial-sampling-induced differences are relatively small compared to those arising from other sources of uncertainty, including the effects of applying different data calibration methods, those of using vector data from different instruments, or those arising from application of different NLFF methods to identical input data.

Context. In order to better understand the underlying processes and prerequisites for solar activity, it is essential to study the time evolution of the coronal magnetic field of solar active regions (ARs) associated with flare activity. Aims. We investigate the coronal magnetic energy and helicity budgets of ten solar ARs around the times of large flares. In particular, we are interested in a possible relation of the derived quantities to the particular type of the flares that the AR produces, namely, whether they are associated with a CME or whether they are confined (i.e., not accompanied by a CME). Methods. Using an optimization approach, we employed time series of 3D nonlinear force-free magnetic field models of ten ARs, covering a time span of several hours around the time of occurrence of large solar flares (GOES class M1.0 and larger). We subsequently computed the 3D magnetic vector potentials associated to the model 3D coronal magnetic field using a finite-volume method. This allows us to correspondingly compute the coronal magnetic energy and helicity budgets, as well as related (intensive) quantities such as the relative contribution of free magnetic energy, EF/E (energy ratio), the fraction of non-potential (current-carrying) helicity, |HJ|/|HV| (helicity ratio), and the normalized current-carrying helicity, |HJ|/ϕ′2. Results. The total energy and helicity budgets of flare-productive ARs (extensive parameters) cover a broad range of magnitudes, with no obvious relation to the eruptive potential of the individual ARs, that is, whether or not a CME is produced in association with the flare. The intensive eruptivity proxies, EF/E and |HJ|/|HV|, and |HJ|/ϕ′2, however, seem to be distinctly different for ARs that produce CME-associated large flares compared to those which produce confined flares. For the majority of ARs in our sample, we are able to identify characteristic pre-flare magnitudes of the intensive quantities that are clearly associated with subsequent CME-productivity. Conclusions. If the corona of an AR exhibits characteristic values of ⟨|HJ|/|HV|⟩ > 0.1, ⟨EF/E⟩ > 0.2, and ⟨|HJ|/ϕ′2⟩ > 0.005, then the AR is likely to produce large CME-associated flares. Conversely, confined large flares tend to originate from ARs that exhibit coronal values of ⟨|HJ|/|HV|⟩ ≲ 0.1, ⟨EF/E⟩ ≲ 0.1, and ⟨|HJ|/ϕ′2⟩ ≲ 0.002.

Sunspot light bridges are known to exhibit a variety of dynamic and persistent phenomena such as surges, small-scale jets etc. in the chromosphere and transition region. While it has generally been proposed that magnetic reconnection is responsible for this small-scale dynamism, persistent flaring activity lasting several hours from the same spatial location on a sunspot light bridge, has rarely been reported. We combine observations from the Atmospheric Imaging Assembly and the Helioseismic Magnetic Imager on board the Solar Dynamics Observatory to investigate homologous flaring activity over a small sunspot light bridge in an emerging flux region. The homologous flares all produced broad, collimated jets, including a B6.4 class flare. The jets rise at a speed of about 200 km/s, reach projected heights of about 98 Mm, and emerge from the same spatial location for nearly 14 hrs, after which they cease completely. A non-linear force free extrapolation of the photospheric magnetic field shows a low-lying flux rope connecting the light bridge to a remote opposite-polarity network. The persistent flares occur as a result of the rapid horizontal motion of the leading sunspot that causes the relatively vertical magnetic fields in the adjacent umbra to reconnect with the low-lying flux rope in the light bridge. Our results indicate that the flaring ceases once the flux rope has lost sufficient twist through repeated reconnections.

We present a detailed analysis of an eruptive event that occurred on early 2019 March 8 in active region AR 12734, to which we refer as the International Women's day event. The event under study is intriguing in several aspects: 1) low-coronal eruptive signatures come in ”pairs” (a double-peak flare, two coronal dimmings, and two EUV waves); 2) although the event is characterized by a complete chain of eruptive signatures, the corresponding coronagraphic signatures are weak; 3) although the source region of the eruption is located close to the center of the solar disc and the eruption is thus presumably Earth-directed, heliospheric signatures are very weak with little Earth-impact. We analyze a number of multi-spacecraft and multi-instrument (both remote-sensing and in situ) observations, including Soft X-ray, (extreme-) ultraviolet (E)UV), radio and white-light emission, as well as plasma, magnetic field and particle measurements. We employ 3D NLFF modeling to investigate the coronal magnetic field configuration in and around the active region, the GCS model to make a 3D reconstruction of the CME geometry and the 3D MHD numerical model EUHFORIA to model the background state of the heliosphere. Our results indicate two subsequent eruptions of two systems of sheared and twisted magnetic fields, which merge already in the upper corona and start to evolve further out as a single entity. The large-scale magnetic field significantly influences both, the early and the interplanetary evolution of the structure. During the first eruption the stability of the overlying field was disrupted which enabled the second eruption. We find that during the propagation in the interplanetary space the large-scale magnetic field, i.e. , the location of heliospheric current sheet between the AR and the Earth likely influences propagation and the evolution of the erupted structure(s).

Aims. We study the relative helicity of active region (AR) NOAA 12673 during a ten-hour time interval centered around a preceding X2.2 flare (SOL2017-09-06T08:57) and also including an eruptive X9.3 flare that occurred three hours later (SOL2017-09-06T11:53). In particular, we aim for a reliable estimate of the normalized self-helicity of the current-carrying magnetic field, the so-called helicity ratio, |HJ|/|H𝒱|, a promising candidate to quantity the eruptive potential of solar ARs. Methods. Using Solar Dynamics Observatory Helioseismic and Magnetic Imager vector magnetic field data as an input, we employ nonlinear force-free (NLFF) coronal magnetic field models using an optimization approach. The corresponding relative helicity, and related quantities, are computed using a finite-volume method. From multiple time series of NLFF models based on different choices of free model parameters, we are able to assess the spread of |HJ|/|H𝒱|, and to estimate its uncertainty. Results. In comparison to earlier works, which identified the non-solenoidal contribution to the total magnetic energy, Ediv/E, as selection criterion regarding the required solenoidal quality of magnetic field models for subsequent relative helicity analysis, we propose to use in addition the non-solenoidal contribution to the free magnetic energy, |Emix|/EJ, s. As a recipe for a reliable estimate of the relative magnetic helicity (and related quantities), we recommend to employ multiple NLFF models based on different combinations of free model parameters, to retain only those that exhibit smallest values of both Ediv/E and |Emix|/EJ, s at a certain time instant, to subsequently compute mean estimates, and to use the spread of the individually contributing values as an indication for the uncertainty.

We compare the coronal magnetic energy and helicity of two solar active regions (ARs), prolific in major eruptive (AR 11158) and confined (AR 12192) flaring, and analyze the potential of deduced proxies to forecast upcoming flares. Based on nonlinear force-free (NLFF) coronal magnetic field models with a high degree of solenoidality, and applying three different computational methods to investigate the coronal magnetic helicity, we are able to draw conclusions with a high level of confidence. Based on real observations of two solar ARs we checked trends regarding the potential eruptivity of the active-region corona, as suggested earlier in works that were based on numerical simulations, or solar observations. Our results support that the ratio of current-carrying to total helicity, vertical bar H-J vertical bar vertical bar H-nu vertical bar, shows a strong ability to indicate the eruptive potential of a solar AR. However, vertical bar H-J vertical bar vertical bar H-nu vertical bar does not seem to be indicative for the magnitude or type of an upcoming flare (confined or eruptive). Interpreted in the context of earlier observational studies, our findings furthermore support that the total relative helicity normalized to the magnetic flux at the NLFF model's lower boundary, H nu/phi(2), represents no indicator for the eruptivity.